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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We introduce a notion of integration on the category of proper birational maps to a given variety \(X\), with values in an associated Chow group. Applications include new birational invariants; comparison results for Chern classes and numbers of nonsingular birational varieties; ``stringy'' Chern classes of singular varieties; and a zeta function specializing to the topological zeta function.
In its simplest manifestation, the integral gives a new expression for Chern-Schwartz-MacPherson classes of possibly singular varieties, placing them into a context in which a ``change-of-variable'' formula holds. Chow group; birational invariants; Chern classes; singularities; stringy invariants. Aluffi, P., Modification systems and integration in their Chow groups, Selecta Math. 11 (2005), 155--202. Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Singularities in algebraic geometry, Characteristic classes and numbers in differential topology, Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) Modification systems and integration in their Chow groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is known that there is at least an invariant analytic curve passing through each of the components in the complement of nodal singularities, after reduction of singularities of a germ of singular foliation in \((\mathbb{C}^2,0)\). Here, we state and prove a generalization of this property to any ambient dimension. singular holomorphic foliations; invariant hypersurfaces; simplicial complexes; desingularization Singularities of holomorphic vector fields and foliations, Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Invariant hypersurfaces and nodal components for codimension one singular foliations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field, and let \(A\) be a finitely generated associative \(k\)-algebra with unit. For each \(d\geq1\) we define the module scheme mod\(_{A}^{d}\) by mod\(_{A}^{d}\left( R\right) =\text{Hom} _{k\text{-alg}}\left( A,\mathbb{M}_{d}\left( R\right) \right) ,\) where \(\mathbb{M}_{d}\left( R\right) \) is the set of \(d\times d\) matrices with entries in \(R\). In particular, mod\(_{A}^{d}\left( k\right) \) can be identified with \(A\)-module structures on \(k^{d};\) furthermore, mod\(_{A}^{d}\) is an affine scheme, say mod\(_{A}^{d}=\text{Spec}\left( k\left[ \text{mod}_{A}^{d}\right] \right)\). The group scheme \(\text{GL} _{d}\) acts on mod\(_{A}^{d}\) by conjugation on its points -- let \(\mathcal{O} _{M}\) denote the \(\text{GL}_{d}\left( k\right) \)-orbit of a fixed \(M\in\)mod\(_{A}^{d}\left( k\right) .\) One can view \(\mathcal{O}_{M}\) as the \(A\)-module structures on \(k^{d}\) isomorphic to \(M.\) \ Understanding the closure of \(\mathcal{O}_{M}\) \ has proved difficult in general.
For \(N\in\)mod\(_{A}^{d},\) given a \(p\times q\) matrix \(\underline{a}\) with coefficients in \(A\) one can naturally construct a \(pd\times qd\) matrix \(N\left( \underline{a}\right) \). Let \(\mathcal{I}_{M}\subset k\left[ \text{mod}_{A}^{d}\right] \) be the ideal generated by the minors of such matrices of size \(1+\)rk\ \(M\left( \underline{a}\right) ,\) and let \(\mathcal{C}_{M}=\text{Spec}\left( k\left[ \text{mod}_{A} ^{d}\right] /\mathcal{I}_{M}\right) \) -- this is a closed subscheme of mod\(_{A}^{d}\) containing \(\mathcal{\bar{O}}_{M}\) since these minors vanish.
In the work under review, the authors study the properties of this scheme \(\mathcal{C}_{M}.\) Comparisons are made with schemes which arise from quiver representations. As an example, if \(Q\) us an equioriented Dynkin quiver os type \(\mathbb{A}\) then \(\mathcal{C}_{M}=\mathcal{\bar{O}}_{M}\) for \(M\) a representation in rep\(_{Q}^{\mathbf{d}}\left( k\right) \): this is a reformulation of a result from Lakshmibai and Magyar .
Using \(\mathcal{C}_{M}\) instead of the orbit closure allows for a module-theoretic interpretation of the tangent space at some \(N\in\) \(\mathcal{\bar{O}}_{M}.\) This allows for a characterization of the singular locus of \(\mathcal{C}_{M}\) when \(A\) is representation-finite. This is useful when trying to describe the singular locus of \(\mathcal{O}_{M}.\) module schemes; orbit closures; representations of quivers Riedtmann, Christine; Zwara, Grzegorz, Orbit closures and rank schemes, Comment. Math. Helv., 88, 1, 55-84, (2013) Group actions on varieties or schemes (quotients), Singularities in algebraic geometry, Representations of quivers and partially ordered sets Orbit closures and rank schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a normal complex variety, and let \(B=\sum b_i B_i\) be a \(\mathbb Q\)--divisor on \(X\) such that \(K_X+B\) is \(\mathbb Q\)--Cartier. Assume there exists a log resolution \(\pi: Y\to X\). Let \(E_i\) be the irreducible divisors in \(\pi^{-1}(\text{supp} B)\) and the exceptional locus of \(\pi\). The log discrepancies \(a_i\) of \(E_i\) with respect to \((X;B)\) are given by
\[
K_Y=\pi^\ast (K_X+B)+ \sum(a_i-1)E_i,
\]
and assumed to be non--zero. Let \(E_I^0=(\bigcap_{i\in I} E_i)\smallsetminus(\bigcup_{l\notin I}E_l)\). The stringy Euler number of the pair \((X; B)\) is defined as
\[
e(X;B)=\sum\limits_I \chi(E^\circ_I)\prod\limits_{i\in I}\frac{1}{a_i}\;,
\]
here \(\chi\) is as usual the Euler characteristic. The stringy \(E\)-function of \((X; B)\) is defined to be
\[
E(X;B)=\sum\limits_I H(E^\circ_I)\prod\limits_{i\in I}\frac{uv-1}{(uv)^{a_i}-1}\;,
\]
where \(H(E^\circ_I)\in \mathbb Z[u,v]\) is the Hodge polynomial of \(E_I^\circ\). Some results are given (generalizing Batyrev's results on log terminal singularities) to prove that these invariants are well-defined. stringy zeta function; stringy Euler number Veys, W., Stringy zeta functions for Q\textit{-Gorenstein varieties, Duke Math. J., 120, 469-514, (2003)} Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry Stringy zeta functions for \(\mathbb Q\)-Gorenstein varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We determine explicitly the irreducible components of the singular locus of any Schubert variety for \(\text{GL}_n (\mathbb{K})\), \(\mathbb{K}\) being an algebraically closed field of arbitrary characteristic. We also describe the generic singularity along each of them. This was obtained earlier by the author in the covexillary case. The main tool used here is the study of certain quasi-resolutions of non-covexillary Schubert varieties. singular locus of Schubert variety; generic singularity; quasi-resolutions Cortez, Aurélie, Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire, C. R. Acad. Sci. Paris Sér. I Math., 333, 6, 561-566, (2001) Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Generic singularities and quasi-resolutions of Schubert varieties for the general linear group | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Separatrix Theorem of \textit{C. Camacho} and \textit{P. Sad} [Ann. Math. (2) 115, 579--595 (1982; Zbl 0503.32007)]
says that there exists at least one invariant curve (separatrix) passing through the singularity of a germ of holomorphic foliation on a complex surface, when the surface underlying the foliation is smooth or when it is singular and the dual graph of resolution surface singularity is a tree. For the singular case see [\textit{M. Sebastiani}, An. Acad. Bras. Ciênc. 69, No. 2, 159--162 (1997; Zbl 0887.57033)].
The author proves the existence of separatrix even when the resolution dual graph of the surface singular point is not a tree. The main result is as follows.
Let \(\mathcal{F}\) be a singular holomorphic foliation on a normal singular surface \(X\). If the foliation has no saddle-node in its resolution/reduction over the singularity \(p \in X\) and the normal sheaf \(N_F\) is \(\mathbb{Q}\)-Gorenstein, then \(F\) has a separatrix through \(p\). singular holomorphic foliations; invariant curves; birational geometry Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions, Singularities of holomorphic vector fields and foliations, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Dynamical aspects of holomorphic foliations and vector fields Invariant curves for holomorphic foliations on singular surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The existence of a resolution of singularities over a field of characteristic zero has been known since the famous work of \textit{H. Hironaka} [Ann. Math. (2) 79, 109--203, 205--326 (1964; Zbl 0122.38603)]. Hironaka's proof has been simplified during the last years. The actual proof given in the book covers only thirty pages and is based on the approach of \textit{J. Włodarczyk} [J. Am. Math. Soc. 18, No. 4, 779--822 (2005; Zbl 1084.14018)]. The book is addressed to readers who are interested in the subject and want to understand one proof of resolution of singularities in characteristic \(0\). Chapter 1 is devoted to resolution of curve singularities. This can be used as a course for beginners in algebraic geometry because it is very elementary. Chapter 2 needs more technical background. It is about resolution of surface singularities including the Jungian method and the Albanese method using projections. The methods presented in Chapter 3 to prove the existence of resolution of singularities in general are again elementary. Several examples are given to motivate the approach and illustrate the proof. János Kollár succeeded in giving a very clear and understandable proof. It was very nice to read the book. It can be a good basis for a graduate course about this subject. ; Kollár, Lectures on resolution of singularities. Lectures on resolution of singularities. Ann. of Math. Studies, 166, (2007) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry Lectures on resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\xi\) be a plane algebroid (reduced) curve defined over \(\mathbb{C}\), recall that two local curves \(C\), \(C'\) have the same topological type if there exists a local homeomorphism of \(\mathbb{C}^ 2\) which exchanges \(C\) and \(C'\). To determine the topological type of the polar curves of \(\xi\) has been an open problem since M. Noether. In fact the topological type of the polar curves depends on the analytical type of \(\xi\). The author solves this problem for a generic curve in the equisingular class giving the answer via the theory of infinitely near points with imposed singularities. More precisely he defines a cluster of infinitely near points with virtual multiplicities and show that the polar curve of such a generic curve goes through this cluster with effective multiplicities equal to the virtual ones and has not singularities outside this cluster. equisingularity; topological type of the polar curves of algebroid curve; infinitely near points E. Casas-Alvero, Singularities of polar curves, Compositio Math. 89 (1993), 339-359. Singularities of curves, local rings, Infinitesimal methods in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Singularities of polar curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We check that the Hilbert scheme, \({\mathcal H}_{d,g}\), of smooth and connected curves of degree \(d\) and genus \(g\) in projective three-dimensional space over \(\mathbb{C}\) is smooth provided that \(d\leq 11\). The proof uses essentially our good knowledge of curves lying on cubic surfaces and the possibility to endow a curve having a special normal bundle with a double structure of high arithmetic genus. Then we give some partial results in the case of degree 12. Namely, we obtain that \({\mathcal H}_{12,g}\) is smooth for \(g<15\) except cases \(g=11,12\), for which we were able to establish only that \({\mathcal H}_{12,g}\) is smooth in codimension 1. This shows that (12,15) is the lexicographically first pair \((d,g)\) such that \({\mathcal H}_{d,g}\) is singular in codimension 1. space curves; normal bundle; double structure Sébastien Guffroy, Lissité du schéma de Hilbert en bas degré, J. Algebra 277 (2004), no. 2, 520 -- 532 (French, with English summary). Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special algebraic curves and curves of low genus, Plane and space curves, Singularities in algebraic geometry, Infinitesimal methods in algebraic geometry Smoothness of Hilbert scheme in low degree. (Lissité du schéma de Hilbert en bas degré). | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be a normal Cohen-Macaulay domain. A non-commutative crepant resolution of \(A\) (for short NCCR) is an \(A\)-algebra End\(_A(M)=:\Gamma\), where \(M\) is a reflexive \(A\)-module, \(\Gamma\) is maximal Cohen-Macaulay as an \(A\)-module and gl\(\dim(\Gamma_p)=\dim A_p\) for all primes \(p\) of \(A\). Two dimensional rings of finite representation type have NCCR. Several examples of normal Cohen-Macaulay domains having a non-commutative crepant resolution are given in the following way.
Let \((A, m)\) be an excellent normal Cohen-Macaulay domain of dimension \(\geq 2\) with perfet residue field \(k\) having NCCR. Conditions are given that \(A[X_1, \ldots, X_n]_{\mathfrak n}/\langle f\rangle\) has NCCR, \(\mathfrak{n}\) a maximal ideal containing \(\mathfrak{m}\) and \(f\). In case \(A\) contains \(k\) one has to assume that \(k[X_1, \ldots, X_n]/\langle f\rangle\) is smooth. The affine case is also treated. non-commutative crepant; resolution; normal domain; Henselian ring Singularities in algebraic geometry, Noncommutative algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Cohen-Macaulay modules, Homological dimension in associative algebras Examples of non-commutative crepant resolutions of Cohen Macaulay normal domains | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0607.00005.]
Let X be a reduced complex algebraic variety of pure dimension d. The ``Nash modification'' of X is defined as follows: Let G be the Grassmannian of locally free quotients of \(\Omega_ X^ 1\) of rank d, \(\nu: G\to X\) the canonical morphism and \(\tilde X\) the closure of \(\nu^{-1}(smooth\quad points\quad of\quad X)\) in G. Then \(\nu: \tilde X\to X\) is the Nash modification of X. Composing this with the normalization of \(\tilde X\) one gets the ``normalized Nash modification'' of X. Abbreviate by N (resp. NN) a finite succession of Nash modifications (resp. normalized Nash modifications). It is an open problem whether X can be desingularized by N (resp. NN). In his thesis at Harvard in 1985 the author proved that if \(\dim (X)=2\) then X can be desingularized by NN. This result is stated in the paper under review and a plan of the proof is outlined.
The author calls a normal surface singularity a ``sandwiched singularity'' if there exists a birational morphism \(X\to X_ 0\) with \(X_ 0\) a smooth surface. (If X'\(\to X\) is a desingularization of X then we have \(X'\to X\to X_ 0\) with X', \(X_ 0\) smooth, which explains the terminology.) By a result of \textit{H. Hironaka} [in Arithmetic and Geometry, Vol. II, Prog. Math. 36, 103-111 (1983; Zbl 0595.14006)] the problem is reduced to desingularizing a sandwiched singularity by NN. To this end the author first provides a classification of sandwiched singularities. Each sandwiched singularity X is a birational join of (finitely many) primitive sandwiched singularities \(X_ i\), where a primitive sandwiched singularity is one which contains exactly one exceptional curve of the first kind. Moreover, the dual graph of (the exceptional divisor on the minimal resolution of) X is the union of the dual graphs of \(X_ i\). The primitive sandwiched singularities and their dual graphs are classified. Having obtained this classification, the author next generalizes the techniques of \textit{G. González-Sprinberg} [Ann. Inst. Fourier 32, No.2, 111-178 (1982; Zbl 0469.14019)], wherein the desingularization of a rational double point and a cyclic quotient was obtained by NN, and applies the generalized techniques to his setup as given by the dual graph to obtain the desired desingularization. Nash modification; primitive sandwiched singularity Spivakovsky, M. Resolution of singularities. Lectures delivered at the ''Journees Singulieres et Jacobiennes''. Grenoble: Institut Fourier. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Sandwiched surface singularities and the Nash resolution problem | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author introduces an equivalence relation called P-equivalence and uses it to prove that it gives an alternative way to globalize local resolution of singularities (using the multiplicity as an invariant) in characteristic zero. This globalization result is proved in a different way by \textit{O. E. Villamayor U.}, Adv. Math. 262, 313--369 (2014; Zbl 1295.14015)], where he also addresses the problem of proving that there exists a local algorithm of resolution of singularities (in characteristic 0) using the multiplicity as the main invariant.
The paper makes use of Villamayor's resolution algorithm of Rees algebras in characteristic zero defined in [Zbl 1295.14015]. Using this result, the author defines the P-equivalence relation that allows him to globalize the resolution algorithm (defined in a local way). This proof is different from the one given by Villamayor who uses the classical relation introduced by Hironaka. resolution of singularities; multiplicity; Rees algebras; equivalence Mustaţă, M.: Spaces of arcs in birational geometry. In: Lecture notes \textbf{(available at the author's personal web page)} Global theory and resolution of singularities (algebro-geometric aspects), Multiplicity theory and related topics, Singularities in algebraic geometry, Local theory in algebraic geometry, Birational geometry Equivalence and resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We give a survey on results related to the Berglund-Hübsch duality of invertible polynomials and the homological mirror symmetry conjecture for singularities. homological mirror symmetry; singularities; strange duality; invertible polynomials; derived categories; weighted projective lines; Coxeter-Dynkin diagrams; group action; orbifold E-function; Burnside ring; unimodal; bimodal Mirror symmetry (algebro-geometric aspects), Singularities in algebraic geometry, Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Group actions on varieties or schemes (quotients), Mixed Hodge theory of singular varieties (complex-analytic aspects), Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Representations of quivers and partially ordered sets, Frobenius induction, Burnside and representation rings Homological mirror symmetry for singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The theory of singularities is a subject which plays an important role in several areas of the mathematics. For instance in algebraic geometry, complex analysis, representation theory, Lie groups, dynamical systems, etc.
This book focuses on the study of singularities from the viewpoint of algebraic geometry. It is a easily manageable book and, although it does not cover all topics, it contains a lot relevant information on the theory of singularities. The book is a translation of a textbook by the author [Introduction to singularities. (Japanese) (Springer, Tokyo) (1997; Zbl 1308.14002)]. Since then, important advances on the theory of singularities have occurred. These advances are mentioned but they are not deeply explored. The goal of the book is based on the classical classification of two-dimensional singularities and shows a classification of singularities by differential forms. Chapters 6 and 7 are devoted to this issue. Chapter 6 introduces a classification of singularities based on canonical sheaves and Chapter 7 studies two dimensional singularities from the point of view of the canonical sheaf. Higher dimensional singularities, in particular three dimensional ones, are considered in Chapter 8, and Chapter 9 is devoted to deformation of singularities. This chapter presents a table showing the change of properties of singularities under a deformation and it is the last chapter of the book. Chapter 1 lists some of the problems about singularities to which the book is devoted. Finally, chapters 2 to 5 are introductory and provide basic information that makes easier to understand the remaining chapters.
We conclude by noting that this a book that, in a reduced number of pages, summarizes important facts on the theory of singularities and it could be useful to study and understand this subject. normal two-dimensional singularities; three-dimensional singularities; classification by regular differential forms S. Ishii, \textit{Introduction to singularities}, Springer, Tokyo (2014). Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities in algebraic geometry, Deformations of singularities, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Deformations and infinitesimal methods in commutative ring theory, Minimal model program (Mori theory, extremal rays) Introduction to singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The concept of motivic integration was invented by Kontsevich to show that birationally equivalent Calabi-Yau manifolds have the same Hodge numbers. He constructed a certain measure on the arc space of an algebraic variety, the motivic measure, with the subtle and crucial property that it takes values not in \(\mathbb R\), but in the Grothendieck ring of algebraic varieties. A whole theory on this subject was then developed by Denef and Loeser in various papers, with several applications. Batyrev introduced with motivic integration techniques new singularity invariants, the stringy invariants, for algebraic varieties with mild singularities, more precisely log terminal singularities. He used them for instance to formulate a topological mirror symmetry test for pairs of singular Calabi--Yau varieties. We generalized these invariants to almost arbitrary singular varieties, assuming Mori's minimal model program.
The aim of these notes is to provide a gentle introduction to these concepts. Here we merely want to explain the basic concepts and first results, including the \(p\)-adic number theoretic pre-history of the theory, and to provide concrete examples. The text is a slightly adapted version of the `extended abstract' of the author's talks at the 12th MSJ-IRI ``Singularity Theory and Its Applications'' (2003) in Sapporo. At the end we included a list of various recent results. W. Veys, ''Arc spaces, motivic integration and stringy invariants,'' in Singularity Theory and its Applications, Tokyo: Math. Soc. Japan, 2006, vol. 43, pp. 529-572. Arcs and motivic integration, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Local ground fields in algebraic geometry Arc spaces, motivic integration and stringy invariants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0596.00007.]
As the title shows, the paper under review discusses Lipman's proof of the existence of resolutions of singularities of surfaces [see \textit{J. Lipman}, Ann. Math., II. Ser. 151-207 (1978; Zbl 0249.14004)]. The main reason of presenting this proof lies in the fact that, with little extra effort, it works for arithmetic surfaces as well. This paper can be also considered as an excellent introduction to Lipman's original paper. The proof itself consists of three main steps: \((1)\quad reduction\) to rational singularities, \((2)\quad assu\min g\) that the surface Y in question has only rational singularities, reduction to rational double points, and \((3)\quad \exp licit\) resolution of rational double points. The first two steps are very conceptual and involve the dualizing sheaf \(\omega_ Y\) of Y. The main two technical tools are a vanishing theorem and a duality theorem for an arbitrary ''partial'' resolution Y'\(\to Y\) of Y. Some proofs are also included, while others are at least sketched. dualizing sheaf; existence of resolutions of singularities of surfaces; arithmetic surfaces; rational singularities Artin M.(1986). Lipman's proof of resolution of singularities of surfaces. In: Cornell G., Silverman J. (eds) Arithmetic Geometry. Springer, New York, pp. 267--287 Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Lipman's proof of resolution of singularities for surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field and let \(X\) be a separated scheme of finite type over \(k\) of pure dimension \(d\). We study the structure of the fibres of the truncation morphisms from the arc space of \(X\) to jet spaces of \(X\) and also between jet spaces. Our results are generalizations of results of Denef, Loeser, Ein and Mustaţă. We will use them to find the optimal lower bound for the poles of the motivic zeta function associated to an arbitrary ideal. jet spaces; motivic integration; motivic zeta function Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) On the structure of the fibres of truncation morphisms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The existence of resolution of singularities of an algebraic variety in arbitrary dimension over a field of characteristic zero was proved by Hironaka. Thereafter several authors have given different constructive proofs. Once we have achieved resolution for a variety, it is natural to wonder if it is possible to resolve simultaneously the singularities of a family of varieties. This paper states a theory of simultaneous resolution of singularities for infinitesimal deformations of embedded varieties, that is, families of embedded varieties parametrized by \(S=Spec(A)\) where \(A\) is an artinian ring.
This resolution involves algorithmic resolution of an embedded variety of arbitrary dimension over a field of characteristic zero. More precisely, the author uses a variant of Villamayor algorithm of resolution of singularities given in \textit{S. Encinas} and \textit{O. Villamayor} [Prog. Math. 181, 147--227 (2000; Zbl 0969.14007)], with some tools coming from the desingularization given by \textit{J. Wlodarczyk} [J. Am. Math. Soc. 18, No. 4, 779--822 (2005; Zbl 1084.14018)]. This variant is called \textit{VW-algorithm} along the paper.
The author reviews all the notions involved in the construction of Villamayor algorithm and adapts all these notions to the case of deformations of embedded varieties. He has done a formidable work to extend all the concepts to suitable conditions over the special fibers.
The paper starts with a revision of Villamayor algorithm of resolution of \textit{basic objects} (tuples \(B=(W,I,b,E)\) where \(W\) is a smooth variety, \(I\) is a never zero \(W\)-ideal, \(b\) a positive integer number and \(E\) a set of regular hypersurfaces in \(W\) having only normal crossings), making an extension of all notions to the case where \(\mathcal{A}\) is a collection of artinian local rings \((A,M)\) such that the residue field \(k=A/M\) has characteristic zero. So he works in the context of \(A\)-basic objects, that is a basic object over a ring \(A\in \mathcal{A}\). One of the key points is the definition of the \textit{adapted hypersurfaces} playing the analogous role to the hypersurfaces of maximal contact, and the proof of that this notion is stable under permissible transformation. The step that differs from Villamayor algorithm is the use of the \textit{homogenized ideal}, due to Wlodarczyk, instead of the \textit{generalized basic objects} to solve the problem of the patching when there are many adapted hypersurfaces. This is performed passing from the \(A\)-basic object \(B\) to its homogenized \(\mathcal{H}(B)\) to make induction on the dimension of the ambient space.
He proves that the \textit{algorithmic equiresolution} of \(A\)-basic object leads to the \textit{algorithmic equiprincipalization} of triples \((W\rightarrow S,I,E)\) over \(A\in\mathcal{A}\), and hence resolution of embedded varieties over an Artin ring \(A\in \mathcal{A}\). In this case it is said that the relative embedded \(A\)-variety is \textit{algorithmically equisolvable}.
The author also includes several interesting examples along the article, such as the example of an \(A\)-basic object that is not algorithmically equisolvable.
The article is self contained, the author includes the necessary theoretical background and an appendix of revision of useful results that not always appear in the literature. resolution of singularities; simultaneous resolution; deformation Nobile, A.: Algorithmic equiresolution of deformations of embedded algebraic varieties. Revista Matemática Hispanoamericana \textbf{25}, 995-1054 (2009, to appear) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Birational geometry, Families, fibrations in algebraic geometry, Singularities of curves, local rings Algorithmic equiresolution of deformations of embedded varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We initiate the study of Nash blowups in prime characteristic. First, we show that a normal variety is non-singular if and only if its Nash blowup is an isomorphism, extending a theorem by \textit{A. Nobile} [Pac. J. Math. 60, No. 1, 297--305 (1975; Zbl 0324.32012)]. We also study higher Nash blowups, as defined by T. Yasuda. Specifically, we give a characteristic-free proof of a higher version of Nobile's theorem for quotient varieties and hypersurfaces. We also prove a weaker version for \(F\)-pure varieties. Nash blowups; normal varieties; differential operators; methods in prime characteristic Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Positive characteristic ground fields in algebraic geometry, Rings of differential operators (associative algebraic aspects), Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Hypersurfaces and algebraic geometry Nash blowups in prime characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a projective three dimensional variety \(Y\) over \(\mathbb{C}\) with only \(\mathbb{Q}\)-factorial singularities, an elementary contraction is a projective surjective morphism with connected fibres \(\mu: Y \to X\) onto a normal variety \(X\) such that the relative Picard number is \(1\) and \(-K_Y\) is \(\mu\)-ample. Elementary contractions arise naturally in the minimal model program and it is of great interest to classify them [see \textit{S. Cutkosky}, Math. Ann. 280, No. 3, 521-525 (1988; Zbl 0616.14003) and \textit{J. Kollár} and \textit{S. Mori}, J. Am. Math. Soc. 5, No. 3, 533-703 (1992; Zbl 0773.14004)]. When an elementary contraction is birational with a prime divisor as exceptional locus it is called a divisorial contraction. The paper under review is inspired by the desire of classifying divisorial contractions giving rise to quotient terminal singularities. (Given the set up described below, where one starts from a given singularity \((X,P)\), the author suggests the use of the term divisorial extraction.) The precise set up is the following:
Let \((X,P)=\mathbb{C}^3/\mathbb{Z}_r(s,-s,1)\) be a quotient singularity with \(0<s<r\) where \(r\) and \(s\) are relatively prime. Let \(\mu : Y \to X\) be a projective birational morphism, from a variety \(Y\) with only terminal singularities, with exceptional locus \(E\) such that \(\mu\) coincides with the blow up along the generic points of \(\mu(E)\), \(E\) is a \(\mathbb{Q}\)-Cartier divisor, \(-E\) is \(\mu\)-ample and either \(E\) is irreducible or all of its components are mapped by \(\mu\) to curves. Under the above assumptions the author proves that \(\mu\) coincides with the weighted blow up of \(X\) at \(P\) with weights \((s/r, 1- s/,1/r)\). Such a morphism is independent of the choice of coordinates and it is unique for a given singularity. divisorial contraction; terminal singularities; quotient singularities; complex projective threefolds Y. Kawamata, Divisorial contractions to 3-dimensional terminal quotient singularities, Higher-dimensional complex varieties (Trento 1994), De Gruyter, Berlin (1996), 241-246. \(3\)-folds, Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Divisorial contractions to 3-dimensional terminal quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities If \(f:X\to Y\) is a projective morphism, where \(X\) is a smooth symplectic variety and \(Y\) is normal, then \(f\) is called a symplectic resolution and one says that \(Y\) has symplectic singularities. Let \(\sigma \) be a symplectic form on \(X,\) \(\dim X=2n.\) Let \(D\subset X\) be the exceptional set and \(E=f(D)\) is precisely the singular locus of \(Y,\) since \(f\) is a crepant contraction. If \(E\) is irreducible and \((2n-2)\)-dimensional and \(f| _D\to E\) has only 1-dimensional fibres then author classifies such contractions. He proves that if \((X,\sigma)\) is a projective symplectic smooth variety of dimension \(2n\) and if \(f:X\to Y\) is a projective birational morphism to a normal variety \(Y,\) \(\text{Sing}(Y)\) contains an irreducible component \(E\) of dimension \(2n-2\), then \(E\) is regular in codimension 1 and its normalization \(\widetilde E\) is a symplectic variety. symplectic manifolds; birational contractions; projective morphisms Wierzba, J.: Contractions of symplectic varieties. J. Algebraic Geom. \textbf{12}(3), 507-534 (2003) Global theory and resolution of singularities (algebro-geometric aspects), Global theory of symplectic and contact manifolds, Singularities in algebraic geometry, Compact complex \(n\)-folds Contractions of symplectic varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a normal complex variety with at most log-terminal singularities and \(\pi:\overline X\to X\) be a desingularities such that the exceptional locus \(\displaystyle\bigcup^m_{i=1} D_i\) \((D_i\) smooth with normal crossings) has discrepancy \(K_{\overline X}-\pi^* (K_X)=\displaystyle\sum^m_{i=1} a_iD_i\). Let \(I=\{1,\dots, m\}\) and define for \(J\subset I\),
\[
D_J:= \begin{cases} \overline X&\text{for }J= \emptyset \\ \bigcap_{j\in J}D_j \end{cases},\qquad D_J^0=D_J \setminus \bigcup_{ j\in I\setminus J} D_j.
\]
Then the string-theoretic \(E\)-function \(E_{\text{str}} (X,u,v)\) is defined by
\[
E_{\text{str}} (X,u,v)=\sum_{J\subseteq I} E(D^0_J,u,v) \prod_{j\in J}{uv-1 \over(uv)^{a_j+1} -1}.
\]
Here \(E(X,u,v) =\sum e^{pq}(X)u^p v^q\in\mathbb{Z} [u,v]\) is the \(E\)-polynomial with
\[
e^{pq}(X)= \sum(-1)^i h^{pq} (H^i_e (X,\mathbb{C})),\qquad h^{pq}(H^i_e (X,\mathbb{C}))\text{ the Hodge numbers.}
\]
The string-theoretic Euler number is defined by
\[
e_{\text{str}} (X)= \lim_{u,v\to 1}E_{ \text{str}}(X,u,v)
\]
and the string theoretic index by
\[
\text{ind}_{\text{str}} (X)=\min\left\{ \ell\in\mathbb{Z}_+\left|e_{\text{str}} (X)\in {1\over\ell} \mathbb{Z}\right\}\right. .
\]
Let \(X\) be an \(A^{(r)}_{n,\ell}\)-singularity, that is, defined by \(x_1^{n+1} +x_2^\ell +\cdots+x^\ell_{r+1} =0\); then an explicit computation of \(E_{\text{str}} (X,u,v)\), \(e_{\text{str}}(X)\) is given. This is applied to give a counterexample to a conjecture of \textit{V. V. Batyrev} [in: Integrable systems and algebraic geometry Proc. 41st Taniguchi Symp., Kobe 1997, Kyota 1997,World Scientific, 1-32 (1998; Zbl 0963.14015)] concerning the boundedness of the string-theoretic index. string-theoretic Euler number Dais, D.: On the string-theoretic Euler number of a class of absolutely isolated singularities. Manuscripta Math. 105, 143--174 (2001) Singularities of surfaces or higher-dimensional varieties, Computational aspects of higher-dimensional varieties, Mixed Hodge theory of singular varieties (complex-analytic aspects), Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) On the string-theoretic Euler number of a class of absolutely isolated singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is known that the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant birational morphisms in all dimensions. In the present paper we extend this result to the entire class of toric local complete intersection singularities. Our strikingly simple proof makes use of Nakajima's classification theorem [\textit{H. Nakajima}, J. Tôhoku Math. J., II. Ser. 38, 85--98 (1986; Zbl 0604.14044)] and of some techniques from toric and discrete geometry. toric discrete geometry; crepant birational morphisms; minimal models; desingularizations; toric local complete intersection singularities Dimitrios I. Dais, Christian Haase, and Günter M. Ziegler, All toric local complete intersection singularities admit projective crepant resolutions, Tohoku Math. J. (2) 53 (2001), no. 1, 95 -- 107. Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects), Complete intersections, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Singularities in algebraic geometry All toric local complete intersection singularities admit projective crepant resolutions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is a pedagogical survey of proofs of De Jong's theorem ``Any variety \(X\) can be altered into a nonsingular variety'' and of some corollaries. The main ingredients and tools are exposed for ``any student of algebraic geometry''. The main reference is \textit{A. J. De Jong}'s paper: ``Smoothness, semistability and alterations'', Publ. Math., Inst. Hautes Étud. Sci. 83, 51-53 (1996; Zbl 0916.14005)]. Let us recall that an alteration of a variety \(X\) is a variety \(Y\) with a proper, surjective and generically finite morphism \(Y \rightarrow X\). In the introduction, the authors make first a small exposition of Hironaka's and De Jong's theorems then give the skeleton of De Jong's proof. The first part of the paper is a proof of De Jong's theorem (theorem 2.3 here) and of some corollaries in characteristic 0 (desingularization 2.8). The proof is complete up to exercises, a few references and an existence theorem proven in the second part. The second part is a small course on the theory of moduli of curves, a theory which is drastically used by De Jong and which is ``well known by the experts''. Indeed, the difficult point in part I is to show that, if there is a morphism \(f: X \rightarrow P\) where all the fibers are curves, after an alteration of \(P\), you can reach the case where all the fibers are curves with only ordinary nodes as singularities. This is a consequence of theorem 3.6, its proof uses the fact that there is a ``tautological family'' of curves over the compactified moduli space of curves with a level structure. The purpose of the second part is to prove the existence of this tautological family. desingularization; alteration; modification; tautological family Abramovich, D.; Oort, F., Alterations and resolution of singularities, (Resolution of Singularities, Obergurgl, 1997, Progr. Math., vol. 181, (2000), Birkhäuser Basel), 39-108 Global theory and resolution of singularities (algebro-geometric aspects), Families, moduli of curves (algebraic), Singularities in algebraic geometry Alterations and resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f:X\rightarrow Y\) be a projective birational morphism, between complex quasi-projective varieties. Fix a bivariant class \(\theta \in H^0 (X\overset{f}{\rightarrow}Y)\cong Hom_{D^b_c (Y)}(Rf_* \mathbb{A}_X, \mathbb{A}_Y)\) (here \(\mathbb{A}\) is a Noetherian commutative ring with identity, and \(\mathbb{A}_X\) and \(\mathbb{A}_Y\) denote the constant sheaves). Let \(\theta_0 :H^0 (X)\rightarrow H^0 (Y)\) be the induced Gysin morphism. We say that \(\theta\)\textit{ has degree one} if \(\theta_0 (1_X)= 1_Y\in H^0 (Y)\). This is equivalent to say that \(\theta\) is a section of the pull-back \(f^* : \mathbb{A}_Y \rightarrow Rf_* \mathbb{A}_X\), i.e. \(\theta \circ f^* =\mathrm{id}_{\mathbb{A}_Y}\), and it is also equivalent to say that \(\mathbb{A}_Y\) is a direct summand of \(Rf_* \mathbb{A}_X\). We investigate the consequences of the existence of a bivariant class of degree one. We prove explicit formulas relating the (co)homology of \(X\) and \(Y\), which extend the classic formulas of the blowing-up. These formulas are compatible with the duality morphism. Using which, we prove that the existence of a bivariant class \(\theta\) of degree one for a resolution of singularities, is equivalent to require that \(Y\) is an \(\mathbb{A}\)-homology manifold. In this case \(\theta\) is unique, and the Betti numbers of the singular locus \(\mathrm{Sing}(Y)\) of \(Y\) are related with the ones of \(f^{-1}(\mathrm{Sing}(Y))\). projective variety; derived category; Poincaré-Verdier duality; bivariant theory; Gysin morphism; homology manifold; resolution of singularities; intersection cohomology; decomposition theorem Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Topological properties in algebraic geometry, Global theory of complex singularities; cohomological properties, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Poincaré duality spaces, Topological properties of mappings on manifolds Bivariant class of degree one | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The versal discriminant of one-modal isolated singularities \(Z_{k,0}\) and \(Q_{k,0}\) which describes an obstacle for analytic triviality of an unfolding along the moduli is computed. The basic idea of the author is to use blowing ups of certain deformations of \(J_{k,0}\) singularities in order to obtain the deformations of singularities in question. He also underlines that similar notions such as versality discriminant, instability locus, etc. were investigated in papers of Damon and many others [see e.g. \textit{J. Damon}, Singularities, Banach Cent. Publ. 20, 161-167 (1988; Zbl 0675.58008)]. unfoldings; deformations; analytic triviality; moduli; equisingularity; versal discriminant; versality discriminant; instability locus; contact equivalence; liftable vector fields; blowing up; blowing down Equisingularity (topological and analytic), Complex surface and hypersurface singularities, Critical points and critical submanifolds in differential topology, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry On blowing up versal discriminants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a reductive group \(G\), the products of projective varieties homogeneous under \(G\) that are spherical for the diagonal action of \(G\) have been classified by \textit{J. R. Stembridge} [Represent. Theory 7, 404--439 (2003; Zbl 1060.17001)]. We consider the \(B\)-orbit closures in these spherical varieties and prove that under some mild restrictions they are normal, Cohen-Macaulay and have a rational resolution. spherical varieties; normal singularities; rational resolutions; homogeneous spaces Achinger, P., Perrin, N.: Spherical Multiple Flags. arXiv:1307.7236 Compactifications; symmetric and spherical varieties, Linear algebraic groups over arbitrary fields, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Spherical multiple flags | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\subset\text{SL}(3,\mathbb{C})\) be a finite group. A \(G\)-cluster is a \(G\)-invariant subscheme \(Z\subset \mathbb{C}^3\) of dimension zero with global sections \(H^0(\mathcal{O}_Z)\) isomorphic as a \(\mathbb{C}[G]\)-module to the regular representation \(R\) of \(G\). \textit{I. Nakamura} [J. Algebr. Geom. 10, No.4, 757--779 (2001; Zbl 1104.14003)] introduced the moduli space \(G\text{-Hilb}\) of \(G\)-clusters on \(\mathbb{C}^3\) as a natural candidate for a projective crepant resolution of \(\mathbb{C}^3/G\) and proved it for \(G\) abelian. \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] subsequently proved the conjecture for all \(G\) by establishing an equivalence of derived categories: \(D(G\text{-Hilb})\sim D^G(\mathbb{C}^3)\).
This paper generalises the notion of \(G\)-cluster: a \(G\)-constellation is a \(G\)-equivariant coherent sheaf \(F\) on \(\mathbb{C}^3\) with global sections \(H^0(F)\) isomorphic as a \(\mathbb{C}[G]\)-module to the regular representation \(R\) of \(G\). Set:
\[
\Theta:=\left\{\theta\in\text{Hom}(R(G),\mathbb{Q})\mid \theta(R)=0\right\}.
\]
For \(\theta\in \Theta\), a \(G\)-constellation is said to be stable (resp. semistable) if every proper \(G\)-equivariant coherent subsheaf \(0\subset E\subset F\) satisfies \(\theta(E)>0=\theta(F)\) (resp. \(\geq\)). Generalizing ideas of \textit{A. V. Sardo-Infirri} [Resolutions of orbifold singularities and the transportation problem on the McKay quiver, preprint, \url{arXiv:alg-geom/9610005}] and \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, 515--530 (1994; Zbl 0837.16005)], the authors study the moduli spaces \(\mathcal{M}_{\theta}\) (resp. \(\overline{\mathcal{M}}_{\theta})\) of \(\theta\)-stable (resp. semistable) constellations. Note that \(G\text{-Hilb}\cong \mathcal{M}_{\theta}\) for parameter \(\theta\) in the cone \(\Theta_+:= \{\theta\in\Theta\mid \theta(\rho)>0\) if \(\rho\neq \rho_0\}\), where \(\rho_0\) denotes the trivial representation of \(G\). A parameter \(\theta\in \Theta\) is generic if every \(\theta\)-semistable \(G\)-constellation is \(\theta\)-stable.
The method of Bridgeland-King-Reid generalises to show that: If \(\theta\) is generic, there is an equivalence of categories \(D(\mathcal{M}_{\theta})\sim D^G(\mathbb{C}^3)\) and \(\mathcal{M}_{\theta}\rightarrow \mathbb{C}^3/G\) is a projective crepant resolution of singularities. It is then natural to ask whether every projective crepant resolution may be realised as a moduli space \(\mathcal{M}_{\theta}\) for some parameter \(\theta\). The main result of this paper answers this question affirmatively in the abelian case: For a finite abelian subgroup \(G\subset \text{SL}(3,\mathbb{C})\), suppose that \(Y\rightarrow \mathbb{C}^3/G\) is a projective crepant resolution. Then \(Y\cong \mathcal{M}_{\theta}\) for some parameter \(\theta\).
For generic \(\theta\), put \(C:=\{\eta\in \Theta\mid\) every \(\theta\)-stable \(G\)-constellation is \(\eta\)-stable\}. This is a convex polyhedral cone (or chamber) in \(\Theta\). The subset \(\Theta^{\text{gen}}\subset \Theta\) of generic parameter is open, dense and is the disjoint union of finitely many open convex polyhedral cones in \(\Theta\). For generic \(\theta\), the moduli space \(\mathcal{M}_{\theta}\) depends only upon the open chamber \(C\subset \Theta\) containing \(\theta\), so we write \(\mathcal{M}_C\) in place of \(\mathcal{M}_{\theta}\) for any \(\theta\in C\). Then the proof's idea is as follows: Since every projective crepant resolution is obtained by a finite sequence of flops from \(G\text{-Hilb}\), it is enough to show that, if \(Y\cong \mathcal{M}_C\) for some chamber \(C\), then for any flop \(Y'\) of \(Y\) there is a chamber \(C'\) (not necessarily adjacent to \(C\)) such that \(\mathcal{M}_{C'}\cong Y'\). Then the first step is to understand the walls of chambers in \(\Theta\) (\S 3) and then how the moduli \(\mathcal{M}_C\) changes as \(\theta\) passes through a wall from \(C\) to another chamber \(C'\). The method uses the description of chambers in terms of Fourier-Mukai transforms. Hilbert schemes of orbits; constellations; crepant resolution; Fourier-Mukai; toric geometry Craw-Ishii A.~Craw and A.~Ishii, Flops of \(G\)-Hilb and equivalences of derived categories by variation of GIT quotient, Duke Math.\ J., 124 (2004), 259--307. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Geometric invariant theory Flops of \(G\)-Hilb and equivalences of derived categories by variation of GIT quotient | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is an expository article on certain aspects of equisingularity theory. A continuous function \(q: U\to\mathbb{R}\), where \(U\) is an open set in \(\mathbb{R}^2\), is called blow-analytic if there is a sequence of quadratic transformations (i.e., blowing-ups with zero-dimensional centers) \(M_N\to\cdots\to M_0= U\) such that the resulting composed function \(q_N: M_N\to \mathbb{R}\) is analytic (note that \(M_j\) is an analytic manifold for all \(j\)). This notion is strictly weaker than the \(C^1\) condition. For instance, \(q(x, y)= {x^2y\over x^2+ y^2}\) is not \(C^1\) but is blow-analytic. For a meromorphic function \(m\) blow-analyticity at \((a, b)\) is equivalent to arcanalyticity, that is the requirement for of any resulting function \(m(x(t), y(t))\) to be pair of convergent power series \(x(t)\), \(y(t)\), with \(x(0)= a\), \(y(0)= b\), the analytic at \(0\). Two germs of analytic functions \(f\) , \(g\) from \((\mathbb{R}^2,0)\) to \((\mathbb{R},0)\) are said to be blow-analytically equivalent if there is a germ of homeomorphism \(h:(\mathbb{R}^2,0)\to (\mathbb{R}^2, 0)\), satisfying \(gh= f\), such that the components of both \(f\) and its inverse are blow-analytic.
The present paper discusses results about this notion. For instance, if \(f\) and \(g\) are blow-analytically equivalent, then the germs of curve they define have the same multiplicity (T. Fukui). In one-parameter families of weighted homogeneous forms, the blow-analytic equivalence class of the fibers is constant (Fukui-Paunescu). A brief discussion of the generalization to dimension \(> 2\), indicating some new problems that appear, follows. An introductory section on the blowing-up process is included. blowing-up; blow-analytic function; blow-analytic equivalence Equisingularity (topological and analytic), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings, Local complex singularities, Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry An elementary exposé on equisingularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0614.00007.]
In this paper the author studies the behaviour of the singularities of a 3-dimensional variety under a sequence of quadratic transformations. In particular, under certain hypothesis of ``good position'' and with a suitable choice of the local parameters, he gives the construction of a ``Newton polygon'' which extends the classical one. singularities of a 3-dimensional variety; quadratic transformations Global theory and resolution of singularities (algebro-geometric aspects), \(3\)-folds, Singularities in algebraic geometry, Rational and birational maps Techniques pour la désingularisation des champs de vecteurs. (Techniques for the desingularization of vector fields) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A small resolution of a neighbourhood of a node \(p_ 0\) of a 3- dimensional analytic space V is a variety \(\tilde V\) and a map \(\pi: \tilde V\to V\) such that \(\pi^{-1}(p_ 0)\) is isomorphic to \({\mathbb{P}}^ 1\) and \(\pi\) induces an isomorphism of \(\tilde V-{\mathbb{P}}^ 1\) and \(V-p_ 0\). The present paper considers the following case: is the variety \(\tilde V\) projective if it is a small resolution of all singularities of a nodal cubic surface in \({\mathbb{P}}^ 4 ?\)
Previous work of the second author reduces this to deciding the nontriviality of the homology classes of exceptional lines \({\mathbb{P}}^ 1\) on \(\tilde V\) [\textit{J. Werner}, ``Kleine Auflösungen spezieller dreidimensionaler Varietäten'', Bonn. Math. Schr. 186 (1987; Zbl 0657.14021)]. - The paper under review answers the question of projectivity of small resolutions in terms of the following projective invariant of the cubic. Take a coordinate system in \({\mathbb{P}}^ 4\) so that the equation of the cubic in coordinates \(z_ 0,...,z_ 4\) will be \(z_ 4Q+R=0\) where Q (resp. R) is a polynomial of degree 2 (resp. 3) in \(z_ 0,...,z_ 3\) and such that a fixed node of the cubic, say P, will have coordinate (0,0,0,0,1). The associated curve of the cubic relative to the node P is the complete intersection \(Q=0,\) \(R=0\) considered as a curve on the ambient quadric \(Q=0\). the authors classify nodal cubic hypersurfaces in \({\mathbb{P}}^ 4\) into 15 classes (J1,...,J15) according to the defect, which is the difference between the fourth and the second betti numbers of the cubic, the homology class of the associated curve on the ambient quadric and the number of singularities of the associated curve (which are all ordinary double points). In each case they decide if the small resolution is projective, the dependence of the associated curve on the node relative to which this curve was found, the number of planes on the cubic, as well as which of the cubic nodes are coplanar. The cubic (Segre cubic) with the maximal number of nodes 10 corresponds to the class J15 in this classification. nodal cubic threefolds; projectivity of small resolutions; betti numbers; number of singularities; cubic nodes; Segre cubic Hans Finkelnberg and Jürgen Werner, Small resolutions of nodal cubic threefolds, Nederl. Akad. Wetensch. Indag. Math. 51 (1989), no. 2, 185 -- 198. \(3\)-folds, Global theory and resolution of singularities (algebro-geometric aspects), Projective techniques in algebraic geometry, Singularities in algebraic geometry Small resolutions of nodal cubic threefolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities There are two different algorithmic approaches to the resolution of singularities. One is due to \textit{B. Bierstone} and \textit{P. Milman} [Invent. Math. 128, No.2, 207--302 (1997; Zbl 0896.14006)] using a stratification by the Hilbert--Samuel--function at the beginning of each choice--of--center step and the strict transform as the corresponding notion of transform. The other approach is due to \textit{O. Villamayor} [Ann. Sci. Éc. Norm. Supér. (4) 22, No. 1, 1--32 (1989; Zbl 0675.14003)] with many contributions by others is based on the use of the weak transform which picks up extra components lying inside the exceptional divisor at each blowing up. The approaches also differ in their descent in dimension of the ambient space. In this article, a hybrid--type algorithmic approach is proposed which allows the use of the strict transform without the full impact of the complexity of the stratification by the Hilbert--Samuel--function, i.e. the Hilbert--Samuel--stratum is not computed in each step, but instead auxiliary ideals are computed which provide the information whether the Hilbert--Samuel--function of the original ideal dropped, in terms of their order. The new approach is illustrated by two examples. resolution of singularities; weak transform; Hilbert-Samuel function; blowing up Computational aspects of higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) A modified coefficient ideal for use with the strict transform | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let X be a singular space with singular locus \(\Sigma\). The author considers the question if for any perversity \(\bar p\) the dual of the intersection homology group \(IH_*^{\bar p}(X)\) with real coefficients is canonically isomorphic to the \(L^ 2\)-cohomology group \(H^*_{(2)}(X-\Sigma)\). Here X-\(\Sigma\) has a metric, which is associated to \(\bar p\). For example on the interior (0,1)\(\times X\) of a cone over X one can consider the metric \(dr\otimes dr+2^{rc}g,\) where g is a metric on X. In the general case a metric g on \(X-\Sigma\) is associated with \(\bar c=(c_ 2,...,c_ n)\) if the restrictions to local cones near points of \(\Sigma\) are of the above type. Also metrics can be associated to a perversity \(\bar p\). The author gives a positive answer to the question if \(\bar p\leq \bar m\) (middle perversion). If \(\bar p=\bar m\) the statement was already due to J. Cheeger.
The proof is rather technical. The paper is written before the publishing of the article of \textit{M. Goresky} and \textit{R. MacPherson} [Invent. Math. 72, 77-129 (1983; Zbl 0529.55007)]. It is clear that part of the proof could be simplified if the sheaf-theoretic set-up of that paper is used. metrics associated to a perversity; singular space; singular locus; intersection homology group; \(L^ 2\)-cohomology group; local cones Nagase, N, \(L^2\)-cohomology and intersection cohomology of stratified spaces, Duke Math. J., 50, 329-368, (1983) Other homology theories in algebraic topology, Stratifications in topological manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Complex singularities, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) \(L^ 2\)-cohomology and intersection homology of stratified spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) an algebraically closed field, \(\operatorname{char} k=0\). Let \(C\) be an irreducible nonsingular curve such that \(kC=S\cap X\), \(k\in\mathbb{N}\), where \(S\) and \(X\) are two surfaces and all the singularities of \(X\) are of the form \(z^p=x^s-y^s\), \(p\), \(s\) primes, \(s=pt+m\), \(t\in\mathbb{N}\), \(m=1\) or 2. We also study the cases, for \(p\), \(s\) primes, \(p=4r+1\), \(r \in\mathbb{N}\), \(s=pt+2r+1\), \(t \in\mathbb{N}\), and \(p=4r+3\), \(r \in\mathbb{N}\), \(s=pt+2r+2\), \(t \in\mathbb{N}\). We prove that \(C\) can never pass through such kind of singularities of a surface, unless \(k=pa\), \(a\in\mathbb{N} \). We study multiplicity-\(k\) structures on varieties, \(k\in\mathbb{N}\). Let \(Z\) be a reduced irreducible nonsingular \((n-1)\)-dimensional variety such that \(kZ=X\cap S\), where \(S\) is a \((N-1)\)-fold in \(\mathbb{P}^N\), \(X\) is a normal \(n\)-fold with certain type of singularities, like linear compound \(V_{ps}\) singularity or (d,l) complete intersection compound \(V_{ps}\) singularity, We study when \(Z\cap \operatorname{Sing} (X)\ne \emptyset \). These results generalize some results in our work [in: Singularities in geometry, topology, foliations and dynamics. A celebration of the 60th birthday of José Seade. Selected papers based on the presentations at the workshop, Mérida, Mexico, December 8--19, 2014. Cham: Birkhäuser. 125--134 (2017; Zbl 1425.14004)]. Seifert invariants of \(z^p=x^s-y^s\), \(p\), \(s\) primes, \(s=pt+m\), \(t\in\mathbb{N}\), \(m=1, 2\) and for \(p=4r+1\), \(r \in\mathbb{N}\), \(s=pt+2r+1\), \(t \in\mathbb{N}\), and \(p=4r+3\), \(r \in\mathbb{N}\), \(s=pt+2r+2\), \(t \in\mathbb{N}\) are studied (Prop. 34). Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, \(3\)-folds, \(4\)-folds, \(n\)-folds (\(n>4\)), Hypersurfaces and algebraic geometry On certain type of singular varieties with smooth subvarieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a field of characteristic zero, \(L/k\) an \(n\)-dimensional algebraic function field, \(K\) a finite algebraic extension of \(L\), \(\nu\) a zero-dimensional valuation of \(K/k\), and \((R,M)\) a regular local ring, essentially of finite type over \(k\) with quotient field \(K\) and ground field \(k\) such that \(\nu\) has center \(M\) in \(R\). Then for some sequence of monoidal transforms \(R\to R^*\) along \(\nu\), there exists a local domain \(S^*\), essentially of finite type over \(k\) with quotient field \(L\) and ground field \(k\) lying below \(R^*\). When \(n=2\) this is stated by \textit{S. S. Abhyankar} in theorem 4.8 of his book ``Ramification theoretic methods in algebraic geometry'' [Ann. Math. Stud. No. 43 (1959; Zbl 0101.38201)]. The above result is a kind of simultaneous resolution of singularities; other forms are also included. resolution of singularity; birational domination; algebraic function field; valuation; regular local ring; monoidal transforms Cutkosky, S. D.: Simultaneous resolution of singularities. Proc. amer. Math. soc. 128, 1905-1910 (2000) Global theory and resolution of singularities (algebro-geometric aspects), Valuations and their generalizations for commutative rings, Singularities in algebraic geometry Simultaneous resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors have conjectured a very general relation between the topology and the analytic invariants of a complex normal surface singularity \((X,0)\) whose link \(M\) is a rational homology sphere. When \((X,0)\) is a hypersurface in \((\mathbb{C}^3,0)\), given by \(g=0\), this conjecture says that the modified Seiberg-Witten invariant of \(M\) equals \(-\sigma(F)/8\), where \(\sigma(F)\) is the signature of the Milnor fiber of \(g\).
In this paper the special, but highly non-trivial case of suspensions of irreducible plane curve singularities (i.e., \(g(x,y,z)=f(x,y)+z^n)\) is completely settled, giving a positive answer to the above conjecture. The paper has a rather long introduction, which provides an excellent overview of the part and present work done in this fascinating area. normal surface singularities; Reidemeister-Turaev torsion; Casson-Walker invariant; links Némethi, A., Nicolaescu, L.: Seiberg-Witten invariants and surface singularities III.: splicings and cyclic covers. Sel. Math. 11 (3--4), 399--451 (2006) Singularities in algebraic geometry, Complex surface and hypersurface singularities, Invariants of knots and \(3\)-manifolds, Global theory and resolution of singularities (algebro-geometric aspects) Seiberg-Witten invariants and surface singularities: splicings and cyclic covers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0575.00008.]
Let D denote the discriminant locus consisting of all singular hypersurfaces of degree d in \(P^ n\). Let \(P\in D\) correspond to a singular hypersurface of the ''worst'' kind, i.e., a d-fold hyperplane. The authors prove that the projectivized tangent cone to D at P is equal to the dual variety of a (d-1)-fold Veronese embedding of a hyperplane. discriminant locus; d-fold hyperplane; tangent cone; Veronese embedding Roy Smith and Robert Varley, The tangent cone to the discriminant, Proceedings of the 1984 Vancouver conference in algebraic geometry, CMS Conf. Proc., vol. 6, Amer. Math. Soc., Providence, RI, 1986, pp. 443 -- 460. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Multiplicity theory and related topics The tangent cone to the discriminant | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author determines explicitly the irreducible components of any Schubert variety \(\text{GL}_n(K)\) for any algebraically closed field \(K\) and describes the generic singularities along them. Similar results have been given by \textit{L. Manivel} [Int. Math. Res. Not. 2001, 849--871 (2001; Zbl 1023.14022)], \textit{S. Billey} and \textit{G. S. Warrington} [Trans. Am. Math. Soc. 355, 3915--3945 (2003; Zbl 1037.14020)], and \textit{C. Kassel, A. Lascoux} and \textit{C. Reutenauer} [J. Algebra 269, 74--108 (2003; Zbl 1032.14012)]. The methods of the present article are geometric and give a different and useful perspective of the field. Quasi-resolutions play an important role, and may be useful in computing Kazhdan-Lusztig polynomials for arbitrary polynomials. The proof builds heavily upon the results of \textit{A. Cortez} [Adv. Math. 178, 396--445 (2003; Zbl 1044.14026)]. Schubert varieties; generic singularities; quasi-resolutions; linear groups; singular loci Cortez, Aurélie, Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire, Adv. Math., 178, 2, 396-445, (2003), MR 1994224 Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Generic singularities and quasi-resolutions of Schubert varieties for the lineary group. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The existence of a desingularization of quasi-excellent schemes as conjectured by \textit{A. Grothendieck} [Publ. Math., Inst. Hautes Étud. Sci. 20, 101--355 (1964; Zbl 0136.15901); ibid. 24, 1--231 (1965; Zbl 0135.39701); ibid. 28, 1--255 (1966; Zbl 0144.19904); ibid. 32, 1--361 (1967; Zbl 0153.22301)] was shown in the author's work [Adv. Math. 219, No. 2, 488--522 (2008; Zbl 1146.14009)] in 2008. Compared to the analogue for varieties, that result had the following disadvantages: Centers of the necessary blowups in the resolution procedure could be non-regular, and functoriality was not satisfied for the given construction.
The article under review is the first of two papers strengthening the results previously obtained: A desingularization is given by only blowing up regular centers and such that the resulting sequence of blowing ups gives an object which is functorial for regular morphisms. The case treated here is the non-embedded desingularization, whereas for the embedded one the author refers to his forthcoming paper [``Functorial desingularization over \(\mathbb{Q}\): boundaries and the embedded case'', \url{arXiv:0912.2570}].
Main result of the article is
Theorem 1.2.1: For any Noetherian quasi-excellent generically reduced scheme \(X=X_0\) over \(\text{Spec} (\mathbb{Q})\) there exists a blow-up sequence \({\mathcal F} (X): X_n \dashrightarrow X_0\) such that the following conditions are satisfied: {\parindent=8mm \begin{itemize}\item[(i)] the centers of the blowups are disjoint from the preimages of the regular locus \(X_{\mathrm{reg}}\); \item[(ii)] the centers of the blowups are regular; \item[(iii)] \(X_n\) is regular; \item[(iv)] the blow-up sequence \({\mathcal F} (X)\) is functorial with respect to all regular morphisms \(X' \to X\), in the sense that \({\mathcal F} (X')\) is obtained from \({\mathcal F} (X)\times_X X'\) by omitting all empty blowups.
\end{itemize}}
The Construction of \({\mathcal F} \) is done starting with any algorithm \({\mathcal F}_{\mathrm{Var}}\) giving desingularizations for varieties in characteristic 0 and which is functorial for regular morphisms in the sense of (i), (iii) and (iv). Furthermore, \({\mathcal F} \) will be found to satisfy the above condition (ii) if this is the case for the algorithm \({\mathcal F}_{\mathrm{Var}}\). This algorithm is extended to pairs \((X,Z)\) of quasi-excellent schemes \(X\) and Cartier divisors \(Z\) in \(X\) containing the singular locus and isomorphic to a disjoint union of varieties, such that \({\mathcal F}_{\mathrm{Var}} (X,Z) \) desingularizes \(X\). Now the formal completion \({\mathcal X} := \hat{X}_Z\) is algebraized by some \(X'\), and \({\mathcal F}_{\mathrm{Var}} (X')\) gives rise to desingularizations on \(\mathcal X\) (and on \(X\)). The main work remaining now is to show that \({\mathcal F}_{\mathrm{Var}} (\mathcal X) = \widehat{{\mathcal F}_{\mathrm{Var}} (X')}\) is canonically defined by \(X_n\), where \(X_n\subseteq \mathcal X\) is some sufficiently large nilpotent neighborhood of the closed fibre. Algebraization is done using the classical approximation results of \textit{R. Elkik} [Ann. Sci. Éc. Norm. Supér. (4) 6, 553--603 (1973; Zbl 0327.14001)].
From the author's abstract: ``As a main application, we deduce that any reduced formal variety of characteristic zero admits a strong functorial desingularization. Also, we show that as an easy formal consequence of our main result one obtains strong functorial desingularization for many other spaces of characteristic zero including quasi-excellent stacks, formal schemes, and complex or nonarchimedean analytic spaces. Moreover, these functors easily generalize to noncompact settings by use of generalized convergent blow-up sequences with regular centers.'' desingularization of quasi-excellent schemes; nonembedded desingularization; functorial desingularization Temkin, M., Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case, Duke Mathematical Journal, 161, 2207-2254, (2012) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Valuation rings Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article contributes to the discussion on resolution of singularities
in characteristic \(p>0\). Examples are given where after blowing ups centered
at points, certain singularities are transformed into a kind of cycle
leading to an infinite increase of the residual order.
Residual order
is a candidate for adapting the string of invariants originating
from the classical inductive characteristic-zero resolution procedure
to positive characteristic: The authors announce a new proof for
the embedded resolution of surfaces which makes use of it [\textit{H. Hauser} and \textit{S. Perlega}, ``A new proof for the embedded resolution of surface singularities'', manuscript (2018)].
In higher dimension the situation appears to be different.
Let \(f(z,x)=z^{p^e} + F(x_1,\dots ,x_n)\) be purely inseparable, \(e>0\),
\(F\) a formal power series having
\(\mathrm{ord} (F)\geq p^e\). A change of parameters \(z_1=z-g(x_1,\dots ,x_n)\)
allows to replace
\(f\) by \(z_1^{p^e} + g(x_1,\dots , x_n)^{p^e} + F(x_1,\dots ,x_n)\)
and thus to remove
any \(p^e\)-th powers in the expansion of \(F\). This procedure is refered to
as \textit{cleaning} of \(F\) and is supposed to be done now.
Let \(E\) be a strictly normal crossing divisor given as the product of \(x_i\),
\(i\in \Delta \) where \(\Delta \subseteq \{ 1, \dots ,n\}\).
Write \(F=\prod _{i\in \Delta} x_i^{r_i}\cdot G(x_1,\dots ,x_n)\) such that
\(r_i=\mathrm{ord}_{(x_i)} (F)\). Then the residual order is defined to be
\(\mathrm{residual.order}_E(f):= \mathrm{ord} (G)\).
The authors give examples for the increase of \(\mathrm{residual.order}_E(f)\)
after consecutive blowing up of points, where the shape of the
equation remains the same
(apart from the increase of exponents). Thus the residual order can attain
arbitrarily large values. Calculations are done explicitely for certain equations of the following type:
(1)
\(p=2\), \(\mathrm{ord} (f) = 8\) in dimension 5,
(2)
\(p\geq 3\), \(\mathrm{ord} (f) = p^3\) in dimension 4.
The examples are not intended to
disprove the existence of a resolution in positive
characteristic since admissible centers of the blowing ups may have
positive dimension as well. resolution of singularities in positive characteristic; residual order; examples for infinite increase of residual order Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Positive characteristic ground fields in algebraic geometry, Formal power series rings Cycles of singularities appearing in the resolution problem in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A toric variety is an algebraic variety over a field \(k\) containing an algebraic torus \(T_X\) whose natural action on itself extends to an action of \(T_X\) on \(X\). A morphism of toric varieties is equivariant if it is compatible with the action of the respective tori. In this paper, the authors give a constructive procedure to resolve the singularities of a toric embedded variety, in an equivariant way. More precisely, given a closed equivariant embedding \(X \subset M\) of toric varieties, with \(M\) smooth, and both defined over a perfect field \(k\), they show how to chose subvarieties \(D_0,\dots, D_t\), where \(D_0\) is a smooth subvariety of \(M_0=M\), \(D_1\) a smooth subvariety of \(M_1\), the blowing-up of \(M_0\) with center \(D_0\), and so on (thus we obtain a sequence of smooth varieties \(M_0, \ldots, M_{t+1}\), and blowing-up morphisms \(\pi _i:M_{i+1} \to M_i, i=0, \dots, t\)) such that, letting \(X_0 =X\) and \(X_{i+1} \subset M_{i+1}\) be the strict transform of \(X_i \subset M_i\) via \(\pi _i\), we have:
(a) \(M_i\) and \(X_i\) are toric, \(D_i \subset M_i\) is \(T_{M_i}\)-invariant and the inclusion \(X_i \subset M_i\) is equivariant,
(b) if \(C_i = X_i \cap D_i\), then \(C_i\) is \(T_{X_i}\)-invariant, \(X_i\) is normally flat along \(C_i\), and \(C_i \subset {\text{Sing}}(X_i)\) (provided \(X_i\) is not smooth),
(c) \(X_{t+1}\) is smooth and it has normal crossings with the exceptional divisors that appear.
A slight variation of the procedure allows one to get ``canonical equivariant embedded resolution'', i.e., the resolution procedure corresponding to an equivariant embedding \(X \subset M\) of toric varieties as above induces that for \(X'\subset M'\), if \(g:M' \to M\) is an open equivariant embedding and \(X'=g^{-1}(X)\) (not counting isomorphisms in the sequences of blowing-ups).
Toric varieties are locally defined by ``binomial equations'' of the form \({x_1^{\alpha _1} \ldots x_n^{\alpha _n}} - {x_1^{\beta _1} \ldots x_n^{\beta _n}}\) (where \(\min(\alpha _i, \beta _i)=0\), for all \(i\)). More generally, one may consider varieties locally defined by binomial equations as above, if they satisfy some mild additional conditions they are called binomial varieties (but some people use the expression in a slightly different sense). The authors show that their results on desingularization of embedded toric varieties extend, with minor changes, to embedded binomial varieties as well as to generalizations thereof: embedded toroidal and locally binomial varieties. The techniques involved in the main results are essentially similar to those used by the authors (and other mathematicians) in the theory of algorithmic, or canonical, resolution of singularities (specially, those of the authors' paper [Invent. Math. 128, 207--302 (1997; Zbl 0896.14006)] and \textit{J. Wlodarczyk} [J. Am. Math. Soc. 18, 779--822 (2005; Zbl 1084.14018)]). For instance, the main results are obtained from a seemingly more technical result, namely a theorem on the resolution of marked monomial ideals. These are 5-tuples \({\mathcal H} = (M,N,P,H,e)\), where \(P \subset N \subset M\) are smooth, toric varieties, \(H\) is a sum of sheaves of \({\mathcal O}_M\)-ideals, each one locally a product of suitable principal ideals, and \(e\) is a positive integer, satisfying certain conditions. The singular set of such a marked ideal is the set of points of \(P\) at which the order of \(H\) is \( \geq e\). But toric varieties (and their mentioned generalizations) have a strong combinatorial flavor. This is cleverly used by the authors to simplify and improve the presentation. For instance, in a crucial inductive step in the resolution process for marked ideals, one has to use (locally defined) maximal contact subvarieties. In general, these are available in characteristic zero only. But in the toric situation these can be obtained rather easily in any characteristic. (Still one needs the field to be perfect to obtain certain relevant isomorphisms of completions of rings). The authors include some interesting examples and propose some problems. For instance, to generalize the present results to the case of binomial varieties in the sense of \textit{D. Eisenbud} and \textit{B. Sturmfelds} [Duke Math. J. 84, 1--45 (1996; Zbl 0873.13021)]. desingularization; blowing-up; marked ideal; standard basis; Hilbert-Samuel function Bierstone, E., Milman, P.: Desingularization of toric and binomial varieties. J. Alg. Geom. 15, 443-486 (2006) Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Singularities in algebraic geometry, Birational geometry Desingularization of toric and binomial varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main purpose of this paper is to prove some results of \textit{Yu. G. Prokhorov} [in: Algebra, Proc. Int. Algebraic Conf., Moscow 1998, 301-317 (2000; Zbl 1003.14005)] for the non-\(\mathbb{Q}\)-factorial case, namely, the existence theorem for the pure log terminal blow-up (theorem 1.5) and the criterion for weakly exceptional singularity (theorem 2.1). These blow-ups allow us to apply the Shokurov inductive method [see \textit{V. V. Shokurov}, J. Math. Sci., New York 102, No. 2, 3876-3932 (2000; Zbl 1177.14078)] to the study of singularities, or more generally, to contractions of algebraic varieties. It reduces the problems of structure, completability, and exceptionality of a singularity to a single exceptional blow-up divisor. For \(\mathbb{Q}\)-factorial singularities, the pure log terminal blow-up is the only blow-up that allows us to extend the complement from the exceptional divisor to the entire variety in the general case (remark 1.3), and for non-\(\mathbb{Q}\)-factorial log terminal singularities, such blow-ups will differ from the pure log terminal blow-up by a small flop contraction (corollary 1.13). In the study of arbitrary \(\mathbb{Q}\)-Gorenstein singularities, it is practically impossible to separate the class of \(\mathbb{Q}\)-factorial singularities from the others. For this reason, one has to use theorems and constructions valid in the general case. The paper under review specifies the results of \textit{Yu. G. Prokhorov} (loc. cit.) about the inductive method for analysis of arbitrary log canonical singularities. log terminal blow-ups; flop С. А. Кудрявцев, ``О чисто логтерминальных раздутиях'', Матем. заметки, 69:6 (2001), 892 -- 898 Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), Rational and birational maps, Singularities in algebraic geometry Pure log terminal blow-ups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X_{\Sigma (\sigma)} = \text{Spec} \mathbb{C} [\check \sigma \cap \mathbb{Z}^ 3]\) be an affine toric variety given by the monoid algebra \(\mathbb{C} [\check \sigma \cap \mathbb{Z}^ 3]\), \(\check \sigma\) the negative dual cone of a lattice cone \(\sigma \subset \mathbb{R}^ 3\), \(\Sigma (\sigma)\) the fan consisting of the faces of \(\sigma\). The authors classify all pairs \(X_{\Sigma'}\), \(X_{\Sigma (\sigma)}\) which occur in minimal models of equivariant resolutions \(\Phi : X_{\Sigma'} \to X_{\Sigma (\sigma)}\), such that \(X_{\Sigma (\sigma)}\) has only quotient singularities and the regular toric variety \(X_{\Sigma'}\) has Picard number at most 3.
All the possible generators \((a_ 1, a_ 2, a_ 3)\) are given and all the corresponding fans \(\Sigma'\) are drawn. resolution of singularities; affine toric 3-varieties; dual cone; lattice cone; fan; quotient singularities Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects), \(3\)-folds, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Singularities in algebraic geometry, Low codimension problems in algebraic geometry On the resolution of singularities in affine toric 3-varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S\) be a locally noetherian scheme of characteristic \(p\) and \(\mathcal E\) be a non-degenerate \(F\)-crystal of rank \(r\) over \(S\). There is a stratification of \(S\) by reduced and locally closed subschemes \(\{U_\beta\}_{\beta\in{\mathcal B}}\), indexed by the set \(\mathcal B\) of the Newton polygons on \([0,r]\), with the property that a point \(s\) of \(S\) lies in the stratum \(U_\beta\) if the Newton polygon of \(\mathcal E\) at \(s\) is exactly \(\beta\). Let \(\overline{U}_\beta\) be the closure of a stratum and \(\eta\) a generic point of \(\overline{U}_\beta\setminus U_\beta\), the authors prove that \(\dim {\mathcal O}_{\overline{U}_\beta, \eta}=1\). If \(S=\text{Spec } A\) is the spectrum of a complete local noetherian ring with algebraically closed residue field, they prove that an isoclinic (all slopes of the Newton polygon are equal) \(F\)-crystal over \(S\) is isogenous to a constant \(F\)-crystal. This result implies an analogous isogeny theorem for \(p\)-divisible groups (cf. theorem~2.17). These results are used to describe all deformations of simple \(p\)-divisible groups which do not change the Newton polygon. The methods used in the proofs of the above results are also applied to prove a result about resolution of singularities of surfaces in characteristic \(p\). Precisely, the authors prove the following fact.
Let \(A\) be a local complete Noetherian ring, normal of dimension \(2\), with algebraically closed residue field \(k\) of characteristic \(p\) and with \(k\subset A\). Let \(S=\text{Spec } A\) and denote by \(0\) its closed point. Having chosen a resolution of singularities \(\pi:\widetilde S\to S\) [cf. \textit{J. Lipman}, Ann. Math., II. Ser. 107, 151-207 (1978; Zbl 0349.14004)] and identifying \(U=S\setminus\{0\}\) with \(\pi^{-1}(U)\), then the natural map \(H^1_{\text{ét}}(\widetilde S,{\mathbb{Q}}_p)\to H^1_{\text{ét}}(U,{\mathbb{Q}}_p)\) is an isomorphism. For a more detailed description of the results and their consequences, we refer to the introduction of the paper. \(F\)-crystals; Newton polygons; characteristic \(p\); stratification; isogeny theorem; \(p\)-divisible groups; deformations; resolution of singularities [13] A. J. de Jong & F. Oort, `` Purity of the stratification by Newton polygons {'', \(J. Amer. Math. Soc.\)13 (2000), no. 1, p. 209-241. &MR 17 | &Zbl 0954.} Global theory and resolution of singularities (algebro-geometric aspects), \(p\)-adic cohomology, crystalline cohomology, Formal groups, \(p\)-divisible groups, Toric varieties, Newton polyhedra, Okounkov bodies, Finite ground fields in algebraic geometry, Singularities in algebraic geometry Purity of the stratification by Newton polygons | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((C,0)\) be a reduced curve germ in a normal surface singularity \((X,0)\). The main goal is to recover the delta invariant \(\delta (C)\) of the abstract curve \((C,0)\) from the topology of the embedding \((C,0)\subset (X,0)\). We give explicit formulae whenever \((C,0)\) is \textit{minimal generic} and \((X,0)\) is rational (as a continuation of some previous papers by the authors).
Additionally, in this case, we prove that if \((X,0)\) is a quotient singularity, then \(\delta (C)\) only admits the values \(r-1\) or \(r\), where \(r\) is the number or irreducible components of \((C,0)\). (\(\delta(C)=r-1\) realizes the extremal lower bound, valid only for ``ordinary \(r\)-tuples''.) normal surface singularities; delta invariant of curves; Poincaré series; periodic constant; twisted duality; rational surface singularities; Weil divisors; Riemann-Roch Singularities in algebraic geometry, Complex singularities, Global theory and resolution of singularities (algebro-geometric aspects) Local invariants of minimal generic curves on rational surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities ``We study a condition of equisingularity in codimension 1 and characteristic \(p\neq 0.\) The definition is given in terms of equiresolution and coincides with Zariski's definition in characteristic 0.'' equisingularity; equiresolution Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Surfaces algébroïdes de type dimensionnel 1 (caractéristique \(p\neq 0)\). (Algebroid surfaces of dimensionality type 1 (characteristic \(p\neq 0))\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities As main results in this paper, the authors prove a conjecture of Veys on poles of maximal order of topological/motivic zeta functions and a related result concerning the structure, with respect to a natural weight function, of the Berkovich skeleton associated with a degeneration of Calabi-Yau varieties.
Let \(k\) be a field of characteristic zero, \(X\) a connected smooth \(k\)-variety and \(f\) a non-constant regular function on \(X\). The motivic zeta function, and its specialization the topological zeta function, at a point \(x\) of \(f=0\), is an important and rich singularity invariant of \(f\), as is clear from the works of Denef-Loeser and many others. It is obvious from formulae in terms of an embedded resolution of \(f=0\) that poles of these zeta functions have order at most \(n=\dim X\). The conjecture of Veys states that, if \(s_0\) is a pole of order \(n\), then \(-s_0\) is the log canonical threshold of \(f\) at \(x\). From geometric point of view, it is in fact a conjecture on the structure of the dual complex associated to an embedded resolution of \(f=0\), with respect to a natural weight function. The conjecture was only known for \(n=2\) and in the special case of Newton non-degenerate polynomials \(f\). Here the authors prove it in full generality, with as main method of proof a well chosen instance of the Minimal Model Program.
In fact, the main geometric ideas in that proof lead to the second main result. Let now \(X\) be a geometrically connected smooth projective \(k((t))\)-scheme with trivial canonical sheaf, and take a volume form \(\omega\) on \(X\). Let \(\mathcal X\) be a simple normal crossings model of \(X\) over \(k[[t]]\). Consider the Berkovich analytification \(X^{\mathrm{an}}\) of \(X\) with its natural weight function \(\mathrm{wt}_\omega\). The dual complex of the special fibre of \(\mathcal X\) admits a canonical embedding \(Sk(\mathcal X)\) in \(X^{\mathrm{an}}\) on the whole of \(X^{\mathrm{an}}\). (This minimal value corresponds in spirit with the log canonical threshold).
This result confirms the expectation that \(\mathrm{wt}_\omega\) induces some ``flow'' on \(X^{\mathrm{an}}\) in the direction of decreasing values of \(\mathrm{wt}_\omega\) that contracts \(X^{\mathrm{an}}\) to the subspace where \(\mathrm{wt}_\omega\) takes its minimal value. In this direction the authors also prove an additional statement concerning ``simultaneous collapses'' to that subspace, as well as an analogous statement on the dual complex in the first main result. motivic zeta functions; minimal model program; Berkovich spaces; degenerations Nicaise, J; Xu, C, Poles of maximal order of motivic zeta functions, Duke Math. J., 165, 217-243, (2016) Singularities in algebraic geometry, Minimal model program (Mori theory, extremal rays), Rigid analytic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Fibrations, degenerations in algebraic geometry Poles of maximal order of motivic zeta functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We establish characterizations of smoothness for a complex affine algebraic variety at a given Cohen-Macaulay point. Our main result treats the case of surfaces with (at most) rational singularities, by a technique that requires the vanishing of suitable Ext modules rather than of sheaf cohomology groups. We also prove results in arbitrary dimension which, in addition to smoothness, detect curves and surfaces as a global feature. rational surface singularity; non-singular point; Cohen-Macaulay point; module of differentials; vanishing of cohomology Local cohomology and algebraic geometry, Homological functors on modules of commutative rings (Tor, Ext, etc.), Regular local rings, Modules of differentials, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Special surfaces Cohomological characterizations of smoothness and the case of rational surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author gives a concise, complete proof of resolution of singularities of 3-folds in characteristic \(> 5\) using the same main steps as Abhyankar's original (508 pages long) proof of this theorem from the mid-1960s. More precisely, Cutkosky breaks the proof down into the following 5 steps to emphasize the overall structure:
1. proof of embedded resolution of surface singularities and principalization of ideals
2. proof of birational equivalence of a projective variety of dimension \(n\) to an appropriate normal one which does not contain points of multiplicity \(>n!\)
3. proof of local resolution of points of multiplicity not exceeding the characteristic of the ground field
4. patching of local resolutions to produce a nonsingular projective variety which is birationally equivalent to the original one
5. change of the resolution obtained in steps 1-4 to also satisfy the condition that the resolution should be an isomorphism outside the singular locus.
To allow the readers to become familiar with the main constructions and ideas before entering too deep into the technical details, very clear and compact outlines of the proofs of embedded resolution and of the constructions of steps 2-5 precede the complete proofs thereof, which makes the article accessible also to algebraic geometers who are not specialists in the desingularization. resolution of singularities; positive characteristic; threefolds; 3-fold; desingularization S. D.CUTKOSKY,\textit{Resolution of singularities for 3-folds in positive characteristic}, Amer. J. Math. 131 (2009), no. 1, 59--127.http://dx.doi.org/10.1353/ajm.0.0036.MR2488485 Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) Resolution of singularities for 3-folds in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let X be a complete normal algebraic 3-fold of general type defined over \({\mathbb{C}}\) and let \(K_ X\) be the canonical (Weil) divisor of X. Assume that: \((a)\quad X\quad has\) only canonical singularities (i.e. there exists a positive integer \(\rho\) such that \(\rho K_ X\) is a Cartier divisor and there is a proper birational morphism \(f:\quad X'\to X\) from a non-singular 3-fold inducing a natural homomorphism \(f^*{\mathcal O}_ X(\rho K_ X)\to {\mathcal O}_{X'}(\rho K_{X'}))\); and \((b)\quad K_ X\quad is\) numerically effective. The main result of the paper states that under the above assumptions the canonical ring R(X) of X is a finitely generated \({\mathbb{C}}\)-algebra, where, by definition, \(R(X)=R(X')=\otimes_{m\geq 0}H^ 0(X',{\mathcal O}_{X'},(mK_{X'})).\)\ The corresponding result for surfaces, defined over an arbitrary algebraically closed field goes back to \textit{D. Mumford} [Ann. Math., II. Ser. 76, 612-615 (1962); appendix to a paper of \textit{O. Zariski}, ibid. 560-615 (1962; Zbl 0124.370)]. Due to Hironaka and to results of Fujita the author reduces to work with the desingularization \(f:\quad X'\to X\) and to show that the stable base locus of \(f^*\rho K_ X\) is empty. This is achieved by proving the following general result. Let D be a Cartier divisor on X such that \(D^ 3>0\) and assume that both D and \(D- K_ X\) are numerically effective; then the complete linear system \(| mD|\) is base point-free for some positive integer m. Recent progress on the subject, including generalizations, can be found in the recent paper of the author in Complex analysis and algebraic geometry, Proc. Conf., Göttingen 1985, Lect. Notes Math. 1194, 41-55 (1986). numerically effective canonical divisor; algebraic 3-fold of general type; canonical singularities; canonical ring; desingularization Kawamata, Y, On the finiteness of generators of a pluricanonical ring for a \(3\)-fold of general type, Am. J. Math., 106, 1503-1512, (1984) \(3\)-folds, Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Singularities in algebraic geometry On the finiteness of generators of a pluricanonical ring for a 3-fold of general type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [The articles of this volume will not be indexed individually.]
Contents: \textit{T. Ohmoto}, Thom polynomials in symplectic geometry, pp. 1-6; \textit{T. Fukuda}, Topological triviality of real analytic singularities, pp. 7-11; \textit{S. Izumiya}, Completely integrable first order partial differential equations, pp. 12-31; \textit{T. Tomaru}, Minimality of two-dimensional Gorenstein singularities of fundamental genus \(\geq 2\) and Yau sequences (Japanese), pp. 32-42; \textit{M. Tomari}, A characterization of log terminal normal graded rings in terms of Pinkham- Demazure's construction, pp. 43-56; \textit{K. Watanabe}, On the cyclic covering of a ``rational singularity'' (in the case of characteristic \(p > 0)\) (Japanese), pp. 57-70; \textit{F. Sakai}, On the irregularity of cyclic coverings of the projective plane (preliminary version), pp. 71- 80; \textit{F. Sakai}, Alexander polynomials of hypersurfaces (after Dimca) (Japanese), pp. 81-92; \textit{O. Riemenschneider}, Special surface singularities: a survey on the geometry and combinatorics of their deformations, pp. 93-118; \textit{T. Urabe}, On the global theory of singularities, pp. 119-125; \textit{S. Ogata}, On signature defects of cusps (Japanese), pp. 126-130; \textit{M. N. Ishida}, The duality of Tsuchihashi cusp singularities, pp. 131-140; \textit{K. Watanabe}, Three-dimensional hypersurface purely elliptic singularities of \((0,1)\)-type, pp. 141-153; \textit{H. Tsuchihashi}, On the volumes of integral convex polytopes satisfying certain conditions, pp. 154-162; \textit{N. Sasakura}, \textit{Y. Enta} and \textit{M. Kagesawa}, Rank two reflexive sheaves which are constructed from the prime field \(\mathbb{F}_p\), pp. 163-187; \textit{S. Ishii}, A Fano three-fold with the one-dimensional locus of nonrational singularities, pp. 188-197; \textit{S. Yokura}, A generalization of Deligne- Grothendieck-MacPherson's natural transformation \(C_*\) and a conjecture, pp. 198-207; \textit{T. Ohsawa}, On the \(L^2\) cohomology of complex spaces. II., pp. 208-225. Proceedings; Kyoto (Japan); Symposium; Analytic varieties; Singularities Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Complex singularities, Proceedings of conferences of miscellaneous specific interest, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Analytic varieties and singularities. Proceedings of a symposium held at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto, March 23-26, 1992 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0744.00034.]
This paper gives conditions under which a 1-parameter deformation \(\pi:X\to D\) of a normal isolated singularity has a simultaneous canonical model. As it is yet unknown (in dimension at least four) if the total space \(X\) has a canonical model, the following condition, which implies the existence, is always assumed: (FG) the graded ring \(\bigoplus_{m\geq 0}f_ *\omega_{\widetilde X}^ m\) is a finitely generated \({\mathcal O}_ X\)-algebra for some resolution \({f: \widetilde X\to X}\).
For a singularity \(V\) the plurigenus \(\gamma_ m(V)\) is defined as \(\dim\omega_ V^{[m]}/f_ *\omega_ Y^{[m]}\), where \(f:Y\to V\) is a partial resolution with at most canonical singularities. The author investigates the behaviour of \(\gamma_ m(X_ t)\), and she proves that \(\pi:X\to D\) admits a simultaneous canonical model if \(\gamma_ m(X_ t)=e_ tm^ n+O(m^{n-1})\) with \(e_ t\) constant. In the Gorenstein case the existence of a simultaneous canonical model is equivalent to \(\gamma_ m(X_ t)\) constant for all \(m\).
The theory is applied to purely elliptic Gorenstein singularities. It is shown that an (FG)-family \(\pi:X\to D\) of \(n\)-dimensional purely elliptic singularities of type \((0,n-1)\) admits a simultaneous canonical model. deformation of a normal isolated singularity; simultaneous canonical model; plurigenus; purely elliptic Gorenstein singularities Ishii, S.: Simultaneous canonical models of deformations of isolated singularities, Algebraic geometry and analytic geometry (Tokyo, 1990), ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991, pp. 81--100 Global theory and resolution of singularities (algebro-geometric aspects), Formal methods and deformations in algebraic geometry, Local complex singularities, Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) Simultaneous canonical models of deformations of isolated singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author studies the ramification sets of finite analytic mappings and applications of his results and methods to punctual Hilbert schemes and to finite analytic maps. He uses essentially the technique of ``chaining'' which consists in associating to a finite map a sequence (or chain) of sets, which are components of ramification loci of increasing multiplicity, and then in controlling their dimensions. For that purpose he uses a theorem of Grothendieck about the order of connexity of subgerms of an irreducible analytic germ, and also in the projective case the Fulton-Hansen theorem and a theorem of Deligne.
In {\S} 1, the author introduces three different notions of multiplicity. The topological multiplicity, the stable multiplicity and the algebraic one. In {\S} 2, he gives a fairly general lower bound to the dimension of the ramification set \(T^{d+1}(f)\), the set of points at which the multiplicity is at least \(d+1\). For the topological case he needs a hypothesis about f, called weak multitransversality which guarantees the additivity of multiplicity under deformation. This theorem is proved by a complicated induction involving multiproducts of ramification sets and the theorem of Grothendieck.
In {\S} 3, the author gives applications of {\S} 2, and of the chaining technique to the punctual Hilbert scheme \(Hilb'{\mathcal O}_{X,x}\) which parametrizes in \(Hilb'(X)\) the punctual schemes concentrated at \(x\in X\). The idea consists in identifying the germ of \(Hilb'({\mathcal O}_{X,x})\) at a smoothable element z with the ramification loci an appropriate map obtained by unfolding the equation of z. He thus obtains a lower bound for the local dimension at z of the open set U of smoothable points in X. This bound is (n-1)(\(\ell -1)\) with \(n=\dim (X)\) in the easiest case (X everywhere irreducible). Various, and more complicated results are obtained when we drop the irreducibility hypothesis or consider instead of U the open set of weakly smoothable (i.e. smoothable in a smooth ambient space) element.
Finally in {\S} 4, the author proves similar results for a finite projective morphism \(f:\quad X^ n\to P^ p.\) He generalizes a previous joint result of himself with Lazarsfeld (case \(n=p)\). This consists again in giving cases of non-emptiness for \(T^{d+1}(f)\) under some complicated numerical conditions. ramification sets of finite analytic mappings; punctual Hilbert schemes; ramification loci of increasing multiplicity T. Gaffney, ''Multiple points, chaining and Hilbert schemes,'' Amer. J. Math., vol. 110, iss. 4, pp. 595-628, 1988. Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry Multiple points, chaining and Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let f(x,y,z,t) be an analytic function such that for each t fixed, \(f=0\) defines a surface with isolated singularity at the origin with the same Milnor number \((=equi\sin gular)\). The author gives an example of such a family with the following property: The generic projection is not equisingular but there is a transverse projection which gives an equisingular family. Milnor number; equisingular family Luengo, I.: An example concerning a question of Zariski. Bull. soc. Math. France 113, No. 4, 379-386 (1985) Global theory and resolution of singularities (algebro-geometric aspects), Local complex singularities, Singularities in algebraic geometry An example concerning a question of Zariski | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(W\) be a finite dimensional linear representation of a reductive algebraic group \(G\) over an algebraically closed field \(k\) of characteristic 0. Let \(\mathcal{H}\) denote the invariant Hilbert scheme \(\mathrm{Hilb}^G_{h_W}(W)\) parametrizing \(G\)-stable closed subschemes \(Z\) of \(W\) with \(h_W\) being the Hilbert function of the general fiber of the (categorical) quotient morphism \(\nu : W \rightarrow W/\!\!/G = \mathrm{Spec}(k[W]^G)\). The article under review addresses the following question: in which cases is the Hilbert-Chow morphism from \(\gamma : \mathcal{H} \rightarrow W/\!\!/G\), possibly restricted to the main component, a desingularization of \(W/\!\!/G\)?
This question was studied before only for finite groups \(G\), in which case the \(G\)-Hilbert scheme of \textit{Y. Ito} and \textit{I. Nakamura} [Proc. Japan Acad., Ser. A 72, No. 7, 135--138 (1996; Zbl 0881.14002)] coincides with the main component of~\(\mathcal{H}\). They gave a positive answer for finite groups of \(\mathrm{SL}(2)\), then \textit{T. Bridgeland} et al. [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] for finite subgroups of \(\mathrm{SL}(3)\), and \textit{M. Lehn} and \textit{C. Sorger} [in: Geometric methods in representation theory. II. Selected papers based on the presentations at the summer school, Grenoble, France, June 16 -- July 4, 2008. Paris: Société Mathématique de France. 429--435 (2012; Zbl 1312.14007)] for a single 4-dimensional symplectic group.
The article under review gives first results in the case of infinite group~\(G\). The author considers four classical groups (SL, O, Sp, GL) with chosen series of natural representations. The main theorem states that in cases which are small enough (that is, they satisfy certain bounds on parameters of chosen representations), \(\mathcal{H}\) is a desingularization of \(W/\!\!/G\). The proof is based on a reduction principle, which allows to obtain information on all cases from the description of certain small ones. Two of four series of representations are analysed in the article, the details for remaining two can be found in the author's PhD thesis. algebraic group; quotient; desingularization; Hilbert scheme R. Terpereau, \textit{Invariant Hilbert schemes and desingularizations of quotients by classical groups}, Transform. Groups \textbf{19} (2014), no. 1, 247-281. Algebraic cycles, Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Invariant Hilbert schemes and desingularizations of quotients by classical groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The residue formula for rational differential 1-forms defined on a compact complex algebraic curve is one of fundamental results in the classical analytic and algebraic geometry [\textit{J.-P. Serre}, Algebraic groups and class fields. Graduate Texts in Mathematics, 117, Springer-Verlag, (1988; Zbl 0703.14001)]. In the multidimensional case it is well-known many variants of this formula in various situations [\textit{E. Kunz}, Math. Z. 152, 165--189 (1977; Zbl 0342.14022); \textit{Ph. Griffiths} and \textit{J. Harris}, Principles of algebraic geometry. Wiley Classics Library. New York, NY: John Wiley \& Sons Ltd., (1994; Zbl 0836.14001)], etc. The so-called Parshin-Lomadze Reciprocity Law is a far reaching generalization of the classical residue formula in a highly general algebraic context. The aim of the author is to describe a geometric proof of the latter result for meromorphic differential \(n\)-forms given on an \(n\)-dimensional complex algebraic variety. In fact, he modifies an approach by \textit{J.-L. Brylinski} and \textit{D. A. McLaughlin} [J. Reine Angew. Math. 481, 125--147 (1996; Zbl 0857.11062); Math. Res. Lett. 3, No.1, 19--30 (1996; Zbl 0866.19002)] based on the following idea: the residue of such \(n\)-form is represented by an integral over a smooth cycle corresponding to a certain class of the \(n\)-dimensional local homology group of the flag associated with a smooth stratification of the variety. Making use of resolution of singularities and elementary properties of the standard blowing-up process, the author reduces the general case of stratification with singular strata to the smooth case considered in [loc. cit.]. The note under review contains a preliminary version of the complete proof in the complex algebraic case; all details will appear somewhere later. residue formula; reciprocity laws; local homology groups; stratified spaces; Whitney stratification; Leray coboundary operators; resolution of singularities; blowing-ups M. Mazin, Geometric theory of Parshin's residues, Mathematical Reports of the Canadian Academy of Science, 2010. Singularities in algebraic geometry, Residues for several complex variables, Global theory and resolution of singularities (algebro-geometric aspects), Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) Geometric theory of Parshin residues | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A new proof of Hironaka's theorem on resolution of singularities is given. There are already several different constructive approaches [cf. \textit{E. Bierstone} and \textit{P. Milman}, Invent. Math. 128, 207--302 (1997; Zbl 0896.14006) or \textit{O. Villamayor}, Ann. Sci. Éc. Norm. Supér., IV. Sér 22, 1--32 (1989; Zbl 0675.14003)]. The resolution process is based on the choice of an invariant which measures the singularities and drops under blowing up the maximal stratum of this invariant. The choice of the invariant and the way to compute it makes the difference between the approaches to resolve singularities. The definition of the invariant is quite involved. It is defined inductively using the knowledge of the resolution process up to this moment. The induction defining the invariant is given by the intersection of the variety with a so-called hypersurface of maximal contact. The choice of this hypersurface is not canonical and some effort is needed to define the invariant in a canonical way.
The approach of this paper is similar to the approach of Bierstone and Milman but based additionally on two observations. The resolution process defined as a sequence of blowing-ups the ambient spaces can be applied simultaneously to a class of equivalent singularities obtained by simple modifications, i.e. to resolve a singularity it is allowed to tune it before starting: In the equivalence class a convenient representative given by a so-called homogenized ideal is chosen. The restrictions of homogenized ideals to different hypersurfaces of maximal contact define locally analytically isomorphic singularities. resolution of singularities; algorithmic resolution Włodarczyk, Jarosław, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc., 18, 4, 779-822, (2005) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Local complex singularities, Modifications; resolution of singularities (complex-analytic aspects) Simple Hironaka resolution in characteristic zero | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Development of higher dimensional algebraic geometry in the last ten years frequently led to consider pairs \((X,D)\) where \(X\) is an algebraic variety, \(D\) is a \(\mathbb{Q}\)-linear combination of divisors and both \(X\) and \(D\) may be singular. The class of all such pairs is currently called the log category. The paper under review is a nicely written survey on the major results concerning the log category and their applications, including new results as well as many new simpler proofs of old results. It can be read by algebraic geometers that are non-familiar with techniques of higher dimensional geometry.
After presenting generalizations of the Kodaira vanishing theorem (section 2) the basic definitions concerning the log category appear in section 3, including discrepancy, a measure of how singular a pair \((X,D)\) is. Sections 4 and 5 are devoted, respectively, to Bertini-type theorems about singularities of generic members of linear systems and to studying linear systems \(K_X+L\), \(K_X\) the canonical system and \(L\) ample. Section 6 deals with the construction of divisors in a fixed numerical equivalence class, which are rather singular at a point \(x\) but not too singular near \(x\). In section 7 singularities of a pair \((X,D)\) are compared to those of \((H,D\mid H)\), \(H \subset X\) a hypersurface (``inversion of adjunction''). Section 8 to 10 introduce the log canonical threshold, a new measure of the singularities of \((X,D)\), suitable for the case in which no information is given by discrepancy, and relate it to previously know invariants. Finally, section 11 presents a new proof of the rationality of canonical singularities avoiding the use of Grothendieck's general duality. log category; discrepancy; log canonical threshold; rationality of canonical singularities J. Kollár, Singularities of pairs, Algebraic geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math. 62, American Mathematical Society, 1997, p. 221-287 Singularities in algebraic geometry, Divisors, linear systems, invertible sheaves, Global theory and resolution of singularities (algebro-geometric aspects), \(n\)-folds (\(n>4\)), Singularities of surfaces or higher-dimensional varieties Singularities of pairs | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities On the most useful connections between algebra and geometry is given by the relationship between the Grothendieck group of vector bundles, \(K_ 0(X)\), and the Chow ring, \(CH^{\bullet}(X)\), of a smooth quasi- projective variety X. Here we will recall some of the aspects of this relationship, and describe a method for extending these ideas to the realm of singular varieties. Grothendieck group; Chow ring; singular varieties (Equivariant) Chow groups and rings; motives, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Grothendieck groups (category-theoretic aspects) The Chow ring of a singular variety | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For an arbitrary scheme \(W\), the \(m\)-th jet scheme \(W_m\) parametrizes morphisms \(\text{Spec} \mathbb{C} [t]/(t^{m+1})\to W\). Main result:
If \(X\) is a smooth variety, \(Y\subset X\) a closed subscheme, and \(q>0\) a rational number, then:
(1) The pair \((X,q\cdot Y)\) is log canonical if and only if \(\dim Y_m\leq (m+1) (\dim X-q)\), for all \(m\).
(2) The pair \((X,q\cdot Y)\) is Kawamata log terminal if and only if \(\dim Y_m< (m+1)(\dim X-q)\), for all \(m\).
The main technique we use in the proof of this result is motivic integration, a technique due to Kontsevich, Batyrev, and Denef and Loeser. As a consequence of the above result, we obtain a formula for the log canonical threshold:
Corollary. If \(X\) is a smooth variety and \(Y\subset X\) is a closed subscheme, then the log canonical threshold of the pair \((X,Y)\) is given by \(c(X,Y)= \dim X-\sup_{m\geq 0} {\dim Y_m\over m+1}\).
We apply this corollary to give simpler proofs of some results on the log canonical threshold proved by \textit{J.-P. Demailly} and \textit{J. Kollár} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 34, No. 4, 525-556 (2001; Zbl 0994.32021)] using analytic techniques. jet schemes; log canonical threshold; motivic integration; Kawamata log terminal M. Mustaţǎ, Singularities of pairs via jet schemes, \textit{J. Amer. Math. Soc.} 15 no. 3 (2002) 599-615 (electronic). Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Minimal model program (Mori theory, extremal rays) Singularities of pairs via jet schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Review of [Zbl 1273.14004; Zbl 1277.14002]. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, Springer, 1977. External book reviews, Foundations of algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities in algebraic geometry, Curves in algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Varieties and morphisms, Divisors, linear systems, invertible sheaves, Rational and birational maps, Schemes and morphisms, Parametrization (Chow and Hilbert schemes), Transcendental methods, Hodge theory (algebro-geometric aspects), Kähler manifolds, Uniformization of complex manifolds, Complex manifolds Book review of: I. R. Shafarevich, Basic algebraic geometry. Volume 1. Varieties in projective space, and Volume 2. Schemes and complex manifolds. Transl. from the Russian by Miles Reid. Third edition. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article presents some interesting results on crepant resolutions of singularities of complex algebraic varieties. The point of view is not the ``classical'' one, where a \textit{crepant resolution} of a variety \(X\) is a proper birational morphism \(\pi: {\tilde X} \to X\), with \(\tilde X\) smooth, such that \({\pi}^{\star}(\omega _X) = \omega _{\tilde X}\), but rather the categorical (or abstract) one. In the classical context, often a variety does not admit a crepant resolution, but in the abstract one better results are expected.
The precise definition of a categorical resolution of a variety \(X\) is rather technical, but basically it consists of a certain category \({\mathcal I}\) together with a functor \(\pi_{\star}: \mathcal I \to D(X)\) (the target is the derived category of quasi coherent sheaves on \(X\)), satisfying certain properties. Sometimes, when the functor is clear from the context, it is not specified. If a further condition (involving the bounded subcategory \(D^b (X)\) of \(D(X)\)) is satisfied, the categorical resolution is said to be \textit{strongly crepant}. (There is a variation of this notion, namely \textit{weakly crepant resolution}). It is known that a morphism \(\pi :\tilde X \to X\) is a crepant resolution of singularities if and only if the induced morphism \({\mathbf R}{\pi}_{\star}:D(\tilde X) \to D^b(X)\) is a strongly crepant resolution of singularities. But in general, categorical crepant resolutions do not ``come'' from classical, or geometric, ones.
The main theorems proved in the paper are:
(1) Let \(V\) be a smooth quasi-projective variety and \(G\) a finite subgroup of \(\mathrm{Aut}(V)\), such that the dualizing sheaf of \(V\) is \(G\)-equivariantly locally trivial. Then \(D_G(V)\) (the derived category of \(G\)-equivariant quasi-coherent sheaves on \(V\)) is a categorical strongly crepant resolution of \(X=V/G\).
(2) Let \(X\) be a quasi projective variety with normal Gorenstein quotient singularities, \(\mathcal X\) a smooth Deligne-Mumford stack whose coarse moduli space is \(X\), and whose dualizing bundle is the pull-back of that of \(X\). Then \(D(\mathcal X)\) is a categorical crepant resolution of \(X\).
Theorem (1) was known, but here a more ``elementary'' proof of it is given. Actually, more is proven about (1) or (2). Namely, the existence of a sheaf of algebras which allows the author to show that a non commutative crepant resolution of \(X\) (in the sense of \textit{M. Van den Bergh} [Duke Math. J. 122, No. 3, 423--455 (2004; Zbl 1074.14013)] is available.
Other results are also discussed, indicating a connection of this work with certain cases of the categorical McKay correspondence. categorical crepant resolution; derived category; stack; quotient singularity; categorical McKay correspondence Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Generalizations (algebraic spaces, stacks), Group actions on varieties or schemes (quotients), Noncommutative algebraic geometry, Derived categories, triangulated categories Categorical crepant resolutions for quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A particular case in the superstring theory where a finite group \(G\) acts upon the target Calabi-Yau manifold \(M\) in the theory seems to attract both physicists' and mathematician's attention. Define the ``orbifold Euler characteristic'': \(\chi (M,G)= {1\over |G |} \sum_{gh=hg} \chi (M^{\langle g, h\rangle})\), where the summation runs over all the pairs \(g,h\) of commuting elements of \(G\), and \(M^{\langle g,h \rangle}\) denotes the subset of \(M\) of all the points fixed by both of \(g\) and \(h\).
Vafa's formula-conjecture. If a complex manifold \(M\) has trivial canonical bundle and if \(M/G\) has a (nonsingular) resolution of singularities \(\widetilde {M/G}\) with trivial canonical bundle, then we have \(\chi (\widetilde {M/G} ) = \chi (M,G)\).
In the special case where \(M= \mathbb{A}^n\) an \(n\)-dimensional affine space, \(\chi (M,G)\) turns out to be the number of conjugacy classes, or equivalently the number of equivalence classes of irreducible \(G\)-modules. If \(n=2\), then the formula is therefore a corollary to the classical McKay correspondence.
Let \(G\) be a finite subgroup of \(SL(2, \mathbb{C})\) and \(\text{Irr} (G)\) the set of all equivalence classes of nontrivial irreducible \(G\)-modules. Let \(X=X_G: =\text{Hilb}^G (\mathbb{A}^2)\), \(S=S_G: =\mathbb{A}^2/G\), \({\mathfrak m}\) (resp. \({\mathfrak m}_S)\) the maximal ideal of \(X\) (resp. \(S)\) at the origin and \({\mathfrak n}: ={\mathfrak m}_S {\mathcal O}_{\mathbb{A}^2}\). Let \(\pi: X\to S\) be the natural morphism and \(E\) the exceptional set of \(\pi\). Let \(\text{Irr} (E)\) be the set of irreducible components of \(E\). Any \(I\in X\) contained in \(E\) is a \(G\)-invariant ideal of \({\mathcal O}_{\mathbb{A}^2}\) which contains \({\mathfrak n}\).
Definition: \(V(I): =I/({\mathfrak m} I+{\mathfrak n})\).
For any \(\rho\), \(\rho'\), and \(\rho''\in \text{Irr} (G)\) define \(E(\rho): =\{I\in \text{Hilb}^G (\mathbb{A}^2)\); \(V(I)\) contains a \(G\)-module \(V(\rho)\}\)
\(P(\rho, \rho'): =\{I\in \text{Hilb}^G (\mathbb{A}^2)\); \(V(I)\) contains a \(G\)-module \(V(\rho) \oplus V(\rho')\}\)
\(Q(\rho, \rho', \rho''): =\{I\in \text{Hilb}^G (\mathbb{A}^2)\); \(V(I)\) contains a \(G\)-module \(V(\rho) \oplus V(\rho') \oplus V(\rho'')\}\).
Main theorem: (1) The map \(\rho \mapsto E(\rho)\) is a bijective correspondence between \(\text{Irr} (G)\) and \(\text{Irr} (E)\).
(2) \(E(\rho)\) is a smooth rational curve for any \(\rho\in \text{Irr} (G)\).
(3) \(P(\rho, \rho)= Q(\rho, \rho',\rho'') = \emptyset\) for any \(\rho,\rho', \rho''\in \text{Irr} (G)\). Hilbert schemes; orbifold Euler characteristics; irreducible components of exceptional set; superstring theory; McKay correspondence Ito, Y., Nakamura, I.: McKay correspondence and Hilbert schemes. Proc. Japan Acad. Ser. A Math. Sci., 72, 135--138 (1996) Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, Global theory and resolution of singularities (algebro-geometric aspects) McKay correspondence and Hilbert schemes | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The purpose of this article is to give a short proof for the González- Sprinberg-Verdier formula [cf. \textit{G. Gonzalez-Sprinberg}, Astérisque 82-83, 7-32 (1981; Zbl 0482.14033)] for the local Euler obstruction of an equidimensional analytic set. As an application the author computes the local Euler obstruction of a \(surface\quad X\) in \(P_ 3({\mathbb{C}})\) at a \(point\quad o\) in terms of the section of X by a generic plane and of the section of X by a generic plane passing through o. González-Sprinberg-Verdier formula; Euler obstruction Sebastiani ( M. ) .- Sur la formule de Gonzalez-Verdier , Bull. Braz. Math. Soc. 16 ( 1985 ), no. 1, p. 31 - 44 . MR 819804 | Zbl 0628.14008 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Obstruction theory in algebraic topology Sur la formule de Gonzalez-Verdier. (On the Gonzalez-Verdier formula) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors give an explicit elementary proof of a local version of resolution of singularities in characteristic zero. ``Local'' means that the centres of blowing up are chosen locally, so that a finite number of finite sequences of local blowings-up may be required to cover a neighbourhood of a given point.
Those investigators who want to start the theory of (resolution of) singularities, and who want to find the next materials, shall look over this paper. I think that this paper feast their eyes upon. initial exponents; Hilbert-Samuel function; uniformization; resolution of singularities; blowings-up Bierstone, E.; Milman, P., \textit{uniformization of analytic spaces}, J. Amer. Math. Soc., 2, 801-836, (1989) Modifications; resolution of singularities (complex-analytic aspects), Local complex singularities, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Uniformization of analytic spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite subgroup of \(\text{SL}(n,\mathbb{C})\). \(G\) acts on \(\mathbb{C}^n\) freely outside a finite collection of linear subspaces of codimension \(\geq 2\). The \(G\)-Hilbert scheme \(\text{Hilb}^G(\mathbb{C}^n)\) parameterizing \(G\)-clusters on \(\mathbb{C}^n\) has been introduced by \textit{I. Nakamura} [J. Algebr. Geom. 10, No.4, 757--779 (2001; Zbl 1104.14003)] as a natural candidate to provide crepant resolutions of the quotient singularity \(\mathbb{C}^n/G\).\newline For \(n=2\), \(\text{Hilb}^G(\mathbb{C}^2)\) is the minimal resolution of \(\mathbb{C}^2/G\) [\textit{Y. Ito} and \textit{I. Nakamura}, Proc. Japan Acad., Ser. A 72, No.7, 135--138 (1996; Zbl 0881.14002)] and for \(n=3\), by the theorem of \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)], \(\text{Hilb}^G(\mathbb{C}^3)\) is smooth and is a crepant resolution of \(\mathbb{C}^3/G\). But for \(n\geq 4\), \(\text{Hilb}^G(\mathbb{C}^n)\) is not always smooth and the quotient \(\mathbb{C}^n/G\) might have no crepant resolution at all.\newline This paper is mostly concerned with the family of examples given by abelian groups:
\[
A_r(4)=\{g\in\text{SL}(4,\mathbb{C})\,| \,g \text{ diagonal}, g^{r+1}=1\},\quad r\geq 1.
\]
The main result is:
\(\text{Hilb}^{A_r(4)}(\mathbb{C}^4)\) is a smooth toric variety with canonical bundle \(\omega={\mathcal O}_{\text{Hilb}^{A_r(4)}(\mathbb{C}^4)}(\sum\limits_{k=1}^mE_k)\) with \(m=r(r+1)(r+2)/6\), where \(E_k\)'s are disjoint smooth exceptional divisors isomorphic to \(\mathbb{P}_1\times\mathbb{P}_1\times\mathbb{P}_1\). Blowing down \(E_k\) to some factors \(\mathbb{P}_1\times\mathbb{P}_1\) for each \(k\), it gives rise to crepant resolutions of \(\mathbb{C}^4/A_{r}(4)\), all of them differ by a sequence of flops of \(4\)-folds.
The result is proved in Section 4 (Theorem 4.1) by a deep study of the toric structure of \(\text{Hilb}^{A_r(4)}(\mathbb{C}^4)\) and Gröbner basis techniques. The easier special case \(r=1\) is treated in Section 3 (Theorem 3.5).\newline In Section 5, the authors compute a non-abelian case. Consider the alternating group \(\mathcal{A}_4\) acting by permutation on \(\mathbb{C}^4\) and restricted to \(\mathbb{C}^3\) considered as the standard representation. They apply their method to give a constructive proof of the known smooth and crepant structure of \(\text{Hilb}^{{\mathcal A}_4}(\mathbb{C}^3)\). The method could be further developed to investigate new higher-dimensional cases. Hilbert scheme of orbits; quotient singularities; 4-folds, toric geometry Parametrization (Chow and Hilbert schemes), \(4\)-folds, Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies, Representations of finite symmetric groups, Global theory and resolution of singularities (algebro-geometric aspects) On hypersurface quotient singularities of dimension \(4\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors propose two alternative invariants for prove the resolution of singularities of a singular surface in \(\mathbb A^3\) defined over an algebraically closed field of characteristic \(p >0\) by:
\[
G(x,y,z)=x^p+F(y,z),
\]
where \(F\) is a polynomial of order greater than \(p\) at \(0\). The results were known since \textit{S. S. Abhyankar} [Ann. Math. (2) 63, 491--526 (1956; Zbl 0108.16803)] but the proof here uses the classical invariants from characteristic zero. But this invariants increase at kangaroo points, so they modify with a ``bonus'' such that they drop. The proof consists in monomialize \(F\) without power of \(p\) and apply a combinatorial game to decrease the order of \(G\). To monomialize \(F\) they study its image under three kinds of blowups: \((y,z)\mapsto (yz+tz,z)\): the translational move, \((yz)\mapsto (yz,z)\): the horizontal move and \((y,z)\mapsto (y,yz)\): the vertical move; and they look the effect on the Newton polygon and their invariants to define the ``bonus''. purely inseparable surfaces; resolution of singularities; positive characteristic; blowups; embedded resolution; Newton polygon Hauser, H; Wagner, D, Alternative invariants for the embedded resolution of purely inseparable surface singularities, L'Enseign Math., 60, 177-224, (2014) Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Alternative invariants for the embedded resolution of purely inseparable surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A \textit{Rees algebra} over a ring \(B\) is a finitely generated graded \(B\)-subalgebra of the polynomial ring \(B[W]\). This concept globalizes to give the notion of Rees algebra (or \textit{Rees sheaf of algebras}) over a base scheme \(V\), say \(\mathcal G \subset \mathcal O _V[W]\). \textit{O. Villamayor} [Ann. Sci. Éc. Norm. Supér. (4) 22, No. 1, 1--32 (1989; Zbl 0675.14003); Ann. Sci. Éc. Norm. Supér. (4) 25, No. 6, 629--677 (1992; Zbl 0782.14009)] introduced the concept several years ago, with the intention to apply it to the theory of resolution of singularities of algebraic varieties. For this purpose, the most interesting situation is that where \(B\) is a regular, finitely generated algebra over a perfect field \(k\), or \(V\) is a smooth algebraic variety over such \(k\). Henceforth, we suppose these assumptions valid.
In Part I, the authors review the basic theory and present some new results. Among other things, given a Rees algebra \(\mathcal G \subset {\mathcal O}_V[W]\), with \(V\) a smooth variety over a perfect field \(k\), they recall the notions of order of \(\mathcal G\) at \(x \in V\), zero set and singular locus of \(\mathcal G\), and integral closure \(\overline {\mathcal G}\) of \(\mathcal G\) in \({\mathcal O}_V[W]\). They define \textit{differential Rees algebra}, i.e., one closed under the action of differential operators, and recall the construction of the minimal differential Rees containing a given \(\mathcal G\), denoted by \({\mathbb D}(\mathcal G)\). They also define the concept of \textit{weakly equivalent} (w.e.) Rees algebras: essentially, Rees algebras (over \(V\)) \(\mathcal G\) and \(\mathcal K\) are w.e. if Sing(\(\mathcal G\)) = Sing(\(\mathcal K\)) and this equality is preserved when we take, successively, suitable ``transforms'', in a sense explained in the text. They relate Rees algebras to \textit{pairs}, that is ordered couples \((I,b)\) where \(I\) is a coherent sheaf of ideals of \(\mathcal O _V\) and \(b \geq 0\) an integer. One may associate to a pair \((I,b)\) a Rees algebra \({\mathcal O}_V[IW^b]\), and any Rees algebra is closely related to one of this type.
Their main new result (Theorem 3.11, called the \textit{canonicity principle}) says that Rees algebras (both over \(V\)) \(\mathcal G\) and \(\mathcal K\) are w.e. if and only if \(\overline{{\mathbb D}(\mathcal G)}=\overline{{\mathbb D}(\mathcal K)}\). Theorem 3.11 is a an immediate consequence of a more general result (Theorem 3.10), expressed in terms of certain inclusions, whose proof is rather involved.
In Part II they discuss some applications of Theorem 3.11. It has been known for a long time (thanks to Hironaka's efforts) that resolution of singularities of an algebraic variety \(X\) (say, embedded in a smooth \(V\)) follows if we can resolve, in a suitable sense, certain pair \((I,b)\) associated to \(X\). But this is a local process: \(I\) is not defined on the whole \(V\), but on an étale neighborhood of a point \(x \in V\). Moreover, the pair \((I,b)\) is not unique. To verify that this process globalizes has been traditionally a hard gluing problem, requiring complicated methods. The authors reinterpret the theory in terms of Rees algebras, and use their main result to give a simple solution to the mentioned gluing problem.
They also review a method to resolve Rees algebras (or, equivalently, pairs), valid in characteristic zero, which yields partial results in positive characteristic. Again one has to face a ``gluing problem'', to verify that certain procedures are well-defined. The authors once more use the canonicity principle to deal with this question.
They conclude the paper by discussing an example showing that an inductive process to lower the multiplicity of a hypersurface, valid over fields of characteristic zero, may fail in positive characteristic. Rees algebra; equivalence; integral closure; differential operators; resolution of singularities Bravo, A.; García-Escamilla, M. L.; Villamayor U., O. E., On Rees algebras and invariants for singularities over perfect fields, Indiana Univ. Math. J., 61, 3, 1201-1251, (2012) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Integral closure of commutative rings and ideals, Derivations and commutative rings On Rees algebras and invariants for singularities over perfect fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Nous étudions une conditions d'équisingularité définie pour une famille de singularités de surface normale par l'existence d'une résolution simultanée très faible et par une condition supplémentaire sur les faisceaux pluricanoniques relatifs. Nous donnons dans le cas d'une famille de singularités rationnelles une condition nécessaire et suffisante portant sur les singularités des fibres pour avoir équisingularité. équisingularity; simultaneous resolution of a family of singularities of a normal surface; pluricanonical sheaf Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Résolution simultanée d'une famille de singularités rationnelles de surface normale. (Simultaneous resolution of a family of singularities of a normal surface) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A stratification of the set of critical points of a map is universal in the class of stratifications satisfying the classical Thom and Whitney-a conditions if it is the coarsest among all such stratifications. We show that a universal stratification exists if and only if the 'canonical subbundle' of the cotangent bundle of the source of the map (constructed via operations introduced by Glaeser) is Lagrangian. The proof relies on a new Bertini-type theorem for singular varieties proved via an intriguing use of resolution of singularities. Many examples are provided, including those of maps without universal stratifications. Thom stratification; Whitney-a stratification; Bertini-type theorem; desingularization; Gauss regular varieties Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Critical points of functions and mappings on manifolds Universal stratifications and a Bertini-type theorem | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The study of singular points of algebraic curves in the complex plane has a long history. Its beginnings can be traced back to Sir Isaac Newton, and the algebraic geometers of the nineteenth and early twentieth century developed it into a fascinating, already remarkably rich theory. One of the major achievements, during this period, was the resolution of singularities of such curves initiated by Max Noether.
From the 1920s on, the then new topological methods were applied to the local study of singularities of curves, knots and links. In the second half of the twentieth century, the newly developing singularity theory in higher dimensions also propelled the study of the singular points of plane curves, and the developments in this area have been tremendous since the late 1960s. In the course of its long history, singularity theory of plane curves has grown into a meeting point for many different disciplines of mathematics, including algebra, complex analysis, algebraic geometry, topology, and combinatorics. The interaction between ideas, methods and techniques from these various sources makes the study of singularities of plane curves particularly fascinating, enlightening, abundant and fruitful. Moreover, this subject provides a beautiful testing ground for geometric ideas, in general, and a perfect topic for developing a profound understanding of the principles of modern geometry, likewise.
The book under review, written by one of the leading experts in singularity theory, is a highly welcome attempt to present a systematic, comprehensive, versatile and up-to-date account of the present state of art of this venerable area within mathematics. Based on an M.Sc. course taught a number of times (since 1975) at the University of Liverpool, it has partly the character of an introductory textbook, and can be used as such, but it also discusses more recent, advanced and intradisciplinary topics from the forefront of current research in the singularity theory of plane curves. Thus the text, consisting of eleven chapters, is virtually divided into two main parts.
The first five chapters are kept to the level of the underlying M.Sc. course and, therefore, are more introductory and elementary in nature. They are meant to form the core of the book, providing the foundations of the classical theory of plane curve singularities. As for this part, the author has chosen the concept of equisingularity, i.e., the most important equivalence relation for singularities, as the general leitmotif for his approach. Equisingularity can be characterized from numerous different points of view, and the development of the distinct ideas and methods leading to that same concept is taken as the frame for an introduction to curve singularities. This is the mean feature of this approach, and of the book as a whole, that the author emphasizes the equivalence of differing concepts and methods from the beginning on, thereby demonstrating their appearance and power in an integrated account.
Chapter 1 compiles the necessary preliminary material: the definition of algebraic curves in the plane, intersection numbers, resultants and discriminants, manifolds and the implicit function theorem, polar curves and inflection points. All this is treated as basically familiar background material and not covered in every detail. The story starts with Chapter 2, where parametrizations of curves via Puiseux power series, branches of curves, multiplicities and tangent lines to curves are discussed. This is used in Chapter 3 to describe the resolution of curve singularities, including the blow-up process, the notion of infinitely near points, invariants of singularities, and the graph-theoretic interpretation of the configurations arising in the resolution process. Chapter 4 deals with the theory of contact of two branches of a curve, the Eggers tree associated with a branch, computing intersection numbers for curves with several branches, and the equivalent characterizations of the concept of equisingularity in the whole framework developed so far. Chapter 5 turns to the topological aspects of curve singularities, with a special emphasis on knots, links and the classical Alexander polynomial. Equisingularity is then reconsidered from this topological point of view.
The second part of the book, which comprises the remaining six chapters, is written at a more sophisticated level, gives introductions to a number of topics of current research, and even offers several new results of the author. In these more advanced chapters, the topological aspects of curve singularities play a dominant role.
Chapter 6 is devoted to the Milnor fibration, Milnor numbers, and the Euler characteristic of a fibration. The latter is used for several instructive calculations of Milnor numbers. Chapter 7 is entitled ``Projective curves and their duals''. The author gives proofs of the general Plücker theorems for singular plane curves, treats Klein's equation by using Euler characteristics of constructible functions, analyzes the singularities of a dual curve, and surveys some known results about curves with so-called maximal singularities.
The following three chapters are very up-to-date and lead up to the calculation of the monodromy of the Milnor fibration. Chapter 8 introduces calculations and notation for later use, including several numerical invariants of singularities and their representation using exceptional cycles on resolution trees. This chapter also contains an introduction to the topological zeta function à la Denef-Loeser.
Chapter 9 discusses the application of W. Thurston's decomposition theorems for 3-manifolds and for homeomorphisms of surfaces to the Milnor fibration. This chapter offers a novel view to the topology of curve singularities, with a number of results published for the first time. Among other things, the author presents a finiteness criterion for the monodromy and a close relation between the Eggers tree, the resolution graph and the Eisenbud-Neumann diagram of a singularity. Chapter 10 continues with new results in calculating the monodromy, mainly by using Seifert matrices and (again), the Thurston decomposition theorems. In addition, it is shown how to classify Seifert forms over a field, and how some of the numerical invariants required for the classification over the rational number field can be calculated.
The final Chapter 11 touches upon the more algebraic aspects of curve singularities. Ideals in the local ring of a singularity are related to the exceptional cycles studied earlier, and a link between ideals and Enriques's clusters of infinitely near points is established in form of a Galois correspondence. The discussion concludes with brief treatments of jets, equisingularity classes and the determinacy of functions.
Each chapter of the book comes with a section on ``Notes'' and ``Exercises''. The notes include historical remarks, references, comments on related material not covered in the text, and some hints for further research. The exercises form a balanced mixture of routine problems on applying the results in the text to concrete examples, on the one hand, and more challenging problems related to an alternative approach to a topic treated before, on the other. Also, the entire text is relaxed by numerous illustrating and instructive examples, and the bibliography is with more than 200 references more than ample.
All in all, this book, being partly an introductory textbook and partly an advanced research monograph, is extremely comprehensive, profuse and versatile. It contains a wealth of information, both classical and topical, on the attractive and evergreen area of plane algebraic curves, and it offers a lot of new insights to all kinds of readers.
The text reflects the author's great expertise in the field in a masterly way, and that just as much as his passion for the subject and his cultured attitude. His style of writing mathematics is utmost pleasant, nowhere formal, very user-friendly, throughout motivating and highly inspiring. No doubt, this book will quickly become a widely used standard text on singularities of plane curves, and a valuable reference book, too. textbook; algebraic curves; singularities; resolution of singularities; topology of singularities; monodromy C. T. C. Wall, \textit{Singular Points of Plane Curves}, Cambridge University Press, New York, 2004. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Singularities in algebraic geometry, Singularities of curves, local rings, Plane and space curves, Global theory and resolution of singularities (algebro-geometric aspects), Monodromy on manifolds Singular points of plane curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review consists of two parts. In the first part, the authors review some basic material about affine toric varieties and then they study more general ones. Their main concern are those toric varieties, not necessarily normal, that can be covered by a finite number of affine open sets, each one invariant under the action of the torus. For these, they provide a combinatorial description, in terms of fans where to each of their cones a finitely generated semigroup is attached, subject to suitable gluing conditions. Henceforth, ``toric variety'' will mean one in this sense. They discuss a number of properties of these varieties and other related concepts, such as invertible sheaves, ampleness, projectivity and blowing-ups with equivariant sheaves of ideals as centers. The blown-up variety is again toric. They emphasize the mentioned combinatorial description in their study.
In the second part they study in detail certain blowing-ups, namely those whose center is the \textit{logarithmic Jacobian ideal} of the toric variety \(X\). The resulting variety is again toric, and if the base field has characteristic zero, this process agrees with the Semple-Nash modification of \(X\), where each point is replaced by the limit positions of tangent spaces at nearby regular points. This is more commonly called the Nash modification (or blow-up) of \(X\), but it seems that this process appeared for the first time in [\textit{J. G. Semple}, Proc. Lond. Math. Soc. (3) 4, 24--49 (1954; Zbl 0055.14505)].
They prove that, starting from a toric variety \(X\) and a monomial valuation \(V\) of maximal rank of its function field, dominating a point \(x\) of \(X\), successive blowing-ups centered at log Jacobian ideals uniformize the valuation. That is, we reach the situation of a toric variety \(X'\) and a regular point \(x' \in X'\), lying over \(x\), such that the valuation dominates \(x'\). The proof is complicated, and throughout they use the combinatorial description of the first part. In particular, in characteristic zero the uniformization is obtained by repeated application of the Semple-Nash modification.
In the final sections of the article they re-interpret their uniformization result in terms of the Zariski-Riemann space of the fan associated to the toric variety \(X\), which involves the preorders of an underlying lattice, introduced in [\textit{G. Ewald} and \textit{M.-N. Ishida}, Tohoku Math. J. (2) 58, No. 2, 189--218 (2006; Zbl 1108.14039)].
In general the authors work over an arbitrary base field, and some results are still valid over more general rings. toric geometry; Semple-Nash modification; logarithmic Jacobian ideal; monomial valuation; uniformization González, P.D., Teissier, B.: Toric geometry and the Semple-Nash modification. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A Matemáticas 108(1), 1-48 (2014) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Valuations and their generalizations for commutative rings, Modifications; resolution of singularities (complex-analytic aspects) Toric geometry and the Semple-Nash modification | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(E_i\) be an irreducible component of the exceptional divisor of a resolution of singularities of a singular algebraic veriety \(X\). The component \(E_i\) is called \textit{essential} if for any other divisorial resolution of \(X\), \(E_i\) up to birational transform, is an irreducible component of the exceptional divisor of the second resolution. Denote by \(X_\infty^{\mathrm{sing}}\) the set of arcs whose ``origin'' belongs to the singular locus of \(X\). In 1963 \textit{J. Nash} [Duke Math. J. 81, No. 1, 31--38 (1995; Zbl 0880.14010)] proved that \(X_\infty^{\mathrm{sing}}\) has finitely many irreducible components \(F_1,\ldots,F_r\). Let \({\mathcal N}:\{F_1,\ldots,F_r\}\to \{\text{essentional divisors}\}\) be the map sending \(F_i\) to the exceptional divisor \(E_i\) such that the generic arc of \(F_i\) has lifting to the resolution passing throught a general point of the component \(E_i\). J.~Nash (in the above mentioned paper) proved that the map \({\mathcal N}\) is injective and posed the question whether the map is surjective. This question is known as the \textit{Nash Problem of arcs}.
From the abstract ``The goal of this paper is to give an historical overview of the Nash problem of arcs in arbitrary dimension, as well as its affirmative solution in dimension two by \textit{J. Fernández de Bobadilla} and \textit{M.~Pe~Pereira} [Ann. Math. (2) 176, No. 3, 2003--2029 (2012; Zbl 1264.14049)] and a negative solution in higher dimension by \textit{S. Ishi} and \textit{J. Kolár} (in dimension greater or equal to 4) [Duke Math. J. 120, No. 3, 601--620 (2003; Zbl 1052.14011)] and by \textit{T. de Fernex} (in dimension 3) [Compos. Math. 149, No. 9, 1519--1534 (2013; Zbl 1285.14013)].'' Nash's problem on arcs; surfaces; singularities; resolution of singularities; space of arcs; wedges Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Arcs and motivic integration The Nash problem and its solution | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article under review studies the Bertini type theorems for varieties with mild singularities in characteristic \(p>0\). In the case of characteristic zero, it is known that the `mild' singularities remain `mild' after cutting by general hyperplanes. Therefore it is natural to ask the same question in characteristic \(p>0\). In this article, the authors prove that it is true for certains \(F\)-singularities.
Recall that the notion strongly \(F\)-regular (resp. sharply \(F\)-pure singularities) is the moral equivalent of log terminal singularities (resp. log canonical singularities). Here is an important special case of the main theorem in this article: for a projective variety \(X\) over an algebraically closed field \(k\), if \(X\) is \(F\)-pure (respectively, strongly \(F\)-regular), then so is a general hyperplane section of a very ample line bundle.
The authors also prove that \(F\)-injective singularities doesnot satify Bertini's theorem. In fact, based on a work of Cumino, Greco and Manaresi [\textit{C. Cumino} et al., Proc. Am. Math. Soc. 106, No. 1, 37--42 (1989; Zbl 0699.14063)], the authors proved that there exists a \(F\)-injective projective surface whose general hyperplane section is not \(F\)-injective. Bertini theorems; \(F\)-singularities; characteristic \(p>0\) Schwede, K.; Zhang, W., Bertini theorems for \textit{F}-singularities, Proc. Lond. Math. Soc. (3), 107, 4, 851-874, (2013) Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Multiplier ideals, Global theory and resolution of singularities (algebro-geometric aspects) Bertini theorems for \(F\)-singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities An analytically irreducible hypersurface germ \((S,0)\subset(\mathbb{C}^{d+1},0)\) is quasi-ordinary if it can be defined by the vanishing of the minimal polynomial \(f\in\mathbb{C}\{X\}[Y]\) of a fractional power series in the variables \(X=(X_1,\dots,X_d)\) which has characteristic monomials, generalizing the classical Newton-Puiseux characteristic exponents of the plane-branch case (\(d=1\)). We prove that the set of vertices of Newton polyhedra of resultants of \(f\) and \(h\) with respect to the indeterminate \(Y\), for those polynomials \(h\) which are not divisible by \(f\), is a semigroup of rank \(d\), generalizing the classical semigroup appearing in the plane-branch case. We show that some of the approximate roots of the polynomial \(f\) are irreducible quasi-ordinary polynomials and that, together with the coordinates \(X_1,\dots,X_d\), provide a set of generators of the semigroup from which we can recover the characteristic monomials and vice versa.
Finally, we prove that the semigroups corresponding to any two parametrizations of \((S,0)\) are isomorphic and that this semigroup is a complete invariant of the embedded topological type of the germ \((S,0)\) as characterized by the work of Gau and Lipman. quasi-ordinary singularities; topological type; semigroup; discriminant; hypersurface singularity; Newton polyhedra P. D. González Pérez, The semigroup of a quasi-ordinary hypersurface , J. Inst. Math. Jussieu 2 (2003), 383--399. Complex surface and hypersurface singularities, Toric varieties, Newton polyhedra, Okounkov bodies, Singularities in algebraic geometry, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Global theory and resolution of singularities (algebro-geometric aspects) The semigroup of a quasi-ordinary hypersurface. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author essentially proves two results. First he describes the nilpotent orbits of a semisimple Lie algebra over an algebraically closed field of characteristic zero, which have the Gorenstein property. --- Secondly he considers a certain normal algebraic variety \(X\). He obtains a necessary and sufficient condition for the existence of resolvable rational singularities on \(X\) in function of the dual Grothendieck sheaf of \(X\). nilpotent orbits of a semisimple Lie algebra; resolvable rational singularities D. I. Panyushev, Rationality of singularities and the Gorenstein property of nilpotent orbits, Funktsional. Anal. i Prilozhen. 25 (1991), no. 3, 76 -- 78 (Russian); English transl., Funct. Anal. Appl. 25 (1991), no. 3, 225 -- 226 (1992). Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Simple, semisimple, reductive (super)algebras, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Rationality of singularities and the Gorenstein property for nilpotent orbits | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be an \(n\)-dimensional projective variety having at most log-terminal singularities and let \(E\) be an ample vector bundle of rank \(r\) on \(X\). The author proves that:
(1) If \(r= n+1\) and \(c_1 (X)= c_1 (E)\) then \((X, E) \simeq (\mathbb{P}^n, {\mathcal O}_P (1)^{n+ 1})\); and
(2) If \(r\geq n+1\) then \(K_X+ c_1 (E)\) is ample unless \((X, E)\simeq (\mathbb{P}^n, {\mathcal O}_P (1)^{n+ 1})\).
If \(X\) is smooth, the results where already known [cf. \textit{Y.-G. Ye} and \textit{Q. Zhang}, Duke Math. J. 60, No. 3, 671-687 (1990; Zbl 0709.14011) and \textit{T. Peternell} [Math. Z. 205, No. 3, 487-490 (1990; Zbl 0726.14034)]. However, the argument used in the smooth case do not work in the singular one. log-terminal singularities Qi Zhang, Ample vector bundles on singular varieties, Math. Z. 220 (1995), no. 1, 59 -- 64. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Ample vector bundles on singular varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This lecture note examines the 6 types of ``quadrilateral'' singularities \(J_{3,0}\), \(Z_{1,0}\), \(Q_{2,0}\), \(W_{1,0}\), \(S_{1,0}\), \(U_{1,0}\): if \(X\) is a class of them, \(PC(X)\) denotes the set of Dynkin graphs \(G\) with components of type \(A\), \(D\), \(E\) having the following property: There is a fibre \(Y\) in the versal deformation of a singularity in \(X\), such that \(Y\) has only rational double points, and \(G\) is given as a (disjoint) union of their Dynkin graphs. Quadrilateral singularities are of modality 2, and \(PC(X)\) is studied for the case, if \(X\) is one of the relevant normal forms [cf. \textit{V. I. Arnold}, Invent. Math. 35, 87- 109 (1976; Zbl 0336.57022)].
Due to Looijenga, \(PC(X)\) can be studied using the lattice embedding of the associated root-lattice into the even unimodular lattice with signature (19,3). Using Nikulin's results for such embeddings, this gives a possibility to determine whether or not \(G\) belongs to \(PC(X)\). The book under review gives a systematic treatment of all cases; technical tools are the root systems \(A,\dots,F\) as well as the nonreduced root systems \(BC\) arising in the relevant constructions. Using Dynkin graphs (in a terminology slightly different from the standard one), the description of \(G \in PC(X)\) (in the cases of \(X = J_{3,0}\), \(Z_{1,0}\), \(Q_{2,0})\) is given by the following theorem: \(G\) belongs to \(PC(X)\) iff it is in a list of exceptions or can be obtained by applying elementary or ``tie transformations'' (in the sense, studied by the author in a previous paper) twice to some of a certain list of essential basic Dynkin graphs and if \(G\) contains no short root. This theorem can be interpreted in the language of elliptic \(K3\)-surfaces: It describes the possible combinations of singular fibres on \(K3\)-surfaces with a singular fibre of type \(I^*_ 0\) (in Kodaira's notation). -- The remaining cases of \(X = W_{1,0}\), \(S_{1,0}\), \(U_{1,0}\) require further efforts. Appearing graphs can be characterized by additional rules (presence of ``obstruction components'' and/or ``dual elementary transformations'').
The book starts with an introduction to quadrilateral singularities: In chapter 1 Looijenga's results (which are basic to reduce the problem to lattice embeddings) are reviewed.
After an introduction to lattices, another section is devoted to a theory of root systems, adapted to the situation considered later. Further, the technical tools for manipulating graphs are introduced, followed by a section, where conditions are given for a Dynkin graph \(G\) to be in \(PC(X)\). The chapter concludes explaining Coxeter-Vinberg graphs associated with hyperbolic spaces.
Chapter 2 deals with the first three types of quadrilateral singularities, as indicated above, whereas chapter 3 and 4 are devoted to the study of the cases \(X = W_{1,0}\), \(S_{1,0}\), \(U_{1,0}\) respectively.
In an appendix, similar questions for plane sextic curves are considered: This is a revised version of the authors' earlier paper [in Singularities, Proc. IMA Particip. Inst. Conf., Iowa City 1986, Contemp. Math. 90, 295-316 (1989; Zbl 0698.14023)], where the above methods are applied to study conditions for a Dynkin graph to correspond to a configuration of \(A,D,E\)-singularities on a sextic curve.
This book gives insight to deep properties of deformations of a class of bimodal singularities. The author points out essential ideas stemming from a discussion at Oberseminar Brieskorn (University of Bonn), especially from \textit{F. J. Bilitewski}, who considered several of the cases treated here, already. The methods employed may be useful to study other types of singularities as well. union of Dynkin graphs; quadrilateral singularities; tie transformations; singular fibres on \(K3\)-surfaces; versal deformation of a singularity; modality; root systems; Dynkin graphs; bimodal singularities T. Urabe, ''Dynkin graphs and triangle singularities,'' In:Proc. of Workshop on Topology and Geometry, Hanoi, Vietnam, March (1993). Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Complex surface and hypersurface singularities, Global theory of complex singularities; cohomological properties, Singularities in algebraic geometry Dynkin graphs and quadrilateral singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author introduces symplectic singularities and classifies symplectic singularities of the simplest type. A normal variety is said to have symplectic singularities if its smooth part carries a closed symplectic form, whose pull back to any resolution extends to a global holomorphic 2-form. The relation between symplectic singularities and symplectic complex (hyperKähler) manifolds is analogous to that between rational Gorenstein singularities and Calabi-Yau manifolds.
The main theorem states that a germ of an isolated singularity with a smooth projective tangent cone is symplectic if and only if it is analytically isomorphic to the germ \((\overline{\mathcal O}_{\text{min}}, 0)\), where \(\overline{\mathcal O}_{\text{min}}= {\mathcal O}_{\text{min}}\cup\{0\}\) is the closure of a smallest non-zero nilpotent orbit \({\mathcal O}_{\text{min}}\) for the adjoint action of a simple complex Lie algebra. As the next step in the classification of symplectic singularities the author proposes isolated singularities with trivial local fundamental group. symplectic singularities; normal variety; rational Gorenstein singularities; Calabi-Yau manifolds A. Beauville, \textit{Symplectic singularities}, \textit{Invent. Math.}\textbf{139} (2000) 541 \textit{Invent. Math.}\textbf{139} (2000) 541 [math/9903070]. Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry Symplectic singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(M\) be a smooth algebraic variety over a field \(k\), algebraically closed and of characteristic zero. Let \(X\subset M\) be a hypersurface, i.e., a closed subscheme locally defined by one equation. Then the singular locus \(Y\) of \(X\) has a natural scheme structure defined by the jacobian ideal.
The author defines a class \(\mu_{\mathcal L}(Y)\) in the Chow group of \(Y\), depending on the data \((M,X)\): he then shows that the class only depends on \(Y\) and on the line bundle \({\mathcal L}:={\mathcal O}_M(X)|_Y\), and gives explicit methods for computing it in the case where \(Y\) is itself smooth.
In \S 2 it is proven that \(\deg(\mu_{\mathcal L}(Y))\) is Parusiński's generalized Milnor number; the class is then used to study local properties near \([X]\) of the dual variety of \((M,{\mathcal O}_M(X))\), i.e., the variety of sections of \({\mathcal O}_M(X)\) defining singular hypersurfaces, recovering and partially extending previous results of Ein, Holme, Landman, Parusiński and Zak.
In \S 3 it is shown that the \(\mu\) class imposes strong restrictions on which schemes \(Y\) can appear as singular subschemes of a hypersurface. hypersurface singularity; Milnor number; jacobian ideal; Chow group Paolo Aluffi, ``On the singular schemes of hypersurfaces'', Duke Math. J.80 (1995) no. 2, p. 325-351 Hypersurfaces and algebraic geometry, Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes) Singular schemes of hypersurfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the situation of finite dimensional modules over tame quiver algebras the degener\-ation-order coincides with the hom-order and with the ext-order. Therefore, up to common direct summands, any minimal degeneration \(N\) of a module \(M\) is induced by a short exact sequence with middleterm \(M\) and indecomposable ends \(U\) and \(V\) that add up to \(N\). We study these ``building blocs'' of degenerations and in particular the codimensions for the case where \(V\) is regular. We show by theoretical means that the classification of all the ``building blocs'' is a finite problem without affecting the codimension or the type of singularity. With the help of a computer we have analyzed completely this case: The codimensions are bounded by 2, so that the minimal singularities are known by \textit{G. Zwara} [Manuscr. Math. 123, No. 3, 237-249 (2007; Zbl 1129.14006)]. Dynkin quivers; tame quiver algebras; singularities of representations of quivers I. Wolters, On deformations of the direct sum of a regular and another indecomposable module over a tame quiver algebra, Dissertation Berg. Univ. Wuppertal, 2008, 151 pages, available from Internet: URN: urn:nbn:de:hbz:468-20090102. Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Research exposition (monographs, survey articles) pertaining to associative rings and algebras On deformations of the direct sum of a regular and another indecomposable module over a tame quiver algebra. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We construct a determinantal resolution of singularities for the universal subscheme in \(\mathcal S \times H_{d + 1} \) and prove that it is isomorphic to the variety of total pairs \(\widetilde{\mathcal S \times H_d }\). algebraic surfaces; punctual Hilbert scheme; Ioneda pairing Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Determinantal varieties Determinantal resolution of the universal subscheme in \({\mathcal S}\times H_{d+1}\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathfrak g\) be a simple complex Lie algebra and \(G\) its adjoint Lie group. For a nilpotent element \(x \in \mathfrak g\) one can consider the nilpotent orbit \(\mathcal O_x := G \cdot x\). The minimal orbit \(\mathcal O_{min}\) of \(\mathfrak g\) is the unique orbit which is contained in the closure of any other non-zero nilpotent orbit. Its closure \(\bar{\mathcal O}_{min}\) is normal with an isolated singularity. Minimal orbit closures and their singularities have been studied intensively by many authors over the last ten years. They are expected to be closely related to symplectic singularities, i.e. those singular symplectic varieties which admit a resolution of singularities by a symplectic manifold.
More precisely one expects that the following conjecture holds: an isolated symplectic singularity of dimension at least four which admits a symplectic resolution is analytically locally isomorphic to \(\bar{\mathcal O}_{\min} \subset \mathfrak{sl}(V)\). This conjecture was proven in dimension four by \textit{J. Wierzba} and \textit{J. A. Wiśniewski} [Duke Math. J. 120, No. 1, 65--95 (2003; Zbl 1036.14007)], but it is open in general.
In the paper under review the author gives a summary of results on nilpotent orbits from the point of view of the minimal model program, but the main focus is an extensive list of related conjectures and open problems. nilpotent orbits; Springer map; Slodowy slice; minimal model program; Fano contact manifold Global theory and resolution of singularities (algebro-geometric aspects), Coadjoint orbits; nilpotent varieties, Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry, Rational and birational maps, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Symplectic manifolds (general theory), Geometry and quantization, symplectic methods, Fano varieties Geometry of nilpotent orbits: results and conjectures | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Multiplier ideals are fundamental objects in birational geometry. It has been expected that multiplier ideals satisfy the subadditivity formula, which compares the multiplier ideal of the product of ideals \(I_j\) and the product of multiplier ideals of \(I_j\). The subadditivity formula on smooth varieties was obtained in [\textit{J.-P. Demailly, L. Ein} and \textit{R. K. Lazarsfeld}, Mich. Math. J. 48, Spec. Vol., 137--156 (2000; Zbl 1077.14516)]. It was generalized to the case of Q-Gorenstein varieties in [\textit{S. Takagi}, Am. J. Math. 128, No. 6, 1345-1362 (2006; Zbl 1109.14005)] and [\textit{E. Eisenstein}, ``Inversion of subadjunction and multiplier ideals'', \url{arXiv:1104.4840}]. In the paper under review, the author further generalizes the formula to the log-\(\mathbb{Q}\)-Gorenstein setting. multiplier ideals; test ideals Takagi, S., \textit{A subadditivity formula for multiplier ideals associated to log pairs}, Proc. Amer. Math. Soc., 141, 93-102, (2013) Multiplier ideals, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) A subadditivity formula for multiplier ideals associated to log pairs | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S\) be a smooth complex algebraic surface. Given positive integers \(n_1<n_2<\cdots <n_k\), let \(S^{[n_1, n_2, \dots, n_k]}\) denote the nested Hilbert scheme parameterizing nested 0-dimensional sub-schemes of \(S\): \(\xi_{n_1}\subseteq \xi_{n_2}\subseteq \cdots \xi_{n_k}\) of length \(n_i\). The nested Hilbert schemes are natural analogues for the Hilbert schemes \(S^{[n]}\) of points, and some of them have played an important role in the study of syzygies. The present well-written paper gives a quite comprehensive study of \(S^{[n, n+1, n+2]}\).
The first main result is a new proof of the irreducibility of \(S^{[n, n+1, n+2]}\) due to \textit{N. Addington} [Algebr. Geom. 3, No. 2, 223--260 (2016; Zbl 1372.14009)]. The idea here is to realize \(S^{[n, n+1, n+2]}\) as \(\mathbb{P}(\mathscr{I}_{Z_{[n, n+1]}})\), where \(Z_{[n, n+1]}\) is the subscheme of \(S\times S^{[n, n+1]}\) parameterizing triples \((p, \xi_{n}, \xi_{n+1})\) with \(p\in \text{Supp}(\xi_{n+1})\), and use a criterion of \textit{G. Ellingsrud} and \textit{S. A. Strømme} [Trans. Amer. Math. Soc. 350, No. 6, 2547--2552 (1998; Zbl 0893.14001)]. Along the way, with more care the authors establish an estimate on codimension of certain strata of \(Z_{[n, n+1]}\) in \(S\times S^{[n, n+1]}\), which is quadratic in the minimal number \(i\) of generators for the localized ideal. While a linear estimate is sufficient in the criterion mentioned above, the quadratic one is of great interest on its own. Via forgetful and residual point maps, the irreducibility of \(S^{[n, n+2]}, S^{[1, n, n+1, n+2]}, S^{[1, n+1, n+2]}\) and \(S^{[1, n, n+2]}\) are deduced from that of \(S^{[n, n+1, n+2]}\).
The second one is an explicit construction of a family of nested subschemes, indicating that \(S^{[1, 2, \dots, 23]}\) is reducible. As a corollary, \(S^{[n_1, n_2, \dots, n_k]}\) is reducible whenever \(k\ge 23\).
The third one is that \(S^{[n, n+1, n+2]}\) is a local complete intersection and has klt singularities. The proof involves showing that a two-step blowup gives a (small) resolution of singularities of \(S^{[n, n+1, n+2]}\).
In the end, the Picard group and the canonical divisor of \(S^{[n, n+1, n+2]}\) are computed in case \(S\) is regular. nested Hilbert schemes; irreducibility; singularities; Picard group Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry Irreducibility and singularities of some nested Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For an \(m\)-primary ideal \(I\) in the local ring of a normal surface singularity, we define \(v\)-superficial elements, which are not far from being superficial in the sense of P. Samuel. Requiring a further condition of transversality, we define ``general elements'' of \(I\), which, as we prove, are characterized by the minimality of the Milnor number in \(I\). In particular, elements with \(\mu\) minimum are \(v\)-superficial. superficial elements; normal surface singularity; minimality of the Milnor number Bondil ( R. ) , Lê ( D.T. ). - Caractérisations des éléments superficiels d'un idéal , C.R. Acad. Sci. Paris , t. 332 , Sér. I , p. 717 - 722 ( 2001 ). MR 1843194 | Zbl 1006.14013 Singularities of surfaces or higher-dimensional varieties, Regular local rings, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Characterizations of superficial elements of an ideal | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper is in some sense a follow-up to the paper ``Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes'' by the first author et al. [``Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes'', Preprint, \url{arXiv:0905.2191}]. In that paper, it was proved that every reduced excellent Noetherian scheme \(X\) of dimension at most two admits a resolution of singularities via finitely many ``permissible'' blowups. Their resolution procedure (called the CJS algorithm) is ``canonical'' in a precise sense; in particular it commutes with localization and completion. The proof of termination is by contradiction, which is unusual in this business.
The present paper introduces a new local invariant \(\iota(X, x)\) for schemes satisfying the above assumptions. In Theorem A, it is shown that this invariant decreases at each step of the CJS algorithm. This provides a new proof of termination of the CJS algorithm which is closer in spirit to the classical approach used in resolution theory. surface singularities; resolution of singularities; invariants for singularities; Hironaka's characteristic polyhedra Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) A strictly decreasing invariant for resolution of singularities in dimension two | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be the Jacobian algebra obtained from the quiver \(Q\) defined by the vertices and oriented edges of a dimer model. If the dimer model satisfies a consistency condition, then \(A\) is a non-commutative crepant resolution of its centre \(Z(A)\), which is the coordinate ring of a 3-dimensional toric Gorenstein singularity. Let \(X=\text{Spec } Z(A)\). Every projective crepant resolution of the singularity \(X\) is obtained as a fine moduli space of stable representations of \(A\) with dimension vector \((1, \dots , 1)\) denoted by \(Y\). For a certain choice of stability parameters depending on a choice of distinguished vertex \(0\in Q\), the dual of the tautological bundle on \(Y\) defines an equivalence of derived categories \(\Psi:D^b(\text{mod-}A)\to D^b(\text{coh}(Y ))\). In the special case, where the dimer model tiles the torus with triangles, then \(A\) is the skew group algebra for a finite abelian subgroup \(G \subset \mathrm{SL}(3, \mathbb{C})\), and it can be arranged \(Y\) to be the \(G\)-Hilbert scheme and the equivalence above to coincide with the derived equivalence of Bridgeland-King-Reid from the McKay correspondence.
Let \(S_i\) denote the simple \(A\)-module corresponding to vertex \(i\) in \(Q\). The main result of the paper under review proves that for any \(i\neq 0\), the object \(\Psi(S_i)\) is quasi-isomorphic to a shift of a coherent sheaf, and the derived dual of \(\Psi(S_0)\) is quasi-isomorphic to the shift by 3 of the push-forward of the structure sheaf of the fiber of \(Y\to X\) over the unique torus-invariant point. In particular, \(\Psi(S_0)\) is a pure sheaf if and only if the fiber is equidimensional. One ingredient of the proof involves establishing a link between the objects \(\Psi(S_i)\) that have non-vanishing cohomology in degree zero and certain walls of the GIT chamber containing the stability parameter. This result in combination with other known results provide the Geometric Reid's recipe which is the dimer model analogue of the Geometric McKay correspondence in dimension three proven by Logvinenko. Geometric Reid's recipe provides a description of the objects \(\Psi(S_i)\). dimer models; crepant resolution; quiver representation Bocklandt, R; Craw, A; Quintero Vélez, A, Geometric reid's recipe for dimer models, Math. Ann., 361, 689-723, (2015) McKay correspondence, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Global theory and resolution of singularities (algebro-geometric aspects), Geometric invariant theory, Derived categories and associative algebras, Representations of quivers and partially ordered sets Geometric Reid's recipe for dimer models | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is an announcement of conjectures and results concerning the generating series of Euler characteristics of Hilbert schemes of points on surfaces with simple (Kleinian) singularities. For a quotient surface \(\mathbb C^2/G\) with \(G<\mathrm{SL}(2,\mathbb C)\) a finite subgroup, we conjecture a formula for this generating series in terms of Lie-theoretic data, which is compatible with existing results for type \(A\) singularities. We announce a proof of our conjecture for singularities of type \(D\). The generating series in our conjecture can be seen as a specialized character of the basic representation of the corresponding (extended) affine Lie algebra; we discuss possible representation-theoretic consequences of this fact. Our results, respectively conjectures, imply the modularity of the generating function for surfaces with type \(A\) and type \(D\), respectively arbitrary, simple singularities, confirming predictions of \(S\)-duality. Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry, Group actions on varieties or schemes (quotients) Euler characteristics of Hilbert schemes of points on surfaces with simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We survey the problems of resolution of singularities in positive characteristic and of local and global monomialization of algebraic mappings. We discuss the differences in resolution of singularities from characteristic zero and some of the difficulties. We outline Hironaka's proof of resolution for positive characteristic surfaces, and mention some recent results and open problems.
Monomialization is the process of transforming an algebraic mapping into a mapping that is essentially given by a monomial mapping by performing sequences of blow ups of nonsingular subvarieties above the target and domain. We discuss what is known about this problem and give some open problems. Cutkosky, Steven Dale, Ramification of valuations and counterexamples to local monomialization in positive characteristic, (2014), preprint Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Resolution of singularities in characteristic \(p\) and monomialization | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K\) be a \(p\)-adic field. We explore Igusa's \(p\)-adic zeta function, which is associated to a \(K\)-analytic function on an open and compact subset of \(K^{n}\). First we deduce a formula for an important coefficient in the Laurent series of this meromorphic function at a candidate pole. Afterwards we use this formula to determine all values less than \(-1/2\) for \(n=2\) and less than \(-1\) for \(n=3\) which occur as the real part of a pole. Segers, D.: On the smallest poles of Igusa's p-adic zeta functions. Math. Z. 252, 429--455 (2006) Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Congruences in many variables, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Zeta functions and \(L\)-functions On the smallest poles of Igusa's \(p\)-adic zeta functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study the normal bundles of the exceptional sets of higher dimensional isolated simple small singularities when the Picard groups of the exceptional sets are of rank one and their normal bundles have certain good filtrations. In particular, we prove numerical inequalities satisfied by normal bundles of exceptional sets. Moreover, we generalize Laufer's results on rationality and embedding dimension of singularities to higher dimension. normal bundle; small singularity; exceptional set; embedding dimension Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Adjunction problems Normal bundles on the exceptional sets of simple small resolutions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author classifies simple hypersurface K3 singularities (X,x) defined by a non-degenerated polynomial \(f(z_ 0,z_ 1,z_ 2,z_ 3)\) and he studies the minimal resolution \(\pi: (\tilde X,E)\to (X,x)\) and the singularities on the exceptional divisor E.
A three-dimensional singularity (X,x) is a simple K3 singularity if (X,x) is quasi-Gorenstein and if the exceptional divisor E of any minimal resolution is a normal K3 surface, where a minimal resolution \(\pi: (\tilde X,E)\to (X,x)\) is a proper morphism with only terminal singularities on \(\tilde X,\) with \(\tilde X\simeq X\setminus \{x\}\) and with \(K_{\tilde X}\) nef with respect to \(\pi\).
- The simple K3 singularities could be regarded as three-dimensional generalizations of simple elliptic singularities.
If the simple K3 singularity (X,x) is defined by a non-degenerated polynomial f(z), then \((1,1,1,1)\in \Gamma(f)\). The weight \(\alpha =\alpha(f)=(\alpha_ 1,\alpha_ 2,\alpha_ 3,\alpha_ 4)\) of the quasi-homogeneous polynomial \(f_{\Delta_ 0}\) associated to the face \(\Delta_ 0\) containing (1,1,1,1) verifies \(\sum^{4}_{i=1}\alpha_ i =1\). - Then to classify the simple K3 singularities we need to study the set \(W_ 4\) of weights: \(W_ 4=\{\alpha \in {\mathbb{Q}}^ 4_+| \quad \sum^{4}_{i=1}\alpha_ i =1,\alpha_ 1\geq...\geq \alpha_ 4\) and \((1,1,1,1)\in Int(C(\alpha))\},\) where \(C(\alpha\)) is the closed cone in \({\mathbb{R}}^ 4\) generated by the set \(T(\alpha)=\{\nu \in {\mathbb{Z}}^ 4_ 0| \alpha.\nu =1\}.\)
The author shows that the cardinality of \(W_ 4\) is 95, and for each weight \(\alpha\) he gives a quasi-homogeneous f of weight \(\alpha\) which defines a simple K3 singularity and such that \(\Delta_ 0=\Gamma (f)\) is the convex hull of \(T(\alpha\)). Then he constructs a minimal resolution \(\pi: \tilde X\to X\) using torus embedding: if the weight \(\alpha(f)=(p_ 1/p,...,p_ 4/p)\), where \(p_ 1,...,p_ 4\) are relatively prime integers, the filtered blow-up with weight \((p_ 1,...,p_ 4)\), \(\Pi: (V,F)\to ({\mathbb{C}}^ 4,0)\), induces a minimal resolution of (X,x). In the last part the author shows that under some conditions on the nondegenerate polynomial f defining the simple K3 singularity (X,x), the type and the number of the singularities on E are determined by the weight \(\alpha(f)\), independently of f. type of singularities; simple hypersurface K3 singularities; minimal resolution; exceptional divisor; number of the singularities; weight Yonemura, T., Hypersurface simple \textit{K}3 singularities, Tohoku Math. J. (2), 42, 3, 351-380, (1990), MR 1066667 Singularities of surfaces or higher-dimensional varieties, \(K3\) surfaces and Enriques surfaces, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Hypersurface simple K3 singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let {\(\Lambda\)} be a concealed canonical algebra and d the dimension vector of a {\(\Lambda\)}-module which is periodic respect to the action of the Auslander-Reiten translation \(\tau\). In the paper, we investigate the union of the closures of the orbits of the \(\tau\)-periodic {\(\Lambda\)}-modules of dimension vector \(\mathbf d\). We show that this set is closed and regular in codimension one. concealed canonical algebra; module variety; regularity in codimension one; periodic module Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry The closure of the set of periodic modules over a concealed canonical algebra is regular in codimension one | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, the author gives interesting formulas for the Hodge numbers of a nodal hypersurface in a smooth complex projective fourfold. Let \(X\) be a smooth complex projective fourfold and let \(Y\) be a nodal hypersurface in \(X\) such that:
A1: the line bundle \({\mathcal M}:={\mathcal O}_X(Y)\) is ample,
A2: \(H^2\Omega^1_X=0,\)
A3: \(H^3(\Omega^1_X \otimes{\mathcal M}^{-1}) =0\).
Denote by \(\widetilde Y\) a big resolution of \(Y\) and \(\widehat Y\) a small one.
Theorem 1:
\[
\begin{aligned} & h^{11}(\widetilde Y)=h^{11}(X) +\mu+\delta,\\ & h^{12}(\widetilde Y)=h^0({\mathcal M}^{\otimes 2}\otimes K_X)+ h^3{\mathcal O}_x-h^0( {\mathcal M}\otimes K_X)-h^3\Omega^1_X-h^4(\Omega^1_X\otimes{\mathcal M}^{-1})-\mu +\delta\end{aligned}
\]
Theorem 2:
\[
h^{11}(\widehat Y)=h^{11}(\widetilde Y)-\mu,\quad h^{12} (\widehat Y)=h^{12}(\widetilde Y),
\]
where \(\mu\) is the number of nodes and \(\delta\) the defect of \(Y\). big resolution; small resolution; Hodge numbers of a nodal hypersurface; number of nodes; defect S. Cynk, ''Defect of a nodal hypersurface,'' Manuscripta Math. 104(3), 325--331 (2001). Singularities of surfaces or higher-dimensional varieties, Transcendental methods, Hodge theory (algebro-geometric aspects), Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, \(3\)-folds Defect of a nodal hypersurface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Noetherian dimer algebras form a prominent class of examples of noncommutative crepant resolutions (NCCRs). However, dimer algebras that are noetherian are quite rare, and we consider the question: how close are nonnoetherian homotopy dimer algebras to being NCCRs? To address this question, we introduce a generalization of NCCRs to nonnoetherian tiled matrix rings. We show that if a noetherian dimer algebra is obtained from a nonnoetherian homotopy dimer algebra \(A\) by contracting each arrow whose head has indegree 1, then \(A\) is a noncommutative desingularization of its nonnoetherian centre. Furthermore, if any two arrows whose tails have indegree 1 are coprime, then \(A\) is a nonnoetherian NCCR. Representations of quivers and partially ordered sets, Noncommutative algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Nonnoetherian homotopy dimer algebras and noncommutative crepant resolutions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Triangulated categories of singularities may be seen as a categorical measure for the complexity of the singularities of a Noetherian scheme \(X\). For a commutative Noetherian ring \(R\), the category \(\mathcal D_{sg}(R)=\frac{\mathcal D^b(\mathrm{mod}-R)}{\mathrm{Perf}(R)}\), where \(\mathrm{Perf}(R)\) is the subcategory of perfect sheaves, i.e., complexes which are quasi-isomorphic to bounded complexes of finitely generated projective modules, is called the \textit{singularity category} of \(R\)
If \(X\) has isolated Gorenstein singularities \(x_1,\dots, x_n\), it is known that the triangulation of the singularity category is equivalent to the direct sum of the stable categories of maximal Cohen-Macaulay \(\widehat{\mathcal O}_{x_i}\)-modules.
\textit{M. Van den Bergh} [Duke Math. J. 122, No. 3, 423--455 (2004; Zbl 1074.14013)] defined noncommutative analogues of (crepant) resolutions (NC(C)R) of singularities which gives pure commutative results. Also, moduli spaces of quiver representations give useful techniques to obtain commutative resolutions from noncommutative ones.
Combining these concepts, the first author together with \textit{I. Burban} [Adv. Math. 231, No. 1, 414--435 (2012; Zbl 1249.14004)] developed the theory of \textit{relative singularity categories}. The idea is that these categories measure the difference between the derived category of a noncommutative resolution (NCR) and the smooth part \(K^b(\mathrm{proj}-R)\subseteq D^b(\mathsf{mod}-R)\) of the derived category of the singularity. This article is about the relation between relative and classical singularity categories.
Using the theories developed here, the authors obtain a purely commutative result. This is in cooperation with Iyama and Wemyss [\textit{M. Kalck} et al., Compos. Math. 151, No. 3, 502--534 (2015; Zbl 1327.14172)], and consists of decomposing \textit{O. Iyama} and \textit{M. Wemyss}' \textit{new triangulated category} [Ill. J. Math. 55, No. 1, 325--341 (2011; Zbl 1258.13015)] for complete rational surface singularities into blocks of singularities categories of ADE singularities. Moreover, \textit{L. Thanhoffer de Völcsey} and \textit{M. Van den Bergh} [``Explicit models for some stable categories of maximal Cohen-Macaulay modules'', Preprint, \url{arXiv:1006.2021}] have proved that the stable category of a complete Gorenstein quotient singularity of Krull dimension 3 is a generalized cluster category. In this article, the authors recover these results using different techniques.
Let \(k\) be an algebraically closed field, let \((R,\mathfrak m)\) be a commutative local complete Gorenstein \(k\)-algebra such that \(k\cong R/\mathfrak m\), and let \(\mathsf{MCM}(R)=\{M\in\mathsf{mod}-R|\mathrm {Ext}^i_R(M,R)=0\mathrm{ for all }i>0\}\) be the full subcategory of \textit{maximal Cohen-Macaulay} \(R\)-modules. Let \(M_0=R,\;M_1,\dots,M_t\) be pairwise non-isomorphic indecomposable \(\mathrm{MCM}\;R\)-modules, put \(M=\bigoplus_{i=1}^tM_i\), and let \(A=\mathrm {End}_R(M)\). If \(\mathrm {gldim}(A)<\infty\), \(A\) is a \textit{noncommutative resolution} (NCR) of \(R\). A particular case influencing the article, is the case where \(R\) has a finite number of indecomposable MCMs and \(M\) is their sum. Then \(\mathrm {End}_R(M)\) is the \textit{Auslander algebra} \(\mathrm {Aus}((\mathsf{MCM})\) which is a NCR.
There is a fully faithful triangle functor \(K^b(\mathrm{proj}-R)\rightarrow D^b(\mathsf{mod}-A)\) whose essential image is \(\mathsf{thick}(eA)\subseteq D^b(\mathsf{mod}-A)\), where \(e\in A\) is the idempotent corresponding to the projection on \(R\).
The \textit{classical singularity category} is defined as \(D_{sg}(R)=D^b(\mathrm{mod}-R)/K^b(\mathrm{proj}-R)\), and the authors define the \textit{relative singularity category} as the Verdier quotient category \(\Delta_R(A)=\frac{D^b(\mathrm{mod}-A)}{K^b(\mathrm{proj}-R)}\cong\frac{D^b(\mathrm{mod}-A)}{\mathrm{thick}(A)}.\) The main result in the present article relates these two categories:
``Theorem. Let \(R\) and \(R^\prime\) be MCM-representation finite complete Gorenstein \(k\)-algebras with Auslander algebras \(A=\mathrm{Aus}(\mathrm{MCM}(R))\) and \(A^\prime=\mathrm{Aus}(\mathrm{MCM}(R^\prime))\), respectively. Then the following statements are equivalent:
(i) There is an equivalence \(\underline{\mathrm{MCM}}(R)\cong\underline{\mathrm{MCM}}(R^\prime)\) of triangulated categories.
(ii) There is an equivalence \(\Delta_R(A)\cong\Delta_{R^\prime}(A^\prime)\) of triangulated categories.
The implication \((ii)\Rightarrow (i)\) holds more generally for non-commutative resolutions \(A\) and \(A^\prime\) of arbitrary isolated Gorenstein singularities \(R\) and \(R^\prime\) respectively.''
The authors use Knörrer's periodicity theorem to give nontrivial examples for (i) in the theorem.
The result is proved in a differential algebra framework. To every Hom-finte idempotent complete algebraic triangulated category \(\mathcal T\) with finitely many indecomposable objects satisfying conditions which holds for \(\mathcal T=\underline{\mathsf{MCM}}(R)\), there is an associated \textit{dg Auslander algebra} \(\Lambda_{dg}(\mathcal T)\), completely determined by \(\mathcal T\).
A recollement of categories is a collection of additive functors with certain properties, and recollements generated by idempotents, Koszul duality and the fractional Calabi-Yau property is used to prove the existence of an equivalence of triangulated categories \(\Delta_R(\mathrm{Aus}(\mathrm{MCM}(R)))\cong\mathrm{per}(\Lambda_{dg}(\underline{\mathrm{MCM}}(R))).\) This equivalence and its like, is then exploited to prove the theorem.
The article contains an appendix with a complete list of the graded quivers determining the dg Auslander algebras for ADE-singularities in all Krull dimensions.
The article shows a relation to generalized cluster categories and stable categories of special Cohen-Macaulay modules over complete rational singularities.
The article contains the necessary development of relative singularity categories, an introduction to derived categories, dg algebras and Koszul duality, complete path algebras and minimal resolutions. A nice section on the fractional Calabi-Yau property is given.
All in all, this is a very nice application of noncommutative algebraic geometry, and illustrates how the theory can, or even should, be used to obtain commutative results. isolated singularity; Gorenstein algebra; Gorenstein singularity; non-commutative resolution; singularity category; relative singularity category; perfect sheaves; dg Auslander algebra; ADE singularities; Dynkin diagrams; recollement; complete path algebra M. Kalck and D. Yang, 'Relative singularity categories I: Auslander resolutions', \textit{Adv. Math.}301 (2016) 973-1021. Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Derived categories, triangulated categories Relative singularity categories. I: Auslander resolutions. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This second edition (see [Zbl 1308.14001] for a review of the first edition) is extended by a chapter on recent developments.
After a short motivating first chapter that introduces the questions addressed in this book, follow two chapters dealing with background, one on sheaves, algebraic varieties and analytic spaces, and one on homological algebra and duality. The treatment includes spectral sequences. In these chapters the theorems are formulated in their natural generality. The definition of a singularity is initially given both for analytic spaces and for algebraic varieties over an arbitrary algebraically closed field. After stating Artin's Algebraization Theorem, that an isolated singularity of an analytic space is isomorphic to the germ of an algebraic variety over \(\mathbb{C}\), only the algebraic case is considered. After a chapter defining the canonical divisor for varieties over an arbitrary algebraically closed field the further discussion is restricted to the field of complex numbers.
The book defines log canonical, canonical, log terminal, terminal and rational singularities and provides a characterization of isolated such ones in terms of plurigenera. The classification is refined in the two-dimensional case, and rational surface singularities are described in some detail. Also the results of the Author on two-dimensional Du Bois singularities are introduced. The next chapter considers the analogous
questions for higher dimensional singularities, and in particular for the case of dimension three. It concludes with the list of the famous
95 families of simple \(K3\)-singularities.
The final chapter presents some developments after the publication of the first Japanese version [Zbl 1308.14002] of this book. These concern log discrepancies for pairs and the use of the space of arcs in their description. This opens up the possibility of proving results in positive characteristic. The book closes with a brief introduction to \(F\)-singularities, in positive characteristic. rational singularities; minimal model program; Du Bois singularities; arc spaces; F-singularities Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities in algebraic geometry, Deformations of singularities, Global theory and resolution of singularities (algebro-geometric aspects), Local complex singularities Introduction to singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Cubical hyperresolutions were introduced by V. Navarro Aznar and co-authors in their investigations on mixed Hodge structures [\textit{F Guillén} et al., Hyperrésolutions cubiques et descente cohomologique. (La plupart des exposés d'un séminaire sur la théorie de Hodge-Deligne, Barcelona (Spain), 1982). Berlin etc.: Springer-Verlag (1988; Zbl 0638.00011)]. In this theory, an important role is played by the notion of \textit{n-cubical variety}. This is a contravariant functor \({X}\bullet\) from \(C_{n-1}\) to the category of reduced algebraic schemes over a (fixed) field \(k\), where the objects of \(C_{n-1}\) are subsets \(I\) of \(\{0, 1, \ldots n-1 \}\) and \(\Hom (I,J)\) has one element if \(I \subset J\) and is empty otherwise, morphisms \(f:{ X} \bullet \to { S} \bullet\) are natural transformations of functors. They introduced a notion of \textit{weak resolution} \(f:{ X} \bullet \to { S} \bullet\) of a cubical variety \({ S} \bullet\): all the varieties \(X _I\) are smooth and certain inequalities involving dimensions must be satisfied. In [loc. cit.] a proof of the existence of weak resolutions (in characteristic zero) is presented. This result is used to prove that an algebraic variety \(X\) admits a \textit{cubic hyperresolution}. This means the existence of a \({n+1}\)-cubical variety \({ X} \bullet\), with \(X_{\emptyset}=X\) and \(X_I\) smooth for \(I \not= \emptyset\), satisfying certain conditions. One of them is: \(X \bullet\) is of \textit{cohomological descent}, which means that a certain complex of sheaves of abelian groups associated to \(X \bullet\) is acyclic.
In the present paper the author gives an example of a 1-cubical variety that does not admit a weak resolution. He proposes a slight modification of the definition, called a \textit{very weak resolution}. He shows that with minor changes the arguments of [loc. cit.] produce a correct proof of the fact that a cubical variety admits a very weak resolution, and that this leads to a demonstration of a small variation of the hyperresolution theorem. \(n\)-cubical variety; cohomological descent; weak resolution; very weak resolution; hyperresolution Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Foundations of algebraic geometry, Homological methods in commutative ring theory Some remarks on hyperresolutions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider versal deformation of a reflexive module over a rational surface singularity. Especially, for a rational double point, we determine the closure relation of the stratification of the mini-versal deformation space of a reflexive module with respect to isomorphism classes. We also determine the singularity of the closure of a minimal stratum. As a reusult, we obtain singular spaces together with their desingularizations which are similar to quiver varieties of Dynkin type. For singularities of type \(A\), we obtain the orbit stratifications and desingularizations of the nilpotent varieties and their subvarieties. The existence of an algebraic versal deformation of an isolated singularity was proved by \textit{R. Elkik} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 6, 553-603 (1973; Zbl 0327.14001)] and the same arguments can be applied to our case. In section 2, we define two kinds of deformation functors \(\text{Def}_E\) and \(\text{Def}_E'\) of a module \(E\) over a local ring \({\mathcal O}\) and give a proof of existence of a versal deformation for each of them by following \textit{M. Artin} [Invent. Math. 27, 165-189 (1974; Zbl 0317.14001)] and \textit{R. Elkik} (loc. cit). We will see in section 4 that the miniversal deformation spaces for the two functors have the same reduced parts (but different scheme structures in general) in the case where \({\mathcal O}\) is a rational surface singularity. versal deformation; rational surface singularity; desingularizations; quiver varieties; Dynkin type Ishii, A., Versal deformation of reflexive modules over rational double points, Math. Ann., 317, 2, 239-262, (2000), MR 1764236 Singularities of surfaces or higher-dimensional varieties, Formal methods and deformations in algebraic geometry, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Versal deformation of reflexive modules over rational double points | 0 |
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