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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 In 2000, a theorem of \textit{M. Levine} and \textit{F. Morel} [C. R. Acad. Sci., Paris 332, 723--728 (2001; Zbl 0991.19001)] stated that algebraic cobordism groups are isomorphic to (multiplicative) Grothendieck groups over smooth schemes. In this paper the author extended this theorem to singular schemes. As a consequence, the author provides a new proof of the singular Riemann-Roch theorem of Baum-Fulton-MacPherson and a new type of Riemann-Roch theorem with respect to pullbacks of locally complete morphisms. algebraic cobordism; Grothendieck groups; singuler schemes DOI: 10.4310/HHA.2010.v12.n1.a8 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Riemann-Roch theorems, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) Algebraic cobordism and Grothendieck groups over singular 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 The main result of this paper is:
Consider a fibration \(f: X\to C\) from a nonsingular threefold to a curve. We suppose that a general fiber of \(f\) is a normal surface. Then the following hold:
There does not appear a simple elliptic singularity on a general fiber if \(p\geq 5\).
Under the assumption that the anticanonical divisor of a fiber is ample, a general fiber is non singular if \(p\geq 11\), i.e. it is a del Pezzo surface.
Under the assumption that the general fiber has a trivial dualizing sheaf and has only rational singularities, it is non singular if \(p\geq 23\), i.e; it is either an abelian surface or a \(K3\) surface.
The main observation is that, if the general fiber has singularities isomorphic to hypersurface singularities, the dimension of the space \(T^1\) of first order infinitesimal deformations of the singularity is divisible by \(p\). The proof consist to check this last property for rational double points and simple elliptic singularities. rational singularities; hypersurfaces singularities Masayuki Hirokado, Deformations of rational double points and simple elliptic singularities in characteristic \?, Osaka J. Math. 41 (2004), no. 3, 605 -- 616. Deformations of singularities Deformations of rational double points and simple elliptic singularities in characteristic \(p\) | 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 \(K3\) surfaces, the class of surfaces which are simply connected and that have trivial canonical bundle, were intensively studies over the last 40 years. One of the earliest results is due to Piatetski-Shapiro (and Shafarevich), who proved that their period map is injective. However, there still open questions regarding the geometry of algebraic, projective \(K3\) surfaces.
In this note we present a new restriction on the position of the singularities of branch curves of projective \(K3\) surfaces. Using this restriction, we are able to find a new Zariski pair, when one of the curves is a branch curve of a projective \(K3\) of genus 4 surface, embedded in \(\mathbb{P}^4\), and the other is not.
Reformulating former results on branch curves of projective \(K3\) surfaces, we pose a conjecture about the reducibility of certain varieties of nodal-cuspidal curves, when one of its components is the component of branch curve of a projective \(K3\) of genus \(g\). \(K3\) surfaces and Enriques surfaces, Plane and space curves, Coverings in algebraic geometry, Families, moduli of curves (algebraic) On the singularities of branch curves of \(K3\) surfaces and applications | 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 Dynkin diagram; simple Lie groups; simple singularities; representations of Weyl groups; monodromy transformations Slodowy, P., Four lectures on simple groups and singularities, \textit{Communications of the Math. Institute}, (1980) Semisimple Lie groups and their representations, Research exposition (monographs, survey articles) pertaining to topological groups, Singularities in algebraic geometry, Local complex singularities, Modifications; resolution of singularities (complex-analytic aspects) Four lectures on simple groups and 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 \(k\) be an algebraically closed field of characteristic 0. Let \(G\) be a reductive algebraic linear group over \(k\) with \(\text{Lie} (G)\) simple and let \(\rho: G \to GL(V)\) be a finite dimensional rational representation. The classification of all pairs \((G, V)\) such that the quotient \(V// G\) has an isolated singularity is given. It is also shown that, for pairs \((G, V)\) with this property, the algebra of invariants \(k[V]^G\) has a natural \(\mathbb Z_+\)-grading such that \(\operatorname {Proj}k[V]^G\) is smooth and rational. quotient; isolated singularity; algebra of invariants Group actions on varieties or schemes (quotients), Actions of groups on commutative rings; invariant theory, Linear algebraic groups over arbitrary fields, Geometric invariant theory On quotient-varieties with an isolated singularity | 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 deals with (skew-symmetric) monodromy groups for boundary critical points. In particular, the symplectic variants of Weyl groups \(B_{\mu}\), \(C_{\mu}\), \(F_ 4\), \(G_ 2\) are described, also, the vanishing cycles for \(B_{\mu}\), \(C_{\mu}\). Relations with monodromy groups of usual singularities are given. skew-symmetric monodromy groups for boundary critical points; vanishing cycles ILYUTA (G.G.) . - Monodromy and vanishing cycles of boundary singularities , Functional analysis, t. 19, p. 173-182. MR 87c:32010 | Zbl 0591.32011 Local complex singularities, Singularities in algebraic geometry Monodromy and vanishing cycles of boundary 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 discuss Hironaka's theorem on resolution of singularities in charactetistic 0 as well as more recent progress, both on simplifying and improving Hironaka's method of proof and on new results and directions on families of varieties, leading to joint work on toroidal orbifolds with Michael Temkin and Jarosław Włodarczyk. Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Generalizations (algebraic spaces, stacks), Research exposition (monographs, survey articles) pertaining to algebraic geometry Resolution of singularities of complex algebraic varieties and their families | 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 local model for Hilbert-Siegel moduli schemes with \(\Gamma_1(p)\)-level structures, when \(p\) is unramified in the totally real field. Our key tool is a variant of the ring-equivariant Lie complex defined by Illusie. variété de Shimura; modèle local; complexe cotangent Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties Local model of Hilbert-Siegel moduli schemes with \(\Gamma_1(p)\)-level structures | 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 algebraic variety with an effective \(\mathbb{R}\)-divisor \(\Delta\). Without assuming that \(K_X + \Delta\) is \(\mathbb{R}\)-Cartier, the author of the paper under review defines a discrepancy \(\alpha(P,X,\Delta)\), where \(P\) is any prime divisor, and the notion of \emph{pseudo-lc} singularity thereof (see Definitions 4.1 and 4.2 in the text). Moreover, as the main result he proves (Theorem 1.1 of the paper) that there exists a projective birational morphism \(h: W \longrightarrow X\) such that every \(h\)-exceptional prime divisor \(E_h\) satisfies \(\alpha(E_h,X,\Delta) < -1\), the reduced \(h\)-exceptional divisor \(E_{\text{red}}\) is \(\mathbb{Q}\)-Cartier and the pair \((W, h_*^{-1}\Delta + E_{\text{red}})\) is lc (in the usual sense). In particular, if \(K_X + \Delta\) was pseudo-lc, then the above morphism \(h\) can be taken to be small (so that \(h\) is a \emph{lc modification} in the sense of Definition 18 in [\textit{J. Kollár}, Peking Math. J. 1, No. 1, 81--107 (2018; Zbl 1439.14035)]). As a further analogy with the lc case, the author proves (Theorem 1.4) that the ring \(\bigoplus_{m \ge 0} H^0(X,\mathcal{O}_X(\llcorner m(K_X + \Delta)\lrcorner))\) is finitely generated, and that Kodaira type vanishing theorem (Theorem 1.5) holds for a Weil divisor \(D\) on \(X\) such that \(D - (K_X + \Delta)\) is ample. Note however that the notions of pseudo-lc and lc pairs coincide only for surfaces (see Corollary 4.16 and Examples 4.10, 4.11). The author also gives a general criterion for a pseudo-lc pair to be lc (see Theorem 1.7). The latter is closely related with the properties of \emph{numerically Cartier divisors} (cf. the proof of Corollary 5.2 and [\textit{S. Boucksom} et al., London Math. Soc. Lect. Note Ser. 417, 29--51 (2014; Zbl 1318.14002)]). All proofs are based on a skillful application of MMP with scaling. singularity of pairs; log canonicalization; log canonical criterion Singularities of surfaces or higher-dimensional varieties, Minimal model program (Mori theory, extremal rays) A class of singularity of arbitrary pairs and log canonicalizations | 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 smooth scheme \(X\) over a field \(k\) of positive characteristic is said to be strongly liftable if \(X\) and all prime divisors on \(X\) can be lifted simultaneously to \(W_2(k)\), the ring of Witt vectors of length two of \(k\). The author continues the study of strongly liftable schemes introduced in [Math.\ Res.\ Lett.\ 17, No. 3, 563--572 (2010; Zbl 1223.14026)] in connection with the study of the Kawamata-Viehweg vanishing theorem in positive characteristic. First he shows that any smooth toric variety is strongly liftable. As a corollary he obtains the Kawamata-Viehweg vanishing theorem for smooth projective toric varieties. Second, he proves the Kawamata-Viehweg vanishing theorem for normal projective surfaces birational to a strongly liftable smooth projective surface, with no singularity assumption, extending the corresponding result obtained in the previous paper. Finally, the author deduces the cyclic cover trick over \(W_2(k)\) and uses it to study the behavior of cyclic covers over strongly liftable schemes. Furthermore, the author takes the opportunity to correct some statements in the previous paper. positive characteristic; strongly liftable scheme; Kawamata-Viehweg vanishing theorem Xie, Q. H., Strongly liftable schemes and the Kawamata-Viehweg vanishing in positive characteristic II, Math. Res. Lett, 18, 315-328, (2011) Positive characteristic ground fields in algebraic geometry, Vanishing theorems in algebraic geometry, Arithmetic ground fields for curves, Coverings in algebraic geometry Strongly liftable schemes and the Kawamata-Viehweg vanishing in positive characteristic. II | 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 At its most basic, mirror symmetry is a local isomorphism called the mirror map between the complex moduli on the \(B\)-model and the Kähler moduli on the \(A\)-model. The complex moduli space parametrizes deformations of complex structure (of the Calabi-Yau manifold under consideration). The Kähler moduli parametrizes the complexified Kähler classes of the mirror.
This local isomorphism was originally defined near special points called the large complex structure limits, which can be thought of as having monodromy that is maximally unipotent. Locally, each such degeneration corresponds to the complexified Kähler cone of the mirror to the degenerating family. Conjecturally, the Kähler cones glue together to form the stringy Kähler moduli space, which should be globally isomorphic to the entire complex moduli.
On both sides, there are additional special points, such as the Geepner and conifold points. Different correspondences refer to equivalences of invariants defined for various couples of special points and the term \textit{global mirror symmetry} introduced by \textit{A. Chiodo} and \textit{Y. Ruan} [Invent. Math. 182, No. 1, 117--165 (2010; Zbl 1197.14043)] refers to having a unified picture, i.e., a global isomorphism on the entire complex and stringy Kähler moduli spaces.
The paper under review is the 4th in a series of papers by various combinations of the authors and Kravitz and Ruan, where global mirror symmetry for simple elliptic singularities is investigated.
Start with an invertible simple elliptic singularity (ISES) \(W\). Simple elliptic singularity means that \(W=0\) is the germ of a singularity such that the exceptional divisor of its minimal resolution is an elliptic curve. Such a singularity can always be given by a quasi-homogeneous non-degenerate polynomial \(W\). Invertible means that the exponent matrix of \(W\) is invertible. The classification of such ISES is given in Table 1.1 of the paper and they correspond to Dynkin diagrams of type \(E_6\), \(E_7\) and \(E_8\).
To such a \(W\), consider its miniversal deformation space \(\mathcal{S}\). This is the (local) complex moduli investigated. The authors consider special limit points \(\sigma\in\mathcal{S}\), namely those for which the corresponding deformation of \(W\) no longer has an isolated critical point at \(0\). They explain how to associate to such \(\sigma\) the Saito-Givental ancestor potential \(\mathcal{A}_W^{SG}(\sigma)\).
The global mirror symmetry expectation is introduced in Conjecture 1.3, which states that the mirror of special limit points in \(\mathcal{S}\) should be mirror to either of the two following: {
- The Fan-Jarvis-Ruan-Witten (FJRW) theory of a simple elliptic singularity.
- The Gromov-Witten (GW) theory of an elliptic orbifold \(\mathbb{P}^1\).
} The mirror relationship is described as an equality between the corresponding ancestor potentials. The authors prove that Conjecture 1.3 holds for Geepner points (Theorem 1.4) and for Fermat simple elliptic singularities (Theorem 1.5).
The paper is overall well-written and includes a good amount of background information. It is instructive to see the full picture of global mirror symmetry unfolding. mirror symmetry; global mirror symmetry; simple elliptic singularities; Saito-Givental theory; primitive forms; Gromov-Witten theory; FJRW theory; quasi-modular forms T. Milanov and Y. Shen, Global mirror symmetry for invertible simple elliptic singularities, preprint (2014), . Mirror symmetry (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Singularities in algebraic geometry, Deformations of singularities Global mirror symmetry for invertible simple elliptic 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 long series of remarks about \(K\)-algebras, where \(K\) is an (usually) algebraically closed field of characteristic zero. These remarks relate to the weak nullstellensatz (\(V(I)\) is nonempty for every proper ideal \(I\) of \(K[X_1, \dots, X_n]\)) and use several problems from \textit{W. Fulton}'s 1969 book on algebraic curves [Algebraic curves. New York-Amsterdam: W.A. Benjamin, Inc. (1969; Zbl 0181.23901)] and elementary notions from functional analysis. zeros of polynomials; spectrum; Gelfand-Mazur Theorem; ideals; characters Polynomials over commutative rings, Extension theory of commutative rings, Relevant commutative algebra, General theory of commutative topological algebras, Ideals, maximal ideals, boundaries, Structure and classification of commutative topological algebras Some remarks on Hilbert's (weak) nullstellensatz | 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 extend the vanishing theorem and the Clifford theorem in the following way:
Let \(R\) be a compact genus \(g\) Riemann surface and \(E\) a simple vector bundle of \(\text{rank} r > 1\) on \(R\).
(1) If \(\deg (E) \geq (3r - 1) (g - 1)\), then \(H^ 1 (R,E) = 0\).
(2) If \(\deg (E) \leq (r - 1) (g - 1)\), then \(H^ 0 (R,E) = 0\).
(3) If both \(H^ 1 (R,E)\) and \(H^ 0 (R,E) \neq 0\) and \(p \geq 3\), then
\[
\dim H^ 0 (R,E) \leq \deg (E)/2 - (r - 1) (g - 1)/2 + 1/2.
\]
(4) The above inequalities are the best possible. vanishing theorem; Clifford theorem; Riemann surface Vanishing theorems in algebraic geometry, Riemann surfaces; Weierstrass points; gap sequences, Classification theory of Riemann surfaces, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Vanishing theorem and Clifford theorem for simple vector bundles | 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 present paper, the authors study relations among several numerical invariants associated to a (free) projective hypersurface \(V\), namely the sequence of mixed multiplicities of its Jacobian ideal, the Hilbert polynomial of its Milnor algebra, and the sequence of the exponents when \(V\) is free. Let us recall some basic notions. For a polynomial \(f \in S = \mathbb{C}[x_{0}, \ldots,x_{n}]\) of degree \(d\) the corresponding Jacobian ideal \(J_{f}\) is generated by the partial derivatives \(f_{j}\) of \(f\) with respect to \(x_{j}\), and the graded Milnor (or Jacobian) algebra is defined as \(M(f) = S/J_{f}\). The Hilbert function \(H(M(f)) : \mathbb{N} \rightarrow \mathbb{N}\) of the graded \(S\)-module \(M(f)\) is defined as
\[
H(M(f))(k) = \dim M(f)_{k}.
\]
It is known that there is a unique \(P(M(f))(t) \in \mathbb{Q}[t]\), called the Hilbert polynomial of \(M(f)\), and an integer \(k_{0} \in \mathbb{N}\) such that
\[
H(M(f))(k) = P(M(f))(k)
\]
for all \(k \geq k_{0}\). Now we would like to recall the notion of mixed multiplicities \(\mu^{i}(f)\) of \(J_{f}\) and \(\mathbf{m}\) the maximal homogeneous ideal \((x_{0}, \ldots , x_{n})\); these can be viewed as the coefficients of the (homogeneous) Hilbert polynomial \(P(R(\mathbf{m} | J_{f})) \in \mathbb{Q}[a,b]\) of the standard bigraded algebra
\[
R(\mathbf{m}|J_{f}) = \sum_{(a,b) \in \mathbb{N}^{2}} \mathbf{m}^{a}J_{f}^{b}/\mathbf{m}^{a+1}J_{f}^{b}.
\]
The main result of the paper is the following.
Theorem. Let \(V = V(f) : f=0\) be a free hyperplane arrangement in \(\mathbb{P}^{n}\) and denote by \(D(f)\) the complement \(P^{n} \setminus V\). Then each of the following data determines the others: {\parindent=0.7cm\begin{itemize}\item[--] The exponents \(d_{1} \leq d_{2} \leq \ldots \leq d_{n}\). \item[--] The sequence \(b_{i}(D(f))\) of Betti numbers of the complement \(D(f)\), encoded in the Poincaré polynomial
\[
P(D(f);t) = \sum_{i} b_{i}(D(f))t^{n-i}.
\]
\item[--] The sequence \(\mu^{*}(f)\) of mixed multiplicities. \item[--] The Hilbert function \(H(M(f))\) and the degree \(d\) of \(f\). \item The Hilbert polynomial \(P(M(F))\) and the degree \(d\) of \(f\).
\end{itemize}}
In Section 4, the authors investigate free and nearly free surfaces in \(\mathbb{P}^{3}\) from the persepctive of the so-called homaloidal surfaces. Let us recall that a projective hypersurface \(V : f = 0\) is said to be homaloidal if the degree of the gradient mapping
\[
\Phi_{f} = \text{grad}(f) : \mathbb{P}^{n} \dasharrow \mathbb{P}^{n}, x \mapsto(f_{0}(x) : \ldots : f_{n}(x))
\]
is equal to one. The main result of this section is Corollary 4.7, where the authors present an explicit list of irreducible homaloidal surfaces, depending on the degree \(d\), and emphasizing in each case whether the associated surface is free or nearly free. This allows to predict that there might exist a relation between free and nearly free surfaces and homaloidal surfaces in \(\mathbb{P}^{3}\). mixed multiplicities; Jacobian ideal; Hilbert polynomial; homaloidal surfaces Birational automorphisms, Cremona group and generalizations, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Hypersurfaces and algebraic geometry, Relations with arrangements of hyperplanes Mixed multiplicities, Hilbert polynomials and homaloidal 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 \(X\) be a \(K3\) surface and consider a fine moduli space of sheaves \(M\) on \(X\). One might hope that moduli spaces of sheaves on \(M\) inherit good properties from \(M\) itself. In this paper, the authors show that under certain conditions this does happen as a connected component of specified moduli spaces of sheaves on \(M\) are isomorphic to \(X\) itself.
The first obstacle to solving this problem is providing stable sheaves on \(M\). Assuming that \(M\) has a universal family, one can restrict the universal family to the fibers over a point \(p \in X\) in the product \(X\times M\). The authors show that in some cases, these ``wrong way'' fibers are stable sheaves on \(M\). This paper's main result is the construction of an explicit moduli space \(M\) on a \(K3\) surface of Picard rank 1 or 2 (with a certain specified intersection matrix in both cases) with a universal family such that the ``wrong way'' fibers identify a connected component of a moduli spaces of sheaves on \(M\) with \(X\) itself.
In order to prove this, the authors generalize a technique to prove stability first given by \textit{D. Stapleton} [Algebra Number Theory 10, No. 6, 1173--1190 (2016; Zbl 1359.14040)] and utilize a theorem of \textit{N. Addington} [Algebr. Geom. 3, No. 2, 223--260 (2016; Zbl 1372.14009)] to give the isomorphism. stable sheaves; moduli spaces; universal families; Hilbert schemes Algebraic moduli problems, moduli of vector bundles, \(K3\) surfaces and Enriques surfaces, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Stability of some vector bundles on Hilbert schemes of points on \(K3\) 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 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 This is an expository paper describing the problem of resolution of singularities over an algebraically closed field of arbitrary characteristic. It begins with a description of the history of the problem, briefly discussing work of Hironaka, Abhyankar and especially Eklof for the resolution of singularities of algebraic varieties. The author then turns to the resolution of singularities for algebroidal varieties \(X=\text{Spec}\bigl(K[[Z_ 1,\cdots,Z_ n]]/I\bigr)\), with \(I\) radical and \(K\) an algebraically closed field of arbitrary characteristic. She first deals with the case of hypersurfaces, and then turns to the general case.
The paper is an interesting overview of the work which the author and others have done, but unfortunately there are no references given for the interested reader to look up the details. resolution of singularities; algebroidal varieties Global theory and resolution of singularities (algebro-geometric aspects), History of mathematics in the 20th century, History of algebraic geometry The resolution of singularities in arbitrary 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 We determine the Galois representations inside the \(\ell\)-adic cohomology of some quaternionic and related unitary Shimura varieties at ramified places. The main results generalize the previous works of Reimann and Kottwitz in this setting to arbitrary levels at \(p\), and confirm the expected description of the cohomology due to Langlands and Kottwitz. Shimura varieties; quaternion algebras; cohomology Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties, \(p\)-adic cohomology, crystalline cohomology On the cohomology of some simple Shimura varieties with bad reduction | 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 good resolution of an isolated singularity \(0\) of an algebraic (or analytic) variety \(X\) is one where the exceptional locus is a divisor \(Z\) with simple normal crossings. Such a good resolution is available in characteristic zero. Moreover, one can find one where all the intersections involving components of \(Z\) are irreducible (property I). To a good resolution of an isolated singularity \(x \in X\) one may associate a CW-complex \(\Gamma (Z)\) of dimension \(\leq \dim X - 1\), the dual complex associated to the resolution. Its homotopy type is independent of the chosen good resolution. If this satisfies condition I then \(\Gamma (Z)\) is a simplicial complex. Working over the complex numbers, in this article the author proves the following results about this object.
Theorem 1. If \(0 \in X\) is an isolated rational singularity (i.e., for a resolution \(f:Y \to X\), \({R^i}_{\star} {\mathcal O}_{Y}=0, ~ i>0\)), \(\dim X = n\), then \(H^{n-1}({\Gamma}(Z), C)=0\).
Theorem 2. If \(0\) is an isolated singularity of a hypersurface the \({\Gamma}(Z)\) has the homotopy type of a point.
As a corollary of these results it is proved that \(\Gamma (Z) \) is homotopically equivalent to a point under any one of the following assumptions: either \(0 \in X\) is an isolated rational singularity of dimension 3. or \(0 \in X\) is a 3-dimensional Gorenstein terminal singularity. The proof of Theorem 1 generalizes an argument of \textit{M. Artin}'s seminal paper on rational singularities of surfaces [Am. J. Math. 88, 129--136 (1966; Zbl 0142.18602)], but also involves a technical lemma on the degeneracy of a spectral sequence associated to a divisor with normal crossings in a Kähler manifold. The second theorem cleverly uses the fact that the link of an isolated hypersurface singularity of dimension at least three is simply connected. The paper is well written and has good bibliographical references. rational singularity; hypersurface singularity; good resolution of singularities; dual complex; cohomology; fundamental group Stepanov, D.A.: A note on resolution of rational and hypersurface singularities. Proc. Am. Math. Soc. \textbf{136}(8), 2647-2654 (2008) Singularities in algebraic geometry, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Local theory in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) A note on resolution of rational and hypersurface 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 \(S^{n}(X)\) be the \(n\)-fold symmetric product of a compact connected Riemann surface \(X\) of genus \(g\) and gonality \(d\). We prove that \(S^{n}(X)\) admits a Kähler structure such that all the holomorphic bisectional curvatures are nonpositive if and only if \(n<d\). Let \(\mathcal{Q}_{X}(r,n)\) be the Quot scheme parametrizing the torsion quotients of \(\mathcal{O}^{\oplus r}_{X}\) of degree \(n\). If \(g\geq 2\) and \(n\leq 2g-2\), we prove that \(\mathcal{Q}_{X}(r,n)\) does not admit a Kähler structure such that all the holomorphic bisectional curvatures are nonnegative. Biswas, I.; Seshadri, H., On the Kähler structures over quot schemes, Ill. J. math., 57, 1019-1024, (2013) Divisors, linear systems, invertible sheaves, Negative curvature complex manifolds, Positive curvature complex manifolds, Parametrization (Chow and Hilbert schemes) On the Kähler structures over Quot 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 We determine conditions on \(q\) for the nonexistence of deep holes of the standard Reed-Solomon code of dimension \(k\) over \(\mathbb F_q\) generated by polynomials of degree \(k+d\). Our conditions rely on the existence of \(q\)-rational points with nonzero, pairwise-distinct coordinates of a certain family of hypersurfaces defined over \(\mathbb F_q\). We show that the hypersurfaces under consideration are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of these hypersurfaces, from which the existence of \(q\)-rational points is established. finite fields; Reed-Solomon codes; deep holes; symmetric polynomials; singular hypersurfaces; rational points Cafure, A.; Matera, G.; Privitelli, M.: Singularities of symmetric hypersurfaces and Reed-Solomon codes. Adv. math. Commun. 6, No. 1, 69-94 (2012) Applications to coding theory and cryptography of arithmetic geometry, Symmetric functions and generalizations, Algebraic coding theory; cryptography (number-theoretic aspects), Combinatorial codes Singularities of symmetric hypersurfaces and Reed-Solomon codes | 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 \textit{F. Loray} et al. [Invent. Math. 213, No. 3, 1327--1380 (2018; Zbl 1426.32014)] classified codimension-one foliations \(\mathcal F\) and numerically trivial canonical class on a complex projective manifold \(X\). In particular, they proved that if the singularities of \(\mathcal F\) are canonical then after passing to an étale cover \(X\) becomes a product \(Y\times Z\) of projective manifolds with \(K_Y=0\) and \(\mathcal F\) becomes the pullback of a foliation \(\mathcal G\) on \(Z\) with trivial canonical class. They also gave a precise description of \(Z\) and \(\mathcal G\) in case the general leaf of \(\mathcal F\) is not algebraic.
The main aim of the paper under review is to extend these results to the case when \(X\) is a normal complex projective variety with canonical singularities. This is achieved under further assumption that \(K_{\mathcal F}\) is torsion. The author shows also that numerical triviality of \(K_{\mathcal F}\) implies that \(K_{\mathcal F}\) is torsion in case \(X\) has terminal singularities or \(K_{\mathcal F}\) is Cartier. The general strategy of proofs is motivated by the proofs in [loc. cit.].
In the subsequent paper with W. Ou, the author extends these results to the case when \(X\) has only klt singularities (see [\textit{S. Druel} and \textit{W. Ou}, ``Codimension one foliations with trivial canonical class on singular spaces. II'', Preprint, \url{arXiv:1912.07727}]). foliations; complex projective varieties; canonical singularities; trivial canonical class Singularities in algebraic geometry, Singularities of holomorphic vector fields and foliations, Dynamical aspects of holomorphic foliations and vector fields Codimension \(1\) foliations with numerically trivial canonical class on singular 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 If \((X,p)\) is an isolated Gorenstein threefold singularity admitting a small resolution \(\pi:Y\to X\), i.e. a resolution such that the exceptional set is the union of smooth rational curves, then we know that \((X,p)\) is locally given by the equation \(f(x,y,z)+tg(x,y,z,t)\) where \(f\) defines a rational double point \((S,p)\), and we say that \((X,p)\) is a \(cDV\).
In this article the author studies the converse question of which \(cDV\) singularities admit small resolutions, when \(f\) defines an \(A_ n\) or a \(D_ n\) surface singularity. --- To obtain his results, the author studies the versal deformation of the surfaces \(S\), \(\overline S=\pi^{- 1}(S)\) which is also a surface with only \(RDP\) singularities, and \(S\tilde S\) where \(\lambda:\tilde S\to\overline S\) is the minimal resolution of \(\overline S\). We denote respectively by \({\mathcal B}\), \(\overline{{\mathcal B}}\) and \(\tilde {\mathcal B}\) and \({\mathcal S}\), \(\overline{{\mathcal S}}\) and \(\tilde{\mathcal S}\) the versal deformation spaces and the versal families of \(S\), \(\overline S\) and \(\tilde S\), and we get:
\[
\begin{matrix} \widetilde{\mathcal S} & \longrightarrow & \overline{{\mathcal S}} & \longrightarrow & \strut{\mathcal S} \\ @VVV @VVV @VVV \\ \widetilde{\mathcal B} & \longrightarrow & \overline{{\mathcal B}} & \longrightarrow & \strut{\mathcal B}. \end{matrix}
\]
The \(cDV\) singularity \((X,p)\) is represented by a map \(\nu:D\to{\mathcal B}\), where \(D\) is a smooth curve and the small resolution is represented by lifting \(\nu\) to \(\overline\nu:D\to\overline{{\mathcal B}}\), such that the space \(Y\) given by \(\overline\nu\) is smooth. The author analyses the lifting criterion explicitly in the \(cA_ n\) and \(cD_ n\) cases in terms of the equation of \((X,p)\) and he obtains:
Theorem: If \((X,p)\) is an isolated \(cA_ n\) singularity admitting a small resolution \(Y\to X\), then the exceptional curve in \(Y\) is a chain of \(n\) rational curves, i.e. the surface \(\overline S\) is smooth, and \(X\) has the form \(xy+g(z,t)\), where \(g(z,t)\) has \(n+1\) distinct branches at the origin. --- Conversely, any \(X\) as above admits a small resolution.
Let \((X,p)\) be an isolated \(cD_ n\) singularity given by \(\nu:D\to{\mathcal B}\), where \({\mathcal B}\) is \(k^ n({\mathbf t})=k^ n(t_ 1,\dots,t_ n)\) and \({\mathcal S}\to{\mathcal B}\) is given by: \({\mathcal S}=\{(x,y,z,{\mathbf t})\mid f(x,y,z,{\mathbf t})=x^ 2+y^ 2z-t_ 1z^{n-2}-\cdots-t_{n-1}+2t_ ny=0\}\). If \(t\) is a local parameter on the smooth curve \(D\), we view the \(t_ i\) as analytic functions of \(t\), and we associate to \(f\) the function \(F(z,t)=z^ n+t_ 1z^{n-1}+\cdots+t_{n-1}z+t^ 2_ n\).
Theorem: The \(cD_ n\) singularity admits a small resolution \(Y\to X\) such that the surface \(\overline S\) is smooth if and only if \(F(z,t)\) factors into \(n\) distinct factors, each tangent to \(z=0\) with even multiplicity.
The author studies more precisely the case of a \(cD_ 4\) singularity, he analyses the function \(F(z,t)\) and finds all the singularities that the surface \(\overline S\) could admit. \(cDV\) singularity; Gorenstein threefold singularity; small resolution; versal deformation spaces Katz, S.: Small resolutions of Gorenstein threefold singularities. Algebraic geometry: Sundance 1988, pp. 61-70, Contemp. Math., vol. 116, Am. Math. Soc., Providence, RI (1991) Global theory and resolution of singularities (algebro-geometric aspects), \(3\)-folds, Formal methods and deformations in algebraic geometry, Deformations of singularities, Singularities in algebraic geometry Small resolutions of Gorenstein threefold 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 M. Reid gave a criterion saying which cyclic quotient singularities are terminal singularities. - The authors use a combinatorial lemma - due to G. K. White and reproved in this note - to interpret this criterion in certain cases and they obtain thereby a complete description of isolated terminal cyclic quotient singularities in dimension three and of isolated Gorenstein terminal cyclic quotient singularities in dimension four. canonical singularities; Bernoulli functions; Gorenstein singularities; cyclic quotient singularities; terminal singularities D. R. Morrison and G. Stevens, ''Terminal quotient singularities in dimensions three and four,''Proc. Amer. Math. Soc.,90, No. 1, 15--20 (1984). Singularities in algebraic geometry, Local complex singularities, Group actions on varieties or schemes (quotients), Fibonacci and Lucas numbers and polynomials and generalizations Terminal quotient singularities in dimensions three and four | 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 0619.00007.]
In the first part of this paper the authors prove that if Z is the zero scheme of a section of a rank 2 vector bundle E on an affine scheme X then the dual bundle \(E^*\) admits a section with the same zero scheme Z if one assumes that \(\det(E/Z)\) is trivial. - This theorem is further used to derive results on the minimal number of equations for the set theoretical description of closed subschemes of an affine scheme. For hypersurfaces the authors prove: Let \(Y\subset X=Spec(R)\) be a subscheme defined by a finitely generated locally principal ideal \({\mathcal I}\) of R such that \({\mathcal I}/{\mathcal I}^ 2\) is generated by m elements (m\(\geq 2)\). Then Y can be set theoretically defined by m functions.
For arbitrary codimension: Y defined by a finitely generated ideal \({\mathcal I}\subset {\mathcal R}\) can be set theoretically defined by \(n=\dim (X)\) functions if one assumes that \(Y_ k\), the set of all points \(y\in Y\) such that \({\mathcal I}_ y\) requires at least k generators, has dimension at most n-k for all k, \(1\leq k\leq n-1\) and if \(Y_ n=\emptyset\). In particular the authors prove that if \(Y\subset Spec(R)\) (R an n- dimensional noetherian ring) is a locally complete intersection without zero-dimensional components then Y can be set theoretically defined by n equations. zero scheme of a section of a rank 2 vector bundle; minimal number of equations; set theoretically defined by n equations Complete intersections, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] A theorem on zero schemes of sections in two-bundles over affine schemes with applications to set theoretic intersections | 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 prove that the Hodge-de Rham spectral sequence for smooth proper tame Artin stacks in characteristic \(p\) (as defined by Abramovich, Olsson, and Vistoli [\textit{D. Abramovich} et al., Ann. Inst. Fourier 58, No. 4, 1057--1091 (2008; Zbl 1222.14004); corrigendum ibid. 64, No. 3, 945--946 (2014)]) which lift mod \(p^{ 2 }\) degenerates. We push the result to the coarse spaces of such stacks, thereby obtaining a degeneracy result for schemes which are étale locally the quotient of a smooth scheme by a finite linearly reductive group scheme. de Rham; Hodge; tame stack; linearly reductive Satriano, M.: de Rham Theory for tame stacks and schemes with linearly reductive singularities. Ann. l'Institut Fourier. http://arxiv.org/abs/0911.2056 (2012, to appear) Generalizations (algebraic spaces, stacks), de Rham cohomology and algebraic geometry, Variation of Hodge structures (algebro-geometric aspects), Stacks and moduli problems, \(p\)-adic cohomology, crystalline cohomology de Rham theory for tame stacks and schemes with linearly reductive 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 \(X\) be a nonsingular surface over complex numbers and \(S_n\) be the symmetric group on \(n\) letters. Denote the Hilbert scheme of \(n\) points on \(X\) by \(X^{[n]}\). By a result of Haiman \(X^{[n]}\) can be identified with the fine moduli space of \(S_n\)-clusters in \(X^n\). If we denote by \(\mathcal Z \subset X^{[n]}\times X^n\) the universal family of \(S_n\)-clusters and \(X^{[n]}\overset{q}{\leftarrow} \mathcal Z \overset{p}\rightarrow X^n\) be the projections then, the derived McKay correspondence of \textit{T. Bridgeland} et al. [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] in this set up gives an equivalence of derived categories \(\Phi: D(X^{[n]})\overset{\sim}{\rightarrow} D_{S_n}(X^n)\) of (\(S_n\)-equivariant) coherent sheaves, where \(\Phi=Rp_*\circ q^*\).
Scala showed that for any vector bundle \(F\) on \(X\), the image of the tautological bundle \(F^{[n]}\) on \(X^{[n]}\) under \(\Phi\) is given by an explicit complex \(\mathsf C^\bullet_F\) of (\(S_n\)-equivariant) coherent sheaves concentrated in nonnegative degrees.
The paper under review studies the derived McKay correspondence above in the reverse order by means of \(\Psi=q_*^{S_n}\circ Lp^*\) (which is not the inverse of \(\Phi\)). The main result of the paper is that if one replaces \(\Phi\) by \(\Psi^{-1}\) the images of \(F^{[n]}\) and \(\bigwedge^k L^{[n]}\) where \(L\) is a line bundle are (explicitly given) sheaves (instead of complexes of sheaves).
This enables the author to prove new formulas and also give simpler proofs for existing formulas for homological invariants of tautological bundles and their wedge powers. derived McKay correspondence; Hilbert scheme; tautological bundle McKay correspondence, Parametrization (Chow and Hilbert schemes) Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles | 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 G be a connected group scheme acting on a scheme X, both of finite type over k. Suppose that \(\phi: X\to Y\) is a geometric quotient in the sense of geometric invariant theory [see \textit{D. Mumford} and \textit{J. Fogarty}, ''Geometric invariant theory'' (1982; Zbl 0504.14008)]. We prove that Y is also of finite type over k. In fact, k may be any excellent ring such like \({\mathbb{Z}}\) or a complete local ring. connected group scheme acting on a scheme; geometric quotient Fogarty, J.: Geometric quotients are algebraic schemes. Advan. in math. 48, 166-171 (1983) Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Geometric invariant theory Geometric quotients are algebraic 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 Zariski proved around 1944 that every birational morphism between smooth surfaces over a field \(k\) is a composition of blowing-ups at closed points. Later, around 1966, Shafarevich proved the same theorem for regular schemes of dimension 2 without base field. This generalization is important for arithmetic geometry. Danilov generalized Zariski's theorem to relative dimension 1 by studying the relative canonical divisor. He also left the regular scheme case as an open question. On the other hand, some results in higher dimension appeared around 1981 [see \textit{B. Crauder}, Duke Math. J. 48, 589-632 (1981; Zbl 0474.14005); \textit{M. Schaps}, ibid. 401-420 (1981; Zbl 0475.14008) and Math. Ann. 222, 23-28 (1976; Zbl 0309.14009); \textit{M. Teicher}, ibid. 256, 391-399 (1981; Zbl 0445.14005)]. But all the authors required the algebraic varieties have an algebraically closed based field. This paper is devoted to the generalization of Schaps and Teicher's results to regular schemes without base field. Although the main technical tool is the theory of ramification index of regular local rings, which we shall review in the first section, our central ideas are based on the papers cited above. In section 2, we shall establish some general lemmas of birational morphisms for regular schemes. Section 3 contains the proofs of the following theorems:
Let \(f : X \to Y\) be a proper birational morphism of regular schemes such that \(\dim (f^{-1} (y)) \leq 3\) for any \(y \in Y\) and \(S(f)\) regular. If \(E(f)\) has only two nonsingular components, then \(f\) is a composition of two blowing-ups with regular centers.
Let \(f : X \to Y\) be a proper birational morphism between regular schemes of dimension three such that \(E(f)\) has only three normally crossing nonsingular components. If \(S(f)\) is a regular subscheme of codimension 2, then
(a) \(f\) consists of three blowing-ups, or
(b) \(f\) is formed by blowing up \(S(f)\) and then blowing up two intersecting curves in different orders at different intersection points. birational morphism of regular schemes Sun, X. : On birational morphisms of regular schemes . Preprint. Rational and birational maps, Schemes and morphisms Birational morphisms of regular 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 The singularities of some rational algebraic surfaces in complex projective spaces are studied. Formulas for the degree of the various types of singular loci in terms of invariants of the surface are given. One example studied is the surface \(X\) obtained by projecting the \(d\)--th Veronese embeddings of the plane \(\mathbb P^2\) to \(\mathbb P^3\) from a linear subspace \(L\subset \mathbb P^N\) of dimension \(N-4\), \(N=\left(\begin{smallmatrix} d+2\\2\end{smallmatrix}\right)-1\). Let \(\Gamma\) be the singular locus of \(X\) then \(\Gamma\) is a curve of degree at most \(\frac{1}{2}d(d-1)(d^2+d-3)\). Similarly the projection of the Segre surface, given by the Segre embedding of bidegree \((a,b)\) of \(\mathbb P^1\times \mathbb P^1\) in \(\mathbb P^N\), \(N=(a+1)(b+1)-1\), is studied. Here \(\deg(\Gamma)\leq 2a^2b^2-4ab+a+b\). del Pezzo surface; rational scroll Piene R., Singularities 40 (1983) Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Rational and ruled surfaces Singularities of some projective 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 In this paper, the author gives explicit equations for determinantal rational surface singularities and proves dimension formulae for the \(T^1\) and \(T^2\) of those singularities. determinantal rational surface singularities T.de Jong, Determinantal Rational Surface Singularities.Comp. Math. 113 (1998), 73--96. Singularities of surfaces or higher-dimensional varieties, Determinantal varieties, Rational and unirational varieties Determinantal 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 We present applications of elimination theory to the study of singularities over arbitrary fields. A partial extension of a function, defining resolution of singularities over fields of characteristic zero, is discussed here in positive characteristic. Villamayor U. O.E.: Elimination with applications to singularities in positive characteristic. Publ. Res. Inst. Math. Sci. 44(2), 661--697 (2008) Global theory and resolution of singularities (algebro-geometric aspects) Elimination with applications to singularities 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 Stable envelopes are special classes in the cohomology or \(K\)-theory or elliptic cohomology of geometrically relevant moduli spaces. They govern the enumerative geometry of the space, the associated representation theory of a quantum group, and are related to solutions of some associated differential equations. In all these areas concrete formulas for stable envelopes are wanted.
The paper under review gives concrete formulas for the elliptic stable envelopes when the moduli space is the Hilbert scheme of points on the space.
Stable envelopes are associated to the torus fixed points of the space. The torus fixed points on the Hilbert scheme are parametrized by certain tuples of partitions. The author describes some trees subordinate to those partitions. To each tree a product of elliptic functions (and their inverses) is assigned, and the formula is a sum of such functions for all trees.
The proof technique is abelianization: an appropriate resolution of the attracting set of the fixed point, calculating the relevant class in the resolution, and pushing it forward.
As a byproduct \(K\)-theoretic and cohomological stable envelopes are also obtained, as the trigonometric and rational limits of the elliptic stable envelope formula. quiver variety; stable envelope; elliptic cohomology; Hilbert scheme of points in the plane Parametrization (Chow and Hilbert schemes), Enumerative problems (combinatorial problems) in algebraic geometry, Combinatorial aspects of representation theory, Algebraic moduli problems, moduli of vector bundles, Mirror symmetry (algebro-geometric aspects), Equivariant \(K\)-theory, Analytic spaces Elliptic stable envelope for Hilbert scheme of points in the plane | 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 well-written paper, which is part I of II, the authors study degeneracy loci of morphisms of vector bundles on a smooth ambient space which, as they show, give rise to perfect obstruction theories and, hence, their associated virtual cycles. They manage to calculate these virtual cycles via the Thom-Porteous formula. The authors then give various applications of these results to punctual Hilbert schemes of \textit{nested} subschemes of a fixed projective surface \(S\), and discuss some implications for local Donaldson-Thomas theory.
More precisely, let \(E_{\bullet} = \{E_0 \overset{\sigma}{\rightarrow} E_1 \}\) be a two-term complex of vector bundles on a smooth ambient space \(A\), let \(n = \operatorname{dim} A\), and let \(r_i = \operatorname{rank}(E_i)\). The \(r\)-th degeneracy locus is defined set-theoretically as
\[Z_r = \bigl\{ x \in A \,\colon\, \operatorname{rank}(\sigma|_x) \leq r \bigr\}.\]
It is equipped with the scheme structure defined by the vanishing of the \((r+1) \times (r+1)\)-minors of \(\sigma\), i.e., by the vanishing of \(\Lambda^{r+1}\sigma \colon \Lambda^{r+1}E_0 \to \Lambda^{r+1}E_1\). In this paper, the authors restrict their attention to the smallest \(r\) for which \(Z = Z_r\) is non-empty; the general case is treated in the sequel [\textit{A. Gholampour} and \textit{R. P. Thomas}, Compos. Math. 156, No. 8, 1623--1663 (2020; Zbl 1454.14028)]. Their first main result, Theorem 3.6, yields a perfect obstruction theory on \(Z\) and provides an expression for its associated virtual cycle \(\iota_*[Z]^{\textrm{vir}}\) where \(\iota \colon Z \hookrightarrow A\) is the embedding.
Let \(S\) be a fixed projective surface, let \(n_1,n_2\) be non-negative integers, and let \(S^{[n_i]}\) denote the Hilbert scheme of zero-dimensional length \(n_i\) closed subschemes of \(S\). The easiest case is the 2-step nested punctual Hilbert scheme,
\[S^{[n_1,n_2]} = \bigl\{ I_1 \subseteq I_2 \subseteq \mathcal{O}_S \,\colon\, \operatorname{length}(\mathcal{O}_S / I_i) = n_i \bigr\}.\] It lies in the ambient space \(S^{[n_1]} \times S^{[n_2]}\) as the locus of points \((I_1,I_2)\) for which there is exists a non-zero map \(\mathrm{Hom}_S(I_1,I_2) \neq 0\). Let \(\pi \colon S^{[n_1]} \times S^{[n_2]} \times S \to S^{[n_1]} \times S^{[n_2]}\) be the natural projection, let \(\mathcal{Z}_1, \mathcal{Z}_2\) denote the universal closed subschemes, and let \(\mathcal{I}_1, \mathcal{I}_2\) denote the corresponding universal ideal sheaves. Then \(S^{[n_1,n_2]}\) arises as the degeneracy locus of the complex of vector bundles
\[R\mathcal{H}om_{\pi}(\mathcal{I}_1,\mathcal{I}_2) = R\pi_{*}R\mathcal{H}om(\mathcal{I}_1,\mathcal{I}_2)\] over \(S^{[n_1]} \times S^{[n_2]}\).
The second main result, Theorem 6.3, states that for any smooth projective surface \(S\), the \(2\)-step nested Hilbert scheme \(S^{[n_1,n_2]}\) carries a natural perfect obstruction theory and virtual cycle
\[\bigl[ S^{[n_1,n_2]} \bigr]^{\textrm{vir}} \in A_{n_1+n_2}\bigl( S^{[n_1,n_2]} \bigr)\]
whose push-forward to \(S^{[n_1]} \times S^{[,n_2]}\) equals \(c_{n_1+n_2}\bigl( R\mathcal{H}om_{\pi}(\mathcal{I}_1,\mathcal{I}_2[1] \bigr)\)
Since Theorem 6.3 expresses \([S^{[n_1,n_2]}]^{\text{vir}}\) as a degeneracy locus, the authors are able to use the splitting principle from topology to give a proof of the following vanishing result of Carlsson and Okounkov [\textit{E. Carlsson} and \textit{A. Okounkov}, Duke Math. J. 161, No. 9, 1797--1815 (2012; Zbl 1256.14010)]: For any smooth projective surface \(S\) we have the vanishing
\[c_{n_1+n_2+i}\bigl( R\mathcal{H}om_{\pi}(\mathcal{I}_1,\mathcal{I}_2[1]\bigr) = 0, \qquad i >0, \]
over \(S^{[n_1]} \times S^{[n_2]}\). The methods in this work allow the authors to generalise this result in Theorem 8.3, yielding the vanishing
\[c_{n_1+n_2+i}\bigl( R\pi_{*}\mathcal{L} - R\mathcal{H}om_{\pi}(\mathcal{I}_1,\mathcal{I}_2 \otimes \mathcal{L}) \bigr) = 0, \qquad i >0, \]
over \(S^{[n_1]} \times S^{[n_2]} \times \operatorname{Pic}_{\beta}(S)\). Here \(\beta \in H_2(S,\mathbb{Z})\) is a curve class and \(\mathcal{L} \to S \times \operatorname{Pic}_{\beta}(S)\) is a Poincaré line bundle. The proof relies on a more general notion of nested Hilbert scheme, denoted \(S_{\beta}^{[n_1,n_2]}\) of subschemes \(S \supset Z_1 \supseteq Z_2\) where \(Z_1\) is allowed to be of dimension \(\leq 1\). In the sequel to this paper [\textit{A. Gholampour} and \textit{R. P. Thomas}, Compos. Math. 156, No. 8, 1623--1663 (2020; Zbl 1454.14028)], the authors construct a natural perfect obstruction theory and virtual cycle on \(S_{\beta}^{[n_1,n_2]}\) for any curve class \(\beta\).
Throughout the paper, attention is paid to surfaces for which either or both of \(h^{0,1}(S)\) and \(h^{0,2}(S)\) are non-zero, either of which necessitate further consideration. The final section of the paper provides an alternative, more geometric, approach to the virtual cycle construction by incorporating the Jacobian \(\operatorname{Jac}(S)\) directly in case \(h^{0,1}(S) \neq 0\). degeneracy locus; nested Hilbert scheme; Thom-Porteous formula; local Donaldson-Thomas theory; Vafa-Witten invariants Algebraic moduli problems, moduli of vector bundles, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Applications of global analysis to structures on manifolds Degeneracy loci, virtual cycles and nested Hilbert schemes. I | 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 regular scheme, \(E\) a vector bundle on \(X\). Maruyama and Sumihiro gave two different ways to obtain another vector bundle on \(X\) using \(E\), an effective divisor \(D\) and other data. Both constructions are called ``elementary transformation''. Here the author gives a third construction and shows that it has the main advantages of the union of the previous ones: geometric meaning, easy to construct examples and a general theorem: every vector bundle on \(X\) is obtained, up to a twist by a line bundle, from a trivial vector bundle using elementary transformations. The author extends his construction to reflexive sheaves (weak elementary transformations) and prove (if \(X\) is quasi-projective and algebraic over an algebraically closed field) that every reflexive sheaf on \(X\) is obtained, up to a twist by a line bundle, from a trivial vector bundle using weak elementary transformations. vector bundle; regular scheme; elementary transformation; reflexive sheaf Abe, T., The elementary transformation of vector bundles on regular schemes, \textit{Trans. Am. Math. Soc.}, 359, 9, 4285-4295, (2007) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on surfaces and higher-dimensional varieties, and their moduli The elementary transformation of vector bundles on regular 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 In the paper under review the authors study the Hilbert series of Hodge ideals of hyperplane arrangements. Let \(X\) be a smooth complex algebraic variety and let \(D\) be a reduced effective divisor on \(X\). One can attach to \(D\) a sequence of ideals called the Hodge ideals \(I_{p}(D)\) for \(p\geq 0\). The first ideal for \(p=0\) is the multiplier ideal \(\mathcal{I}((1-\varepsilon)D)\) for \(0<\varepsilon <<1\) and the higher ideals can be viewed as similar but more refined measures of the singularities of \(D\). In the present paper the authors focus on the classes of the Hodge ideals in the Grothendieck group \(K_{0}(X)\) of coherent sheaves on \(X\) as encoded in the generating function \[\sum_{p\geq 0}[I_{p}(D)]y^{p} \in K_{0}(X)[y].\]
The main observation is that this generating function can be described in terms of the motivic Chern class of the inclusion \(j : U \rightarrow X\), where \(U\) is the complement of the support of \(D\). This motivic Chern class is a group homomorphism
\[mC_{y} : K_{0}(\mathrm{Var}/X) \rightarrow K_{0}(X)[y],\]
where \(K_{0}(\mathrm{Var}/X)\) is the Grothendieck group of varieties over \(X\). Let \(X = V\) be a complex vector space of dimension \(n\) and \(D = D_{\mathcal{A}}\) is the divisor corresponding to an arrangement \(\mathcal{A}\) of linear hyperplanes in \(X\). We consider the standard action of \(T = \mathbb{C}^{*}\), then describing \(I_{p}(D_{\mathcal{A}})\) in the Grothendieck group \(K_{0}^{T}(X)\) of \(T\)-equivariant coherent sheaves on \(X\) is equivalent to describing the Hilbert series \(H_{I_{p}(D_{\mathcal{A}})}(t)\) of \(I_{p}(D_{\mathcal{A}})\). Due to the fact that in the setting as above the equivariant motivic Chern class is easy to compute, the first main result describes the generating function of the Hilbert series \(H_{I_{p}(D_{\mathcal{A}})}(t)\) in terms of the Poincaré polynomial \(\pi(\mathcal{A},t)\) of the arrangement.
Theorem A. If \(\mathcal{A}\) is a central hyperplane arrangement of \(d\) in an \(n\)-dimensional complex vector space \(V\) and if \(D_{\mathcal{A}} = \sum_{H \in \mathcal{A}}H\), then
\[\sum_{p\geq 0}H_{I_{p}(D_{\mathcal{A}})}(t) y^{p} = \frac{t^{d}}{(1-t)^{n}(1-t^{d}y)}\cdot \pi(\mathcal{A},(1-t)/t(1-t^{d-1}y)).\] By letting \(y=0\) in the above theorem and recalling the identification of \(I_{0}(D)\) with a multiplier ideal, one obtains the following result.
Corollary. If \(\mathcal{A}\) is a central hyperplane arrangement of \(d\) hyperplanes in an \(n\)-dimensional complex vector space \(V\) and if \(D_{\mathcal{A}} = \sum_{H \in \mathcal{A}}H\), then the Hilbert series of the multiplier ideal \(I = \mathcal{I}((1-\varepsilon)D_{\mathcal{A}})\) with \(0 < \varepsilon \ll1\) is given by
\[H_{I}(t) = \frac{t^{d}}{(1-t)^{n}}\cdot \pi(\mathcal{A},t^{-1}-1).\] generating functions; Hilbert series; hyperplane arrangements; multiplier ideals; motivic Chern classes Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Singularities in algebraic geometry, Relations with arrangements of hyperplanes The Hilbert series of Hodge ideals of hyperplane arrangements | 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 show that any union of slc strata of a Fano log pair with semi-log canonical singularities is simply connected. In particular, Fano log pairs with semi-log canonical singularities are simply connected, which confirms a conjecture of the first author. Fano varieties; semi-log canonical singularities; slc strata; simple connectedness; rational chain connectedness Fano varieties, Minimal model program (Mori theory, extremal rays) Simple connectedness of Fano log pairs with semi-log canonical 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 \(R\) be a regular local ring of dimension \(d\) with maximal ideal \({\mathfrak m}\), and let \(X=\text{Spec}(R)\). Let \({\mathfrak p}\) and \({\mathfrak q}\) be prime ideals of \(R\) such that \({\mathfrak p}+{\mathfrak q}\) is \({\mathfrak m}\)-primary or, equivalently, such that \(R/ {\mathfrak p}\otimes_RR/{\mathfrak q}\) is a module of finite length. Then the intersection multiplicity of \(R/{\mathfrak p}\) and \(R/{\mathfrak q}\) is defined to be
\[
\chi(R/{\mathfrak p},R/{\mathfrak q})=\sum^d_{i=0}(-1)^i\text{length} \bigr(\text{Tor}_i^R(R/{\mathfrak p},R/{\mathfrak q})\bigr).
\]
For the basic properties of intersection multiplicities we refer to the book by \textit{J.-P. Serre}: ``Algèbre local, Multiplicités'', Lect. Notes Math. 11 (1965; Zbl 0142.28603). Serre made several conjectures, of which we state two.
(1) Nonnegativity: \(\chi(R/{\mathfrak p},R/{\mathfrak q})\geq 0\).
(2) Positivity: If \(\dim(R/{\mathfrak p})+\dim(R/{\mathfrak q})=\dim (R)\), then \(\chi(R/{\mathfrak p},R/{\mathfrak q})>0\).
Serre proved these conjectures in the equicharacteristic case using the method of reduction to the diagonal. Recently, Gabber used a construction of de Jong to prove that these multiplicities are always nonnegative, thus establishing one of the conjectures. In his proof, O. Gabber constructed a scheme that can be represented by a bigraded ring and reduced the computation of intersection multiplicities to the computation of an Euler characteristic defined by modules over this ring.
In the paper under review we define Hilbert polynomials for bigraded modules over this type of bigraded ring and show that the Euler characteristic can be computed using these Hilbert polynomials. We then use this construction to give a simple proof of a criterion for positivity proven by \textit{K. Kurano} and \textit{R. C. Roberts} [Compos. Math. 122, No. 2, 165--182 (2000; Zbl 0970.13006)] as follows:
Theorem 3. Let \(C=k [X_0,\dots,X_s,T_0,\dots,T_u,S_1,\dots, S_v]\), and let \(Q\) be a bigraded prime ideal of \(C\). Assume that the dimension of \(C/Q\) is \(u+v+1\), and let \(J=(T_0, \dots,T_u,S_1,\dots, S_v)\). Then the following are equivalent.
(1) \(\chi(C/Q, C/J)>0\).
(2) The \(m^{u+1}n^v\) coefficient of the Hilbert polynomial \(P_{C/Q}\) is positive.
(3) \(Q\cap k[S_1,\dots,S_v]=0\). DOI: 10.1307/mmj/1030132731 Multiplicity theory and related topics, Homological conjectures (intersection theorems) in commutative ring theory, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Intersection multiplicities and Hilbert polynomials. | 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 has two parts. In the first, one studies for non-isolated hypersurface singularities questions as: finite determinacy, unfoldings and deformations, the topology of the nearby fibres. Best results are given when the singular locus has dimension one. The key point is to fix an analytic germ \(\Sigma\) in \(({\mathbb{C}}^ m,0)\) and to look to holomorphic functions which contain \(\Sigma\) in their singular locus and to coordinate transformations which leave \(\Sigma\) invariant. This leads to the study of the corresponding right-equivalence relation and of two algebraic notions: the primitive ideal associated to \(\Sigma\) and Jacobi modules. The primitive ideal of \(\Sigma\) is given exactly by the functions as above, i.e. it is \(\{f| (f)+J_ f\subset I_{\Sigma}\}\) \((J_ f\) is the Jacobi ideal of f). A Jacobi module is nothing else than a quotient \(I_{\Sigma}/J_ f.\)
In the second part (pure algebraic) one studies the depth and the projective dimension of the quotient of two ideals. This topic emerges naturally from the first part and is used in the first part. depth of a module; resolutions of Jacobi modules; non-isolated hypersurface singularities; unfoldings; deformations; Jacobi ideal; projective dimension Pellikaan, G. R.: Hypersurfaces singularities and resolutions of Jacobi modules. (1985) Local complex singularities, Complex singularities, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Singularities of surfaces or higher-dimensional varieties, Deformations of complex singularities; vanishing cycles, Singularities in algebraic geometry Hypersurface singularities and resolutions of Jacobi modules | 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 SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a growth condition. From an arbitrary SI-sequence the authors construct a reduced, arithmetically Gorenstein configuration of linear varieties, \(G\), of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. It is shown that \(G\) has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of the projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. Finally, it is shown that over a field of characteristic zero every set of simplicial polytopes with fixed \(h\)-vector contains a polytope with maximal graded Betti numbers. SI-sequence; Hilbert function; weak Lefschetz property; Gorenstein ideal; graded Betti numbers; simplicial polytopes; Gorenstein liaison; arithmetically Gorenstein subscheme; compressed algebra; Gorenstein configuration of linear varieties Migliore, J.; Nagel, U., Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers, \textit{Adv. Math.}, 180, 1-63, (2003) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Linkage, complete intersections and determinantal ideals, Syzygies, resolutions, complexes and commutative rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Linkage, Configurations and arrangements of linear subspaces, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Special polytopes (linear programming, centrally symmetric, etc.), \(n\)-dimensional polytopes Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers. | 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 See the joint review of part I, II in [Zbl 0122.38603]. Global theory and resolution of singularities (algebro-geometric aspects) Resolution of singularities of an algebraic variety over a field of characteristic zero. II | 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 toric Hilbert scheme is a parameter space for all ideals with the same multigraded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is connected. These graphs are used to exhibit curves in \(\mathbb P^4\) whose associated toric Hilbert schemes have arbitrary dimension. We show that the flip graph maps into the Baues graph of all triangulations of the point configuration defining the toric ideal. Inspired by the recent discovery of a disconnected Baues graph, we close with results that suggest the existence of a disconnected flip graph and hence a disconnected toric Hilbert scheme. Maclagan D., ''Combinatorics of the toric Hilbert scheme'' (1999) Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies Combinatorics of the toric Hilbert scheme | 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 Two varieties \(X\) and \(Y\) of codimension \(c\) in nonsingular ambient variety \(A\) are linked if their union is a complete intersection \(V \subset A\); see the survey of \textit{J. C. Migliore} for the case \(A = \mathbb P^n\) [Introduction to liaison theory and deficiency modules, Progr. in Math. 165, Birkhäuser, Boston, (1988; Zbl 0921.14033)]. For fixed \(X\), a generic link \(Y\) is obtained by choosing a general complete intersection \(V\) containing \(X\), when it is natural to ask what properties of \(X\) carry over to \(Y\). Even if \(X\) is nonsingular, one expects \(Y\) to be only regular in codimension \(2c-1\) (for example, see work of the reviewer [Collect. Math. 49, 447--463; (1998; Zbl 0959.14014)]). On the other hand, \textit{M. Chardin} and \textit{B. Ulrich} have shown that if \(X\) is a local complete intersection and has at worst rational singularities, then the same holds for \(Y\) [Am. J. Math. 124, 1103--1124 (2002; Zbl 1029.14016)].
Using the algebraic form of liaison developed by \textit{C. Huneke} and \textit{B. Ulrich} [Ann. Math. 126, 277--334 (1987; Zbl 0638.13003); Duke Math. J. 56, 415--429 (1988; Zbl 0656.13026); J. Reine Angew. Math. 390, 1--20 (1988; Zbl 0732.13004)], the author investigates Mather-Jacobian (MJ) singularities, which were independently introduced in work of \textit{L. Ein} and \textit{S. Ishii} [Math. Sci. Res. inst. Publ. 68, 125--168 (2015; Zbl 1359.14020)] and \textit{T. de Fernex} and \textit{R. Docampo}[J. Eur. Math. Soc. (JEMS) 16, 165--199 (2014; Zbl 1334.14022)]. The author uses his earlier work [Am. J. Math. 136, 1665--1691 (2014; Zbl 1312.14113)] and \textit{S. Ishii}'s inversion of adjunction [Ann. Inst. Fourier 63, 89--111 (2013; Zbl 1360.14048)] to show that If \(X\) has MJ-canonical (resp. MJ-log canonical) singularities, then so does the generic link \(Y\). He also deduces some inequalities on minimal MJ-log discrepancies. Mather-Jacobian singularities; liaison; general linkage Singularities in algebraic geometry, Linkage, Linkage, complete intersections and determinantal ideals Mather-Jacobian singularities under generic linkage | 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 show the existence of infinitely many links of non-degenerate simple \(K3\) singularities defined by non-quasihomogeneous polynomials such that the second Betti numbers of the links are 17, which do not appear in the case of the singularities defined by quasi-homogeneous polynomials. hypersurface simple \(K3\) singularity; link Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, \(K3\) surfaces and Enriques surfaces, Milnor fibration; relations with knot theory The links specific to 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 The author gives a new description of the graded ring structures for two-dimensional cyclic quotient singularities. This has been explicitly obtained by Pinkham and Demazure using the weight of the variables. Here he uses certain \(\mathbb Q\)-divisors on \(\mathbb P^1\), which are canonically associated to the resolutions of singularities. T. Tomaru: Pinkham-Demazure construction for two dimensional cyclic quotient singularities , Tsukuba J. Math. 25 (2001), 75-83. Singularities in algebraic geometry, Local complex singularities Pinkham-Demazure construction for two dimensional cyclic 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 In the paper under review, the authors classify \textit{nearly Gorenstein} rings for the invariant ring \(R^G\) where \(R=\Bbbk[[ x_1,\dotsc,x_d]]\) is a formal power series ring over an algebraically closed field \(\Bbbk,\) \(G\) is a finite cyclic subgroup of \(\mathrm{GL}(d,\Bbbk)\) acting linearly on \(R\), and \(|G|\) is invertible in \(\Bbbk.\) Since \(G\) is cyclic, the invariant ring is monomial, hence the classification can be described in terms of a numerical criterion.
Nearly Gorenstein property extends the property of being Gorenstein and the authors also provide a family of examples where nearly Gorenstein property fails by defining the residue of the trace ideal of the corresponding canonical module.
Moreover, the authors provide an exhaustive list of cyclic quotient singularities for \(d=3\) and \(4\le |G|\le 7\) and for \(d=4\) and \(4\le |G|\le 6.\) The cases where \(d=2\) or \(|G|\le3\) are already covered by their main result.
Finally, the paper contains an addendum to Ding's classification of nearly Gorenstein quotient singularities when \(d=2.\) quotient singularity; trace ideal; invariant ring; nearly Gorenstein Actions of groups on commutative rings; invariant theory, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Group actions on varieties or schemes (quotients) Nearly Gorenstein cyclic 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 This article studies Cox rings of certain varieties. If \(X\) is an algebraic variety satisfying suitable conditions, there is an associated graded ring \({\text{Cox}}(X)\), essentially \(\bigoplus \Gamma(X, \mathcal O _X(D))\), for \(D\) a representative of an element of the class group \({\text{Cl}}(X)\). Thus, \({\text{Cox}}(X)\) is a multi-graded ring, in general not indexed by \(\mathbb Z\). More precisely, in the present article it is assumed that \(X\) is the minimal resolution of a surface quotient singularity \(Y={\mathbb C}^2/G\), where \(G\) is a subgroup of \(\mathrm{GL}(2,{\mathbb C})\). The main results are as follows, where \(f:X \to Y\) is the mentioned minimal resolution. {\parindent=6mm \begin{itemize}\item[(a)] Let \(n\) be the number of irreducible components of the exceptional divisor of the minimal resolution \(f\). Then there is a hypersurface \(S\) in \({\mathbb C}^{n+3}\) (whose defining equation is described) such that \(S \simeq {\text{Spec}}({\text{Cox}}(X))\). In the construction and proofs the author exploits an action of the ``Picard torus'' \(T={\text{Hom}} ({\text{Pic}}(X), {\mathbb C}^{\star}) \simeq {(\mathbb C ^{\star})}^n\) on \(\mathbb C ^{n+3}\) and uses, among other techniques, a good amount of toric geometry. \item[(b)] A description of the ring \({\text{Cox}} (X)\) is given as a subalgebra of the coordinate ring of the the product of the Picard torus \(T\) a certain singular surface (again with a quotient singularity). The generators of the subalgebra \({\text{Cox}} (X)\) are described. \item[(c)] An explicit description of the minimal resolution as a divisor in a toric variety.
\end{itemize}} These results are completely proved, although the author indicates that (a) could be shown, alternatively, using the methods of [\textit{J. Hausen} and \textit{H. Süß}, Adv. Math. 225, No. 2, 977--1012 (2010; Zbl 1248.14008)]. But she hopes that the techniques developed in the present paper could yield generalizations of her results to higher dimension, as well as a description of \(X\) as a quotient of an open set of \({\text{Spec}} ({\text{Cox}} (X)\), in case \({\text{Cox}}(X)\) is finitely generated.
The article reviews much of the necessary background material and includes some interesting examples. Cox ring; surface quotient singularity; minimal resolution; toric variety Donten-Bury, M, Cox rings of minimal resolutions of surface quotient singularities, Glasg. Math. J., 58, 325-355, (2016) Singularities of surfaces or higher-dimensional varieties, Surfaces and higher-dimensional varieties, Hypersurfaces and algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Group actions on varieties or schemes (quotients), Group actions on affine varieties, Global theory and resolution of singularities (algebro-geometric aspects) Cox rings of minimal resolutions of surface 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 The paper discusses stably trivial torsors for spin and orthogonal groups over smooth affine schemes over infinite perfect fields of characteristic unequal to 2. We give a complete description of all the invariants relevant for the classification of such objects over schemes of dimension at most 3, along with many examples. The results are based on the $\mathbb{A}^1$-representability theorem for torsors and transfer of known computations of $\mathbb{A}^1$-homotopy sheaves along the sporadic isomorphisms to spin groups. Motivic cohomology; motivic homotopy theory, Group schemes On stably trivial spin torsors over low-dimensional 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 \(R=P/(f)\) be an analytic hypersurface ring. The author investigates first the relation between maximal Cohen-Macaulay modules (MCM) over R and over \(R_ 1=P_ 1/(f+y^ 2)\), where \(P_ 1=P<y>\) (and \(P=k<x_ 0,...,x_ n>\), k algebraically closed and char(k)\(\neq 2)\). He proves in \(corollary^ 2.8\) that there are only finitely many isomorphism classes of indecomposable MCM's over R if and only if this is true for \(R_ 1\). In theorem 3.1 it is shown - in a more general frame - that there is a canonical bijection between the sets of isomorphism classes of MCM's over R and over \(R_ 2=P_ 2/(f+y^ 2+z^ 2)\) respectively, where \(P_ 2=P<y,z>.\)
Since the two-dimensional simple singularities, i.e. the rational double points, have only finitely many isomorphism classes of MCM's over their local rings [by \textit{M. Artin} and \textit{J.-L. Verdier}, Math. Ann. 270, 79-82 (1985; Zbl 0553.14001)], one gets by iterated application of corollary 2.8 the main result of this paper: There are only finitely many isomorphism classes of indecomposable MCM's over the local ring of an isolated simple hypersurface singularity \((A_ k, D_ k, E_ 6, E_ 7, E_ 8\) in Arnold's classification). The author also gives a conceptional description of the Auslander-Reiten quivers of the simple plane curve singularities in \(char(k)=0\), showing that these quivers coincide with certain graphs associated to representations of finite reflection groups in \(Gl(2,k).\)
[See also the following review.] analytic hypersurface ring; maximal Cohen-Macaulay modules; isolated simple hypersurface singularity; Auslander-Reiten quivers Knörrer, H., Cohen-Macaulay modules on hypersurface singularities, I. Invent. Math., 88, 153-164, (1987) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Cohen-Macaulay modules on hypersurface singularities. I | 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 relations between certain numerical invariants of a normal complex surface singularity. The main results are characterizations, in terms of the invariants, of minimally elliptic singularities and of rational triple points. pluri-genus; minimally elliptic singularity; rational triple point Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Divisors, linear systems, invertible sheaves Characterizations of minimally elliptic singularities and rational triple points | 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 Manuscr. Math. 56, 333-342 (1986; Zbl 0585.14025) hat \textit{H. Maeda} die stabile Reduktion der Fermatkurve vom Primzahlgrad \(p\) über einem Zahlkörper modulo einer Primstelle \({\mathbf p}\) über \(p\) bestimmt. Kürzlich hat \textit{J. J. van Beele} in seiner Dissertation [``Models und modularity questions concerning Fermat and Klein curves'' (Leiden 1994)] darauf aufmerksam gemacht, daß der Beweis in dieser Arbeit nicht vollständig ist und einen anderen Beweis dieser Aussage gegeben, der auf der von Coleman und McCallum bestimmten stabilen Reduktion der Quotientenkurven der Fermatkurve beruht. In der vorliegenden Arbeit wird eine einfachere Konstruktion des stabilen Modells der Fermatkurve modulo \({\mathbf p}\) dargestellt, und zwar mit Hilfe der Desingularisierungstheorie der exzellenten Flächen nach \textit{H. Hironaka} [im Anhang von: \textit{V. Cossart, J. Giraud} and \textit{U. Orbanz}, ``Resolution of surface singularities'', Lect. Notes Math. 1101 (1984; Zbl 0553.14003)]. stable reduction of the Fermat curve Arithmetic ground fields for curves, Global theory and resolution of singularities (algebro-geometric aspects), Special algebraic curves and curves of low genus, Global ground fields in algebraic geometry, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Singularities of surfaces or higher-dimensional varieties Resolution of surface singularities and stable reduction of Fermat 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 For a projective surjective morphism \(f:X\to Y\) of smooth quasi-projective varieties we denote by \(X^r\) the \(r\)-fold fibre product of \(X\) over \(Y\). Because of the compatibility of the relatively canonical sheaf with flat base change one has an isomorphism \(\omega_{ X^r/Y} \to\omega_{X/Y} \boxtimes \cdots \boxtimes \omega_{X/Y}\). This is essentially used in the proof of the Fujita-Kawamata positivity theorem [\textit{T. Fujita}, J. Math. Soc. Japan 30, 779--794 (1978; Zbl 0393.14006) and \textit{Y. Kawamata}, Compos. Math. 43, 253--276 (1981; Zbl 0471.14022)]. Considering the more general situation of a normal variety \(X\) with at most rational singularities together with an effective Cartier divisor \(D\), there exists an analogue of the Fujita-Kawamata positivity theorem for the sheaf \(f_*(\omega_{X/Y} \{-\frac DN\})\) which follows easily from the original version by use of vanishing theorems. Yet, the problem whether there exists an analogue of the isomorphism above for this situation is interesting in itself, since multiplier ideal sheaves themselves, in particular on singular varieties, are still not understood very well. Whereas on smooth varieties multiplier ideal sheaves are compatible with products, we show that in our more general situation they are compatible with fibre products only under certain conditions. fibre product; fibration; positivity DOI: 10.1142/S0129167X03001855 Divisors, linear systems, invertible sheaves, Singularities in algebraic geometry, Fibrations, degenerations in algebraic geometry, Vanishing theorems in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] A short note on multiplier ideal sheaves 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 The main proposition, Theorem 1.2, is the existence for excellent Deligne-Mumford champ of characteristic zero of a resolution functor independent of the resolution process itself. Received wisdom was that this was impossible, but the counterexamples overlooked the possibility of using weighted blow ups. The fundamental local calculations take place in complete local rings, and are elementary in nature, while being self contained and wholly independent of Hironaka's methods and all derivatives thereof, i.e. existing technology. Nevertheless \textit{D. Abramovich} et al. [``Functorial embedded resolution via weighted blowing ups'', Preprint, \url{arXiv:1906.07106}], have varied existing technology to obtain even shorter proofs of all the main theorems in the pure dimensional geometric case. Excellent patching is more technical than varieties over a field, and so easier geometric arguments are pointed out when they exist. Global theory and resolution of singularities (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to algebraic geometry Very functorial, very fast, and very easy 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 consider a category of equivariant sheaves and its \(K\)-theory on a scheme over an algebraically closed field with a periodic morphism. We prove the localization theorem in the \(K\)-theory for algebraic schemes and with the help of the theorem, we show the Lefschetz-Riemann-Roch theorem for smooth algebraic schemes. The paper generalizes the result of \textit{P. Donovan} [Bull. Soc. Math. Fr. 97(1969), 257-273 (1970; Zbl 0185.494)], who proved the theorem under the assumption of projectivity. \(K\)-theory; equivariant sheaves; Lefschetz-Riemann-Roch theorem Riemann-Roch theorems, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Lefschetz-Riemann-Roch theorem for smooth algebraic 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 \textit{W. D. Neumann} and \textit{J. Wahl} [Math. Ann. 326, No.1, 75--93 (2003; Zbl 1032.14010)] proved that the universal abelian cover of every quotient-cusp is a complete intersection and have conjectured that a similar results holds for any \(\mathbb Q\)-Gorenstein normal surface whose link is a rational homology sphere. Cusps are not included in the scope of this conjecture, nevertheless the author considers a similar question and exhibits a cusp with no Galois cover by a complete intersection. The main techniques are plumbing and study of monodromy for cusp singularities. link of a singularity Singularities in algebraic geometry, Coverings in algebraic geometry, Complex surface and hypersurface singularities, Milnor fibration; relations with knot theory A cusp singularity with no Galois cover by a complete intersection | 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, we will investigate further properties of \(\mathcal A\)-schemes introduces in [Tak]. The category of \(\mathcal A\)-schemes possesses many properties of the category of coherent schemes, and in addition, it is co-complete and complete. There is the universal compactification, namely, the Zariski-Riemann space in the category of \(\mathcal A\)-schemes. We compare it with the classical Zariski-Riemann space, and characterize the latter by a left adjoint. Generalizations (algebraic spaces, stacks) \(\mathcal A\)-schemes and Zariski-Riemann 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 \(X\) be a \(K3\) or abelian surface over the field of complex numbers. Fix a very ample curve \(C\) on \(X\). Let \(\mathcal M\) be the moduli space of pairs of the form \((F, s)\), where \(F\) is a stable sheaf over \(X\) whose Hilbert polynomial coincides with that of the direct image, by the inclusion map \(C\hookrightarrow X\), of a line bundle of degree \(d\) over \(C\), and \(s\) is a nonzero section of \(F\). Assume that \(d\) is so large that \(F\) has a nonzero section. There is a holomorphic \(2\)-form on \(\mathcal M\), namely, the pullback of the Mukai symplectic form on moduli space of stable sheaves over \(X\). On the other hand, there is a map of \(\mathcal M\) to a Hilbert scheme parametrizing \(0\)-dimensional subschemes of \(X\) that sends \((F, s)\) to the divisor, defined by \(s\), on the curve defined by the support of \(F\). The authors prove that the above \(2\)-form on \(\mathcal M\) coincides with the pullback of the symplectic form on the Hilbert scheme. surface; bundle; sheaf; moduli space Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, \(K3\) surfaces and Enriques surfaces, Complex-analytic moduli problems On the symplectic structures on moduli space of stable sheaves over a \(K\)3 or abelian surface and on Hilbert scheme of points | 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 smooth scheme \(X\) over a field \(k\) of positive characteristic is said to be strongly liftable if \(X\) and all prime divisors of \(X\) can be lifted simultaneously to \(W_2(k)\), the ring of Witt vectors of length two of \(k\). The author studies properties of strongly liftable schemes and produces many examples: \(\mathbb A_k^n\), \(\mathbb P_k^n\), smooth projective curves, smooth complete intersections in \(\mathbb P_k^n\) having Picard number \(1\), smooth projective surfaces of the type \(\mathbb P_C(\mathcal E)\), where \(\mathcal E\) is a decomposable rank-2 vector bundle on a smooth projective curve. As an application, he proves that the Kawamata--Viehweg vanishing theorem in positive characteristic holds on a normal projective surface which is birational to a strongly liftable surface. This generalizes a previous result in [Math. Z. 266, No. 3, 561--570 (2010; Zbl 1235.14021)]. positive characteristic; strongly liftable scheme; Kawamata-Viehweg vanishing theorem Xie, Q. H., Strongly liftable schemes and the Kawamata-Viehweg vanishing in positive characteristic, Math. Res. Lett., 17, 2010, 563--572. Positive characteristic ground fields in algebraic geometry, Vanishing theorems in algebraic geometry, Rational and ruled surfaces Strongly liftable schemes and the Kawamata-Viehweg vanishing 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 This paper is a survey on the theory of strongly liftable schemes, which is based on the author's talk at the Sixth International Congress of Chinese Mathematicians in Taipei.
In this paper, we first recall some results on vanishing theorems in algebraic geometry. Secondly, we give a systematic introduction to the theory of strongly liftable schemes and its applications to vanishing theorems in positive characteristic. Finally, some remarks and further problems are put forward. strongly liftable scheme; Kawamata-Viehweg vanishing; positive Vanishing theorems in algebraic geometry, Minimal model program (Mori theory, extremal rays) Strongly liftable schemes and the Kawamata-Viehweg vanishing 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 The author gives a characterization for hypersurface singularity to be stably equivalent to a plane curve singularity in terms of Thom-Boardman symbols and integral closure of ideals. jacobian extension of higher order; hypersurface singularity; plane curve singularity; Thom-Boardman symbols Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Singularities in algebraic geometry, Analytic algebras and generalizations, preparation theorems, Singularities of curves, local rings Caractérisations algébriques des singularités de Thom-Boardman de courbes planes | 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 smooth projective surface and \(X^{[n]}\) the Hilbert scheme of generalized \(n\)-tuples on \(X\). The paper deals with the Chern classes of tautological bundles of the form \(p_*({\mathcal O}_\Xi\otimes q^*E)\), where \(E\) is a vector bundle on \(X\) and \((p,q):\Xi\to X^{[n]}\times X\) is the universal family. It gives an algorithmic description of the action of these Chern classes on the cohomology of the Hilbert schemes within the framework of Nakajima's oscillator algebra [\textit{H. Nakajima}, Ann. Math., II. Ser. 145, No. 2, 379-388 (1997; Zbl 0915.14001)]. This method leads to an identification of the cohomology ring of \((\mathbb{A}^2)^n\) with a ring of explicitly given differential operators on a Fock space.
The paper ends with the computation of the top Segre classes of tautological bundles associated to line bundles on \(X^{[n]}\) up to \(n=7\), extending computations of Severi, LeBarz, Tikhomirov and Troshina, and gives a conjecture for the generating series. Hilbert scheme; Chern classes of tautological bundles; differential operators on a Fock space Lehn, M.: Chern classes of tautological sheaves on Hilbert schemes of points on surfaces. Invent. Math. \textbf{136}(1), 157-207 (1999). arXiv:math/9803091 Parametrization (Chow and Hilbert schemes), Characteristic classes and numbers in differential topology, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Chern classes of tautological sheaves on Hilbert schemes of points on 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 Resolution of singularities of varieties over fields of characteristic zero can be proved by using the multiplicity as main invariant. The proof of this result leads to new questions in positive characteristic. We discuss here results that arise when applying inductive arguments (induction on the dimension of the varieties).
{\parindent=0.5cm Fix a variety \(X^{(d)}\) of dimension \(d\) over a \textit{perfect field} \(k\) or, more generally, a pure-dimensional scheme of finite type over \(k\). Fix a closed point \(x\in X^{(d)}\) of multiplicity \(e>1\). Define a \textit{local simplification of the multiplicity at} \(x\in X^{(d)}\) as a proper birational map, say \(X^{(d)}\leftarrow X^{(d)}_1\), where \(X^{(d)}\) denotes now a neighborhood of \(x\), so that \(X^{(d)}_1\) has multiplicity \(<e\) at any point \(x_1\in X^{(d)}_1\).
Assume, by induction on \(d\), the existence of local simplifications of the multiplicity for schemes over \(k\) of dimension \(d^\prime\), for all \(d^\prime<d\). We prove, under this inductive assumption, that a local simplification at \(x\in X^{(d)}\) can be constructed when \((C_{X,x})_{\mathrm{red}}\) is not regular.
Here, \(C_{X,x}\) denotes the tangent cone of \(x\in X\), and \((C_{X,x})_{\mathrm{red}}\) is the reduced scheme. The paper uses classical results of commutative algebra, and compares the effect of blowing up along equimultiple centers, and along normally flat centers.
} multiplicity; resolution of singularities; singularities in positive characteristic Global theory and resolution of singularities (algebro-geometric aspects) On the simplification of singularities by blowing up at equimultiple centers | 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 Over the complex numbers, the generic vanishing theorems on the cohomology groups \(h^i(X, \omega_X \otimes L)\) where \(L\) is an algebraically trivial line bundle on a smooth projective variety \(X\) with canonical bundle \(\omega_X\) are a fundamental tool in the study of varieties with maximal Albanese dimension. The proof of generic vanishing relies crucially on the results of \textit{J. Kollár} [Ann. Math. (2) 123, 11--42 (1986; Zbl 0598.14015); Ann. Math. (2) 124, 171--202 (1986; Zbl 0605.14014)] on the higher direct images \(R^ia_*\omega_X\) of the Albanese morphism \(a\colon X \to A\), which are known to fail in positive characteristic. In the previous article of \textit{C. D. Hacon} et al. [Duke Math. J. 168, No. 9, 1723--1736 (2019; Zbl 1436.14033)], the structure of Cartier module on \(R^ia_*\omega_X\) is exploited to obtain some generic vanishing statement, which are powerful enough to prove various birational characterisations of (ordinary) abelian varieties.
In this paper, the authors further investigate generic vanishing theorems for Cartier modules on an abelian variety \(A\). The main result states that for a Cartier module \(F_*\Omega_0 \to \Omega_0\) on \(A\) and \(i >0\), then there exists a closed subset \(W_i\) of codimension \(i\) outside of which a Frobenius limit version of vanishing on \(h^i(A, \Omega_0 \otimes L)\) holds. Stronger results saying that \(W_i\) is a torsion translate of abelian subvarieties are proven in the case where \(A\) has no supersingular factors. One important application of this concerns the study of the singularities of theta divisors: an irreducible theta divisor of a principally polarised abelian variety without supersingular factors is strongly \(F\)-regular (generalising the fact that it has canonical singularities in characteristic 0). generic vanishing; theta divisors; positive characteristic Arithmetic ground fields for abelian varieties, Theta functions and abelian varieties, Vanishing theorems in algebraic geometry Generic vanishing in characteristic \(p>0\) and the geometry of theta divisors | 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 classification of maximal Cohen-Macaulay (MCM) modules over Noetherian local rings is a difficult problem in general, and it has a long history; we refer the reader to the book [\textit{G. Leuschke} and \textit{R. Wiegand}, Cohen-Macaulay representations. Providence, RI: American Mathematical Society (AMS) (2012; Zbl 1252.13001)] for a detailed overview of the subject.
Let \(k\) be an algebraically closed field such that \(\text{char}(k) \neq 2\). The goal of the article under review is to study the MCM representation type of rings of the form
\[
k[[x,y,z]]/(xy, y^q - z^2),
\]
and also the ring
\[
k[[x,y,z]]/(xy, z^2)
\]
(additional results in the case \(\text{char}(k) = 2\) are also obtained; see Remark 2.3 of the article under review for details).
The main results of the article are as follows. The authors first prove that the above rings have \textit{tame} MCM representation type (see Section 4 of the article for the definition of tameness). They then go on to give an explicit description of all indecomposable MCM modules over the ring \(k[[x,y,z]]/(xy, z^2)\). Finally, the authors apply the previous results to construct explicit families of \textit{matrix factorizations} associated to the hypersurface ring \(k[[x,y]]/(x^2y^2)\). Matrix factorizations were introduced in [\textit{D. Eisenbud}, Trans. Am. Math. Soc. 260, 35--64 (1980; Zbl 0444.13006)]; roughly speaking, they are the data of two-periodic tails of minimal free resolutions of finitely generated modules over hypersurface rings. maximal Cohen-Macaulay modules; matrix factorizations; MCM representation type Burban, I., Gnedin, W.: Cohen-Macaulay modules over some non-reduced curve singularities. arXiv:1301.3305v1 Cohen-Macaulay modules, Cohen-Macaulay modules in associative algebras, Representation type (finite, tame, wild, etc.) of associative algebras, Singularities in algebraic geometry Cohen-Macaulay modules over some non-reduced curve 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 In this article, we give the structure of codimension one foliations with canonical singularities and numerically trivial canonical class on varieties with klt singularities. Building on recent works of Spicer, Cascini-Spicer and Spicer-Svaldi, we then describe the birational geometry of rank two foliations with canonical singularities and canonical class of numerical dimension zero on complex projective three-folds. foliations; birational geometry; singularities Singularities in algebraic geometry, Singularities of holomorphic vector fields and foliations, Dynamical aspects of holomorphic foliations and vector fields Codimension one foliations with numerically trivial canonical class on singular spaces. II | 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 result of the paper is a simple characterization of normal varieties, having only log canonical, log terminal or rational singularities. The well-known theorem by Kempf states that a normal complex algebraic variety \(X\) with a resolution of singularities \(\pi:X'\to X\) has only rational singularities iff it is Cohen-Macaulay and satisfies \(\pi_*{\mathcal O}_{X'}(K_{X'})\simeq{\mathcal O}_X(K_X)\). In analogy, the authors show that, given a normal complex singularity \((X,x)\) and a reduced Weil divisor \(D\) such that \(K_X+D\) is \({\mathbb Q}\)-Cartier, then \((X,D)\) has a log-canonical singularity at \(x\) iff the stacks \((\pi_*({\mathcal O}(mK_{X'}+(m-1)D')))_x\) and \({\mathcal O}_X(m(K_X+D))_x\) are equal for all \(m\geq 1\), where \(D'=\pi^{-1}(D\cup\text{ Sing}X)_{\text{ red}}\) is a simple normal crossing divisor. Respectively, in case \(D=0\) the above condition characterizes log-terminal singularities. Among applications of the main result, the authors give formulas for plurigenera of isolated singular points of the varieties, satisfying the above conditions, as well as for the logarithmic Kodaira dimension of \(X\backslash\{0\}\) for a normal subvariety \(X\subset{\mathbb C}^n\) with a good \({\mathbb C}^*\)- action. quasihomogeneous singularity; graded algebra; Kodaira dimension Flenner, H; Zaidenberg, M, \textit{log-canonical forms and log canonical singularities}, Math. Nachr., 254/255, 107-125, (2003) Singularities in algebraic geometry, Global theory of complex singularities; cohomological properties, Complex surface and hypersurface singularities, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Log-canonical forms and log canonical 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 aim of this paper is to show that the structures on \(K\)-theory used to formulate Lusztig's conjecture for subregular nilpotent \(\mathfrak{sl}_n\)-representations are, in fact, natural in the McKay correspondence. The main result is a categorification of these structures. The no-cycle algebra plays the special role of a bridge between complex geometry and representation theory in positive characteristic.
For Part I, cf. Compos. Math. 138, No. 3, 337--360 (2003; Zbl 1086.17010). Modular Lie (super)algebras, Singularities of surfaces or higher-dimensional varieties, Twisted and skew group rings, crossed products, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Representation theory for linear algebraic groups Subregular representations of \(\mathfrak{sl}_n\) and simple singularities of type \(A_{n-1}\). II | 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 deals with the scheme structure of the jet schemes of determinantal varieties. Given a variety \(X\) over an algebraically closed field \(k\) of characteristic zero, the study of the arc space and jet schemes \({\mathcal J}_m(X), m\geq 0\) on \(X\) was suggested in \textit{J. F. Nash, jun.} [Duke Math. J. 81, No. 1, 31--38 (1996; Zbl 0880.14010)] as an approach to the study of its singularities. A nice result in \textit{M. Mustaţă} [Invent. Math. 145, No. 3, 397--424 (2001; Zbl 1091.14004)] states that if \(X\) is a local complete intersection variety, then all the jet schemes \({\mathcal J}_m(X)\) are irreducible if and only if \(X\) has only rational singularities. Since determinantal varieties only have rational singularities, it follows that all the jet schemes of the variety of singular square matrices are irreducible. A natural question arises: are the jet schemes of any determinantal variety irreducible?
In this paper, the author proves that the second jet scheme and all the odd jet schemes of \textit{essentially} all determinantal varieties are reducible, thus giving a negative answer to the above question. The idea of the proof is based on the computation of the dimensions of the schemes of jets centered on the singular locus of \(X\). Similar results appear in \textit{T. Košir} and \textit{B. A. Sethuraman} [J. Pure Appl. Algebra 195, No. 1, 75--95 (2005; Zbl 1085.14043)]. In addition, a formula for the number of irreducible components and their dimensions in the case of the variety of matrices with rank at most one is given (cf. Example 4.7 of \textit{M. Mustaţă} [J. Am. Math. Soc. 15, No. 3, 599--615 (2002; Zbl 0998.14009)]). As an application, the log canonical threshold of the pair \((\mathbb{A}^{rs},X)\) is computed. determinantal variety; jet schemes; complete intersections; rational singularities Cornelia Yuen, Jet schemes of determinantal varieties, Algebra, geometry and their interactions, Contemp. Math., vol. 448, Amer. Math. Soc., Providence, RI, 2007, pp. 261 -- 270. Determinantal varieties, Complete intersections, Singularities in algebraic geometry Jet schemes of determinantal 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 k be an algebraically closed field, \(R=k[[x,y]]\), \(m=(x,y)\) the maximal ideal of R and \(h(I)(z)=\sum h_ i(I)z^ i \) the Hilbert function of an ideal I of R of \(colength\quad n\) where \(h_ i(I)=\dim_ k(m^ i/((I\cap m^ i)+m^{i+1}))\). For a fixed polynomial h with nonnegative integer coefficients and \(h(1)=n\) the ideals I with Hilbert function \(h(I)=h\) are parametrized by a locally closed subscheme \(Z_ h\) of the punctual Hilbert scheme \(Hilb^ nR\) and give a stratification \(Hilb^ nR=\cup_{h(1)=n}Z_ h \) [\textit{A. A. Iarrobino}, Mem. Am. Math. Soc. 188 (1977; Zbl 0355.14001), Bull. Am. Math. Soc. 78, 819-823 (1972; Zbl 0268.14002) and \textit{J. Briançon}, Invent. Math. 41, 45-89 (1977; Zbl 0353.14004)].
The author constructs a cellular decomposition of the strata \(Z_ h\) and computes their Betti numbers by modifying the cellular decomposition of \(Hilb^ n{\mathbb{P}}_ 2\) given by \textit{G. Ellingsrud} and \textit{S. A. Strømme} [Invent. Math. 87, 343-352 (1987; Zbl 0625.14002)]. Hilbert stratum; Hilbert function Göttsche L., Manuscripta Math. 66 pp 253-- (1990) Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Betti numbers for the Hilbert function strata of the punctual Hilbert scheme in two variables | 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 Betti numbers of the Hilbert scheme compactification of the space of twisted cubic curves are computed, using a cell decomposition coming from an action of the multiplicative group with isolated fixed points. Due to a small mistake, easily corrected, numbers in the paper are too large. The correct value of the even Betti numbers can be found in a paper by \textit{G. Ellingsrud, R. Piene} and the reviewer [''On the variety of nets of quadrics defining twisted cubis'', Proc. Rocca di Papa conference 1985 (to appear in Lect. Notes Math.)]. The Betti numbers are 1, 2, 6, 10, 16, 19, 22, 19, 16, 10, 6, 2, 1. Betti numbers of the Hilbert scheme compactification of the space; of twisted cubic curves; Betti numbers of the Hilbert scheme compactification of the space of twisted cubic curves Schaub, D.: Sur l'homologie du schéma de Hilbert des cubiques de ? ? 3 de genre arithmetique nul. C.R. Acad. Sci., Paris, t.301, (série I) 307-310 (1985) (Equivariant) Chow groups and rings; motives, Parametrization (Chow and Hilbert schemes), Applications of methods of algebraic \(K\)-theory in algebraic geometry Sur l'homologie du schéma de Hilbert des cubiques de \({\mathbb{P}}^ 3_{{\mathbb{C}}}\) de genre arithmétique nul. (On the homology of the Hilbert scheme of genus 0 cubics in \({\mathbb{P}}^ 3_{{\mathbb{C}}})\) | 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 Fix an algebraically closed field \(k\). A curve singularity is a complete local Noetherian \(k\)-algebra \(S\) of Krull dimension 1 without nilpotent elements and such that \(S/M=k\), where \(M\) is the maximal ideal of \(S\); \(S\) is called plane if \(M\) is generated by 2 elements and it is called one branch curve singularity if it doesn't have zero divisors. \textit{A. Schappert} [Lect. Notes Math. 1273, 168--177 (1987; Zbl 0639.14016)], the first author and \textit{G.-M. Greuel} [in: The Brieskorn anniversary volume. Proceedings of the conference dedicated to Egbert Brieskorn on his 60th birthday, Oberwolfach, Germany, July 1996. Prog. Math. 162, 3--26 (1998; Zbl 0924.14014)] characterized plane curve singularities with no more than 1-parameter families of ideals.
In this paper, the authors give a criterion for a one branch curve singularity to have at most 2-parameter families of ideals. In particular they present a classification of such curve singularities using Arnold's list, see [\textit{V. I. Arnol'd}, \textit{S. M. Gusejn-Zade} and \textit{A. N. Varchenko}, Singularities of differentiable maps. Volume I. Monographs in Mathematics, Vol. 82. Boston-Basel-Stuttgart: Birkhäuser. (1985; Zbl 0554.58001)]. The proof of this result is based on the ``sandwich'' procedure used e.g. in [Zbl 0924.14014)] for the calculation of ideals. curve singularity; ideal; family of ideals; sandwich technique Structure, classification theorems for modules and ideals in commutative rings, Singularities of curves, local rings One branch curve singularities with at most 2-parameter families of ideals | 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 explicit computation of the intersection cohomology IH(X) à la Goresky-MacPherson of a complex space X is usually difficult. Nevertheless, according to Goresky-MacPherson, if a small resolution of the singularities \(\tilde X\to X\) of X exists, then IH(X) is roughly speaking the same as the cohomology \(H(\tilde X)\) of \(\tilde X.\) The author proves by an explicit construction the existence of a small resolution for any Schubert cell and therefore obtains a combinatorial description of the intersection cohomology. intersection cohomology; small resolution for any Schubert cell Zelevinskiĭ, A. V.: Small resolutions of singularities of Schubert varieties. Funct. anal. Appl. 17, No. 2, 142-144 (1983) Global theory and resolution of singularities (algebro-geometric aspects), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, (Co)homology theory in algebraic geometry Small resolutions of singularities of Schubert 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 prove the existence of a generically smooth component \(V\) of the Hilbert scheme \(H(d,g; \mathbb{P}^4)\) of smooth connected nondegenerate curves in the projective 4-space for every degree \(d \geq 21\) and genus \(g \geq 3d-27\) (for which \(H(d,g; \mathbb{P}^4)\) is nonempty), and we compute its dimension. Moreover, in the subrange \(g > 1 + d(d + 3)/14\), we expect the component \(V\) to be among the ``nicest'' and smallest components of \(H(d,g; \mathbb{P}^4)\) because its generic curve sits on the ``most general'' smooth surface allowed. We also prove a corresponding existence result for \(H(d,g; \mathbb{P}^5)\) for every \(d \geq 30\) and \(g \geq 2d - 16\), and we get some partial results for \(H(d,g; \mathbb{P}^n)\), \(n \geq 6\), as well.
To prove these results we develop a new criterion for a smooth curve, sitting on a smooth surface in \(\mathbb{P}^n\), to be unobstructed. Finally we determine a range in the \((d,g)\)-plane where there exist smooth connected curves on a smooth Castelnuovo surface in \(\mathbb{P}^n\) for \(n = 4\) and 5. smooth curve on a smooth surface; Hilbert scheme; nondegenerate curves in the projective 4-space DOI: 10.1515/crll.1996.475.77 Parametrization (Chow and Hilbert schemes), Projective techniques in algebraic geometry, Divisors, linear systems, invertible sheaves, Local deformation theory, Artin approximation, etc., Vanishing theorems in algebraic geometry, Special algebraic curves and curves of low genus, Low codimension problems in algebraic geometry Concerning the existence of nice components in the Hilbert scheme of curves in \(P^ n\) for \(n=4\) and 5 | 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 formula for the local Denef-Loeser zeta function of a superisolated singularity of hypersurface in terms of the local Denef-Loeser zeta function of the singularities of its tangent cone. We prove the monodromy conjecture for some surface singularities. These results are applied to the study of rational arrangements of plane curves whose Euler-Poincaré characteristic is three. topological zeta function; monodromy conjecture; local Denef-Loeser zeta function; superisolated singularity of hypersurface; rational arrangements of plane curves Artal Bartolo, E.; Cassou-Noguès, Pi.; Luengo, I.; Melle Hernández, A., Monodromy conjecture for some surface singularities, Ann. sci. éc. norm. supér. (4), 35, 4, 605-640, (2002) Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Complex surface and hypersurface singularities, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Monodromy conjecture for some 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 Let \(X\) be a complex projective manifold, and let \(\mathcal F \subset T_X\) be a (singular) foliation, that is a saturated subsheaf that is closed under the Lie bracket. The locus where \(\mathcal F \subset T_X\) is not a subbundle is called the singular locus \(\mathrm{sing}(\mathcal F)\) of the foliation, the foliation is regular when \(\mathrm{sing}(\mathcal F)=\emptyset\). The canonical bundle \(K_{\mathcal F}\) of the foliation \(\mathcal F\) is defined as the dual of \(\det(\mathcal F)\). In analogy with the classification of projective manifolds in terms of the canonical bundle, one expects that the positivity properties of \(K_{\mathcal F}\) govern the geometry of the foliation. For foliations on surfaces this program is fully established by the work of \textit{M. Brunella} [Birational geometry of foliations. Reprint of the 2000 edition with new results. Cham: Springer (2015; Zbl 1310.14002)] and \textit{M. McQuillan} [Pure Appl. Math. Q. 4, No. 3, 877--1012 (2008; Zbl 1166.14010)]. This paper contributes to the MMP for foliations on higher-dimensional manifolds by establishing several fundamental results in the case where the canonical bundle is numerically trivial:
The first main theorem states that if the anticanonical bundle \(K_{\mathcal F}^*\) is pseudoeffective and the singular locus \(\mathrm{sing}(\mathcal F)\) is not empty, then the manifold \(X\) is uniruled. If \(K_{\mathcal F}^*\) is not numerically trivial the statement also follows from the recent result of \textit{F. Campana} and \textit{M. Păun} [Publ. Math., Inst. Hautes Étud. Sci. 129, 1--49 (2019; Zbl 1423.14109)], but the case where \(c_1(K_{\mathcal F})=0\) shows a surprising interaction between the singularities of the foliation and the global geometry of \(X\). This interaction can be made more explicit for a foliation that has canonical singularities in the sense of McQuillan: under this assumption the authors prove that the leaves of \(\mathcal F\) are dominated by rational curves if and only if the canonical bundle \(K_{\mathcal F}\) is not pseudoeffective. For the proof of the first main theorem the authors extend a technique introduced by \textit{J.-P. Demailly} [in: Complex geometry. Collection of papers dedicated to Hans Grauert on the occasion of his 70th birthday. Berlin: Springer. 93--98 (2002; Zbl 1011.32019)] to show that if \(K_X\) is pseudoeffective, then the foliation \(\mathcal F\) admits a supplement \(\mathcal G \subset T_X\) such that \(T_X=\mathcal F \oplus \mathcal G\). In particular both subsheaves are subbundles.
The second main theorem gives a classification of codimension one foliations with \(c_1(K_{\mathcal F})=0\). If the foliation is regular, a complete description was established by \textit{F. Touzet} [Ann. Sci. Éc. Norm. Supér. (4) 41, No. 4, 657--670 (2008; Zbl 1166.32014)]. In view of the first main theorem one can thus assume that \(X\) is uniruled and the foliation \(\mathcal F\) has canonical singularities. In this case the authors prove that there exists an étale cover \(\mu: Y \times Z \rightarrow X\) such that the canonical bundle \(K_Y\) is trivial, and the foliation \(\mu^{-1}(\mathcal F)\) is a pull-back of a foliation \(\mathcal G\) on \(Z\) with {\em trivial tangent sheaf}. If the general leaf of the foliation \(\mathcal F\) is not algebraic, the variety \(Z\) is a projective equivariant compactification of a complex abelian Lie group and \(\mathcal G\) is induced by the action of a codimension one subgroup. The proof of this result involves an impressive variety of tools including deformation theory of rational curves along a foliation, the study of transversely projective structures and the reduction of the foliation to fields of positive characteristic. Parts of this very inspiring paper have been generalised recently by \textit{S. Druel} [``Codimension one foliations with numerically trivial canonical class on singular spaces'', Preprint, \url{arXiv:1809.06905}] to foliations on singular spaces. holomorphic foliation; MMP; singular foliation; uniruled manifolds; rational curves Singularities of holomorphic vector fields and foliations, Dynamical aspects of holomorphic foliations and vector fields, Minimal model program (Mori theory, extremal rays) Singular foliations with trivial canonical class | 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 prove a scheme-theoretic version of Mnev-Sturmfels universality, suitable to be used in the proof of Murphy's Law in Algebraic Geometry [\textit{R. Vakil}, Invent. Math. 164, No. 3, 569--590 (2006; Zbl 1095.14006)]. Somewhat more precisely, we show that any singularity type of finite type over \(\mathbb{Z}\) appears on some incidence scheme of points and lines, subject to some particular further constraints. Mnev-Sturmfels universality; Murphy's Law Lee, S.H. and Vakil, R., Mnëv-Sturmfels Universality for Schemes, arXiv:1202.3934v2 (2012). Families, moduli, classification: algebraic theory, Configurations and arrangements of linear subspaces, Local deformation theory, Artin approximation, etc. Mnëv-Sturmfels universality for 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 In this paper the author explains some new techniques for studying singularities of linear systems, with applications to birational maps between 3-fold Mori fibre spaces, and especially the property of birational rigidity. These techniques are closely related and all have something to do with Shokurov's connectedness principle -- see section 3.2. Though they do not as yet form a coherent method, they are intended to replace the combinatorial study of the resolution graph, started by \textit{V. A. Iskovskikh} and \textit{Yu. I. Manin} [Math. USSR, Sb. 15 (1971), 141-166 (1972); translation from Mat. Sb., Nov. Ser. 86(128), 140-166 (1971; Zbl 0222.14009)]. The author's goal is to provide concise but complete proofs of the known criteria for birational rigidity of 3-fold Mori fibre spaces. Chapter 2 contains a brief exposition of the Sarkisov program. Chapter 3 is a study of singularities of linear systems, based on Shokurov connectedness and inversion of adjunction, and represents the technical core of the paper. In chapter 4 are given several new proofs and generalisations of the rigidity theorem for conic bundles, first known by \textit{V. G. Sarkisov} [Math. USSR, Izv. 17, 177-202 (1981); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 44, 918-945 (1980; Zbl 0453.14017), Math. USSR, Izv. 20, 355-390 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 2, 371-408 (1982; Zbl 0593.14034)]. In chapter 5 is re-proved the Pukhlikov's rigidity criterion [see \textit{A. V. Pukhlikov}, Izv. Math. 62, No. 1, 115-155 (1998); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 1, 123-164 (1998; Zbl 0948.14008)] for Del Pezzo fibrations of degree 1 and 2. Chapter 6 represents a review of the main known rigidity theorems for Fano 3-folds: hypersurfaces and complete intersections. As a pattern example, in section 6.2 is proved the [known by \textit{V. A. Iskovskikh}, J. Sov. Math. 13, 815-868 (1980); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 12, 159-236 (1979; Zbl 0415.14025) and by \textit{V. A. Iskovskikh} and \textit{A. V. Pukhlikov}, J. Math. Sci., New York 82, No. 4, 3528-3613 (1996; Zbl 0917.14007)] rigidity of a general smooth 3-fold complete intersection of type (2,3), with the purpose to illustrate the power of the new methods. Throughout the paper are stated various conjectures and open problems. birational rigidity; Sarkisov program; Shokurov connectedness; 3-fold Mori fibre spaces A. Corti, Singularities of linear systems and 3-fold birational geometry, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser. 281, Cambridge University Press, Cambridge (2000), 259-312. \(3\)-folds, Divisors, linear systems, invertible sheaves, Rational and birational maps Singularities of linear systems and 3-fold birational geometry | 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 \(E\) be a vector bundle of rank \(r\) on a smooth curve \(X\). In the Hilbert scheme of rank \(k\) quotients of \(E\), the component Quot\(_{k,d}(E)\) denotes the Hilbert scheme of rank \(k\) quotients of given degree \(d\). When \(d\) is minimal say \(d=f_k:=f_k(E)\), a result due to \textit{S. Mukai} and \textit{F. Sakai} [Manuscr. Math. 52, 251--256 (1985; Zbl 0572.14008)] says that \(\dim\text{Quot}_{k,f_k}(E)\leq k(r-k)\). The paper under review gives an estimate of the dimension of Quot\(_{k,d}(E)\) for all \(d\),
\[
\dim\text{Quot}_{k,d}(E)\leq k(r-k)+(d-f_k)(k+1)(r-k).
\]
The proof is a clever induction of \(d-f_k\) to a Hecke modification \(E'\) of \(E\), which has \(d-f_k(E')<d-f_k(E)\).
By applying the estimate, the author obtains better bound \(N\) such that \(| \Theta_{\mathcal{SU}_X(r,e)}^p| \) is base point free whenever \(p\geq N\), where \(\mathcal{SU}_X(r,e)\) is the moduli space of (equivalent classes of) semistable bundles of rank \(r\) with a fixed determinant of degree \(e\). Similar results for \(\mathcal U_X(r,e)\) (without fixing the determinant) and moduli spaces of sheaves on surfaces are also given in the paper. Hecke modification; theta divisors Popa M.: Dimension estimates for Hilbert schemes and effective base point freeness on moduli spaces of vector bundles on curves. Duke Math J. 107, 469--495 (2001) Vector bundles on curves and their moduli, Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles Dimension estimates for Hilbert schemes and effective base point freeness on moduli spaces of vector bundles on 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 This paper surveys the authors' recent works on Du Bois singularities. Du Bois singularities are introduced by Steenbrink by using Du Bois complex which appears in Hodge theory. They give a characterization of Du Bois singularities in terms of canonical sheaves and show the relation between Du Bois singularities (a Hodge theoretic notion) and log canonical singularities (a birational geometric notion). These singularities are defined over the complex number field.
On the other hand, there are also notions (\(F\)-pure, \(F\)-regular, \(F\)-injective) of singularities over positive characteristic base field by using Frobenius map. They also show the relation between Du Bois singularities and \(F\)-injective singularities by reduction modulo \(p\). Du Bois singularities; log canonical singularities; Frobenius map Kovács, S.J., Schwede, K.E.: Hodge theory meets the minimal model program: a survey of log canonical and Du Bois singularities, Topology of stratified spaces. In: Math. Sci. Res. Inst. Publ., vol. 58, pp. 51-94. Cambridge University Press, Cambridge (2011) Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry, Positive characteristic ground fields in algebraic geometry Hodge theory meets the minimal model program: a survey of log canonical and Du Bois 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 Here we look at the postulation of general unions of zero-dimensional schemes more general than fat points w.r.t. a very positive linear system. Projective techniques in algebraic geometry, Divisors, linear systems, invertible sheaves Zero-dimensional schemes, blowing-up and asymptotic postulation | 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{Kyoji Saito}, \(\theta\)-invariant formulas for extended affine root systems and moduli spaces of elliptic singularities (Japanese) (p. 1-22); \textit{Shigeru Mukai}, On groups of automorphisms of K3 surfaces (Japanese) (p. 23-56); \textit{Masaaki Yoshida}, Differential equations of rank 3 on \({\mathbb{C}}P^ 2\) (Japanese) (p. 57-60); \textit{Iku Nakamura}, Infinitesimal deformations of cusp singularities (p. 61-62); \textit{Jiro Sekiguchi}, A note on the invariant holonomic system (p. 63- 70); \textit{Hiroyasu Tsuchihashi}, 3-dimensional singularities with resolutions whose exceptional sets are toric divisors (p. 71-94); \textit{Fumio Sakai}, Normal surfaces and intersection theory (p. 95-109); \textit{Nobuo Sasakura}, A Čech cohomological method of construction of holomorphic vector bundles (p. 110-144); \textit{Mitsuyoshi Kato}, Complex surfaces corresponding to weighted line configurations on complex projective planes (Japanese) (p. 145-160); \textit{Isao Naruki}, Straying of 3-curves and families of conic sections (Japanese) (p. 161-175); \textit{Kimio Watanabe} and \textit{Shihoko Ishii}, A classification and construction of purely elliptic singularities (Japanese) (p. 176-209); \textit{Mutsuo Oka}, On the resolution of hypersurface singularities (p. 210-256); \textit{Masataka Tomai}, On calculational formulas for geometric genera and elliptic singularities (Japanese) (p. 257-289); \textit{Toshisumi Fukui}, Certain ramified coverings of \(P_ 2({\mathbb{C}})\) (Japanese) (p. 290-304). Singularities; Varieties; Proceedings; Symposium; Kyoto; RIMS Singularities in algebraic geometry, Local complex singularities, Global theory and resolution of singularities (algebro-geometric aspects), Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Deformations of singularities, Deformations of complex singularities; vanishing cycles, Singularities of surfaces or higher-dimensional varieties, Complex singularities, Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces Recent results on singularities of varieties. Proceedings of a Symposium held at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto, March 15-17, 1984 | 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 show that a version of the desingularization theorem of Hironaka \(\mathcal C^{\infty}\) holds for certain classes of functions (essentially, for subrings that exclude flat functions and are closed under differentiation and the solution of implicit equations). Examples are quasianalytic classes, introduced by E. Borel a century ago and characterized by the Denjoy-Carleman theorem. These classes have been poorly understood in dimension \(> 1\). Resolution of singularities can be used to obtain many new results; for example, topological Noetherianity, curve selection, Łojasiewicz inequalities, division properties. quasianalytic; Denjoy-Carleman class; resolution of singularities Bierstone E., Milman P.D.: Resolution of singularities in Denjoy-Carleman classes. Selecta Math. (N.S.) 10(1), 1--28 (2004) Real-analytic and semi-analytic sets, \(C^\infty\)-functions, quasi-analytic functions, Differentiable maps on manifolds Resolution of singularities in Denjoy-Carleman classes | 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 \(V\) be a normal algebraic surface over an algebraically closed field of characteristic zero. Suppose that the singular locus of \(V\) consists of the only point \(P \in V\) which is a quotient singularity. Let \(f \colon X \rightarrow V\) be a minimal resolution and let \(D\) be the reduced exceptional divisor with respect to \(f.\) The singular point \(P\) is called almost minimal when the pair \((X,D)\) is [see \textit{M. Miyanishi} and \textit{S. Tsunoda}, Jap. J. Math., New Ser. 10, 195-242 (1984; Zbl 0596.14023)]. Making use of Mori theory [\textit{S. Mori}, Ann. Math., II. Ser. 116, 133-176 (1982; Zbl 0557.14021)], the author studies such singularities with logarithmic Kodaira dimension \(\overline{\kappa}(X-D) = -\infty\) and obtains a classification theorem when the support of \(D\) is contained in a fiber of a certain \({\mathbb P}^1\)-fibration. Then the theory of rational surfaces with almost minimal singularities is considered. In fact, the author gives a classification of almost minimal pairs \((X,D)\) in the cases where \(\overline{\kappa}(X-D) = 1\) or \(D\) is irreducible. In conclusion some intersecting examples with \(\overline{\kappa}(X-D) = 0\) are described in details. normal surface; rational surface; almost minimal singularity; quotient singularity; minimal resolution; exceptional divisor; canonical divisor; extremal curve; logarithmic Kodaira dimension; Hirzebruch surface; Picard number; dual graph Kojima H., J. Math. Kyoto Univ. 38 pp 77-- (1998) Global theory and resolution of singularities (algebro-geometric aspects), Rational and ruled surfaces, Embeddings in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Almost minimal embeddings of quotient singular points into 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 Let \(X\subset\mathbb P^n\), \(n=3,4\), be a (complex, projective) hypersurface, with only ordinary singularities. The relations between various (numerical) projective invariants of \(X\) were studied since 19'th century, The authors rederive numerous classical relations between these invariants using the universal Thom polynomials of (multi-)singularities of maps.
In more details, the ordinary singularities of \(X\subset \mathbb{P}^3\) are of the three possible types: the double points (\(xy=0\)), the triple points (\(xyz=0\)), the crosscap (\(xy^2=z^2\)). The possible ordinary singularities of \(X\subset \mathbb{P}^4\) contain (in addition) the types \(xyzw=0\), \(w(xy^2-z^2)=0\).
The projective invariants for \(X\subset \mathbb{P}^3\) are: the degree, the Euler characteristic, the rank/class of \(X\) (i.e. the first/second polar), the number of cusps in generic projection \(X\to \mathbb{P}^2\), the degree of the double curve \(D\), the rank of \(D\), the class of immersion of \(D\) in \(X\), the number of triple points, the number of crosscaps. The authors express these invariants in terms of the four basic ones: the degree and pushforwards of Chern classes.
For \(X\subset \mathbb{P}^4\) there are many more invariants, at least twenty. The authors express them in terms of the seven basic ones, as before. Thom polynomials; classical enumerative geometry; projective surfaces Enumerative problems (combinatorial problems) in algebraic geometry, Classical problems, Schubert calculus, Global theory of complex singularities; cohomological properties Classical formulae on projective surfaces and 3-folds with ordinary singularities, revisited | 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 noetherian local ring \(R\) of prime characteristic is said to be \(F\)-rational if every parameter ideal is tighly closed. In an earlier paper, the second author [``\(F\)-rational rings have rational singularities'' (preprint)] showed that \(F\)-rational type implies rational singularities. \textit{A. Conca} and \textit{J. Herzog} [``Determinantal rings of one-sided ladders are \(F\)-rational'' (preprint)] used this result to prove that one-sided ladder determinantal ideals have rational singularities. This can be deduced also from works of \textit{Mulay} and \textit{Ramanathan} on Schubert varieties. Since the classical determinantal varieties have rational singularities, one may ask whether this holds for the larger class of ladder determinantal varieties. The main result of this paper confirms this question for complete intersection ladder determinantal varieties by showing that all ideals of their coordinate rings are tightly closed. tight closure; rational singularities; complete intersection ladder determinantal varieties \beginbarticle \bauthor\binitsD. \bsnmGlassbrenner and \bauthor\binitsK. E. \bsnmSmith, \batitleSingularities of certain ladder determinantal varieties, \bjtitleJ. Pure Appl. Algebra \bvolume101 (\byear1995), no. \bissue1, page 59-\blpage75. \endbarticle \endbibitem Linkage, complete intersections and determinantal ideals, Singularities in algebraic geometry, Determinantal varieties, Integral closure of commutative rings and ideals Singularities of certain ladder determinantal 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 smooth projective complex surface, and denote by \(T=\text{Hilb}^n(X)\) the Hilbert scheme of \(n\) points in \(X\), where \(n\) is a fixed positive integer. Let \(Z\subset T\times X\) be the incidence correspondence, and consider a line bundle \(L\) on \(X\). One defines the tautological bundle \(L^{[n]}\) associated to \(L\) on \(T\) as \(L^{[n]}:= \pi_*(p^*L)\), where \(p:Z\to X\) and \(\pi:Z\to T\) are the natural projections. For any positive integer \(k\), there is a natural morphism \(H^0(X,L)^{\otimes k}\to H^0(\text{Hilb}^n(X), \left(L^{[n]}\right)^{\otimes k})\), which is equivariant with respect to the natural action induced by the symmetric group \(\mathfrak S_k\).
In the paper under review, the author proves that previous morphism is in fact an isomorphism, provided that \(n\geq k\). Taking invariants, one deduces the isomorphism
\[
S^kH^0(X,L) \simeq H^0(\text{Hilb}^n(X), S^kL^{[n]}),
\]
where \(S^k\) means \(k-\)th symmetric power. The proof relies on the study of the stratification of the fibred product \((Z/T)^k\) induced from the canonical stratification of \(X^k\). This argument works well only when \(n\geq k\), and in fact at the end of the paper the author remarks that the result is not true when \(n<k\), and that it does not apply to the computation of the sections of the Donaldson determinant bundle [see \textit{G. Danila}, Ann. Inst. Fourier 50, No. 5, 1323--1374 (2000; Zbl 0952.14010)]. Hilbert scheme of points; projective surface; symmetric group; Donaldson determinant bundle Gentiana Danila, Sections de la puissance tensorielle du fibré tautologique sur le schéma de Hilbert des points d'une surface, Bull. Lond. Math. Soc. 39 (2007), no. 2, 311 -- 316 (French, with English summary). Parametrization (Chow and Hilbert schemes), Vanishing theorems in algebraic geometry Sections of the tensor power of the tautological bundle on the Hilbert scheme of points of 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 We consider a mixed Hodge module \(M\) on a normal surface singularity \((X,x)\) and a holomorphic function \(f:(X,x) \to (\mathbb{C},0)\). For the case that \(M\) has an abelian local monodromy group, we give a formula for the spectral pairs of \(f\) with values in \(M\). This result is applied to generalize the Sebastiani-Thom formula and to describe the behaviour of spectral pairs in series of singularities. series of plane curve singularities; singularity spectrum; surface singularity; mixed Hodge module; spectral pairs A. NéMethi and J. H. M. Steenbrink, Spectral pairs, mixed Hodge modules, and series of plane curve singularities , New York J. Math. 1 (1994/95), 149--177. Singularities of curves, local rings, Transcendental methods, Hodge theory (algebro-geometric aspects), Variation of Hodge structures (algebro-geometric aspects), Families, moduli of curves (algebraic), Singularities of surfaces or higher-dimensional varieties Spectral Pairs, Mixed Hodge Modules, and Series of Plane Curve 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 purpose of this paper is to study the connectedness of the Hilbert scheme \(H_{d,g}\) of degree \(d\) and genus \(g\) curves. Thanks to the methods of triads [cf. \textit{R. Hartshorne, M. Martin-Deschamps} and the author, Math. Ann. 315, No.3, 397--468 (1999; Zbl 0976.14019)], we show that a large class of curves (the curves whose Rao--module is Koszul, i.e. a complete intersection) are in the connected component of extremal curves. This generalizes widely several recent results. [P]\textsc{D. Perrin},\textit{Un pas vers la connexité du schéma de Hilbert: les courbes de Koszul sont dans la composante des extrémales}, Collect. Math.,\textbf{52}, 3 (2001), pp. 295-319. Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) A step towards the connectedness of the Hilbert scheme: The Koszul curves are in the component of extremal 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 article gives a summary of the author's unpublished Ph.D thesis. It is known that Dynkin diagrams can be separated in two classes: the simply laced (or homogeneous) ones \(A_k\) (\(k\geq 1\)), \(D_k\) (\(k\geq 4\)), \(E_6\), \(E_7\) and \(E_8\), and the non-simply laced (or inhomogeneous) ones \(B_k\) (\(k\geq 2\)), \(C_k\) (\(k\geq 3\)), \(F_4\) and \(G_2\).
The aim of the article is to generalise a construction by H. Cassens and P. Slodowy of the semiuniversal deformations of the homogeneous simple singularities to the inhomogeneous ones.
To a homogeneous simple singularity, one can associate the representation space of a particular quiver.
This space is endowed with an action of the symmetry group of the Dynkin diagram associated to the simple singularity which allows the construction and explicit computation of the semiuniversal deformations of the inhomogeneous simple singularities.
By quotienting such maps, deformations of other simple singularities are obtained.
In some cases, the discriminants of these last deformations are computed. simple singularities; quiver representations; root systems; foldings Deformations of singularities, Representations of quivers and partially ordered sets, Root systems Deformations of inhomogeneous simple singularities and quiver representations | 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 his famous paper ``Local Normal Forms of Functions'' [Invent. Math. 35, 87--109 (1976; Zbl 0336.57022)], \textit{V. I. Arnold} presented a classification of hypersurface singularities of moda\-lity smaller or equal to two over the complex numbers together with a singularity determinator (an algorithm to compute the normal form of a given singularity). Arnold gave also a corresponding classification over the real numbers.
The authors developed algorithms to determine the normal form of simple and unimodal singularities over the real numbers. This paper contains the splitting lemma and the case of simple singularities. The algorithms are implemented in the \textsc{Singular} library \texttt{realclassify.lib}. hypersurface singularities; algorithmic classification; real geometry Marais, M., Steenpaß, A.: The classification of real singularities using Singular Part I: splitting lemma and simple singularities. J. Symb. Comput. 68, 61--71 (2015) Singularities in algebraic geometry, Computational aspects of algebraic surfaces, Real algebraic and real-analytic geometry The classification of real singularities using \textsc{Singular}. I: Splitting lemma and 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 Let \((X,x)=\{f=0\}\subset\mathbb{C}^ 3\) be an isolated quasi-homogeneous hypersurface singularity and \(G=\langle(e_ n^{i_ 0},e_ n^{i_ 1},e_ 2^{i_ 2})\rangle\) a cyclic group which is diagonally acting on \((X,x)\), where \(e_ n=\exp(2\pi\sqrt{-1}/n)\). Then the quotient \((x/G,\pi(0))\) is a normal surface singularity with \(\mathbb{C}^*\)-action. In our paper we study those surface singularities. Although they are very special normal surface singularities with \(\mathbb{C}^*\)-action, it is meaningful to study them. Because, if we consider a singularity of such type, we can easily compute the genus \(p_ g\) and determine if it is Gorenstein, though it is not easy to do them for general normal surface singularities with \(\mathbb{C}^*\)-action.
In section 2, we give a method to resolve them; it is obtained by the techniques of P. Orlik, Ph. Wagreich and A. Fujiki. Although the algorithm is not so simple, it is easy to program for the computer. In section 3, we give a formula to compute the geometric genus \(p_ g(X/G,\pi(0))\) and a criterion for \((X/G,\pi(0))\) to be Gorenstein. In section 4, we classify all rational singularities which are obtained as cyclic quotients of the simple elliptic singularity \(\tilde E_ 8\) by a reflection free finite cyclic group \(G\). These singularities are already found as weighted dual graph from other different points of view. Our result gives concrete representations for them. quasi-homogeneous hypersurface singularity; normal surface singularity T. Tomaru, Cyclic quotients of \(2\)-dimensional quasi-homogeneous hypersurface singularities, Math. Z., 210 (1992), 225-244. Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties Cyclic quotients of \(2\)-dimensional quasi-homogeneous hypersurface 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 This paper studies formality of the differential graded algebra \(\mathrm{RHom}^{\bullet}(E,E)\), where \(E\) is a semistable sheaf on a \(K3\) surface. The main tool is Kaledin's theorem on formality in families. For a large class of sheaves \(E\), this DG algebra is formal, therefore we have an explicit description of the singularity type of the moduli space of semistable sheaves at the point represented by \(E\). This paper also explains why Kaledin's theorem fails to apply in the remaining case. Zhang, Z., A note on formality and singularities of moduli spaces, Mosc. Math. J., 12, 863-879, (2012) Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Spin and Spin\({}^c\) geometry, \(K3\) surfaces and Enriques surfaces A note on formality and singularities of moduli 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 \(V\) be an irreducible affine algebraic variety over a field \(k\) of characteristic zero, and let \((f_0,\dots ,f_m)\) be a sequence of elements of the coordinate ring. There is probably no elementary condition on the \(f_i\) and their derivatives which determines whether the blow-up of \(V\) along \((f_0,\dots ,f_m)\) is non-singular. The result of the current paper is that there indeed is such an elementary condition, involving the first and second derivatives of the \(f_i,\) provided we admit certain singular blow-ups, all of which can be resolved by an additional Nash blow-up.
This paper is the promised sequel of an earlier paper [\textit{J. A. Moody}, J. Algebra 189, No. 1, 90-100 (1997; Zbl 0891.14004)], in which the same program was carried out for individual vector fields. Indeed, the current paper generalizes the result of the mentioned paper to algebraic foliations of arbitrary codimension, and the case of codimension zero foliations corresponds to the problem of resolving the singularities of \(V.\)
The results have a close connection with a question of Nash concerning resolutions. We now describe this briefly following a paper by \textit{J. Milnor} [Math. Intell. 17, No. 3, 11-17 (1995; Zbl 0846.01016)] where further references may be found. Let \(r=\dim(V).\) Suppose \(V=V_0 \subset W_0\) is an embedding in a non-singular variety over \(k.\) Then \(V_0\) lifts to a subvariety \(V_1\subset W_1=\text{Grass}_r(W_0)\) of the variety of \(r\)-planes in the tangent bundle of \(W_0\). The natural map \(\pi:V_1\to V_0\) is called the Nash blow-up of \(V_0\). It is the lowest blow-up where \(\pi^*(\Omega_{V_0/k})/\)torsion is locally free. Now we can repeat the process, giving a variety \(V_2\subset W_2=\text{Grass}_r(W_1)\) and so-on, and the question is whether eventually \(V_i\) is non-singular.
There is a particular explicit sequence of ideals \(R=J_0, J_1, J_2,\dots \subset R\) so that \(V_0= \text{Bl}_{J_0}V\), \(V_1=\text{Bl}_{J_1}V\), \(V_2=\text{Bl}_{J_2}V,\dots .\) with \(J_i|J_{i+1}\) for all \(i.\) Applying our earlier paper [\textit{J. A. Moody}, Ill. J. Math. 45, No. 1, 163-165 (2001; Zbl 0989.13001)], \(V_i\) is non-singular if and only if the ideal class of \(J_{i+1}\) divides some power of the ideal class of \(J_i.\)
The present paper brings things down to earth considerably: Such a divisibility of ideal classes implies that for this value of \(i\) and for some \(N\geq r+2\)
\[
J_i^{N-r-2}J_{i+1}^{r+3}=J_i^NJ_{i+2}.
\]
Yet note that this identity in turn implies \(J_{i+2}\) is a divisor of some power of \(J_{i+1}.\) Thus although \(V_i\) may fail to be non-singular, when the identity holds the next variety \(V_{i+1}\) must be non-singular. Thus the Nash question is equivalent to the assertion that the identity above holds for some sufficiently large \(i\) and \(N.\) Nash question; Jacobian condition; non-singular blow-up; Nash blow-up J.A. Moody, On resolving singularities, Maths Institute, Warwick University, 2001, preprint. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) On resolving 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 \(X\) be an irreducible smooth projective surface over a subfield \(k\) of \(\mathbb C\). For a positive integer \(n\), let \(X^{[n]}\) be the Hilbert scheme of artinian closed subschemes of \(X\) with length \(n\); this is an irreducible smooth projective variety of dimension \(2n\). The main result of this paper is that if \(k = \mathbb C\) (resp.~\(k\) is a number field) and the Hodge (resp.~Tate) conjecture is true for all powers of \(X\), then the Hodge (resp.~Tate) conjecture holds for all \(X^{[n]}\). As a consequence, the Hodge and Tate conjectures are established for \(X^{[n]}\) whenever \(X\) is an abelian surface or a Kummer surface. Hodge conjecture; Tate conjecture; Hilbert schemes; Heisenberg algebra Algebraic cycles, Parametrization (Chow and Hilbert schemes), Arithmetic ground fields for abelian varieties, Transcendental methods, Hodge theory (algebro-geometric aspects) On the Hodge conjecture and the Tate conjecture for the Hilbert schemes of an abelian 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 It is known that versal deformation of nonsimple singularities depend on moduli. However it can be topologically trivial along some or all of them. The first step is to determine the versal discriminant, the next one is to integrate them.
The author of this paper shows that in general quasihomogeneous miniversal deformations of \(J_{k,0}\) singularities are topologically trivial along the moduli. The only exceptions are when \(k\geq 3\) and the singularity is harmonic or anharmonic.
The results of this paper are some generalization of the corresponding results of J. Damon and A. Galligo, concerning the case of the Pham singularity (\(J_{3,0}\) in Arnold classification). moduli of singularities; topological trivialization; \(J_{k,0}\) singularities Equisingularity (topological and analytic), Complex-analytic moduli problems, Families, moduli of curves (analytic) On the topological triviality along moduli of deformations of \(J_{k,0}\) 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 smooth scheme \(X\) over a field \(k\) of positive characteristic is said to be strongly liftable over \(W_2(k)\), if \(X\) and all prime divisors on \(X\) can be lifted simultaneously over \(W_2(k)\). In this paper, we first deduce the Kummer covering trick over \(W_2(k)\), which can be used to construct a large class of smooth projective varieties liftable over \(W_2(k)\), and to give a direct proof of the Kawamata-Viehweg vanishing theorem on strongly liftable schemes. Secondly, we generalize almost all of the results in [the author, Math. Res. Lett. 17, No. 3, 563--572 (2010; Zbl 1223.14026); Math. Res. Lett. 18, No. 2, 315--328 (2011; Zbl 1250.14013)] to the case where everything is considered over \(W(k)\), the ring of Witt vectors of \(k\). strongly liftable schemes; Kawamata-Viehweg vanishing; positive characteristic Xie, Q. H.; Wu, J., Strongly liftable schemes and the Kawamata-Viehweg vanishing in positive characteristic III, J. Algebra, 395, 12-23, (2013) Positive characteristic ground fields in algebraic geometry, Vanishing theorems in algebraic geometry Strongly liftable schemes and the Kawamata-Viehweg vanishing in positive characteristic. III | 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 rational projective curve \(C\) of degree \(d\) over the algebraically closed field \(k\) can be given as the closure of the image of a rational map \(\Psi: \mathbb{P}^1\to \mathbb{P}^{n-1}\)\ ,\ \(\Psi(q)=(g_1(q):\ldots:g_n(q))\;,\;g_1, \ldots, g_n\in B=k[x,y]\) homogeneous of degree \(d\). The aim of the authors is to study the curve using the syzygy module \(\left\{\begin{pmatrix} x_1\\\vdots\\x_n\end{pmatrix} \in B^n\;|\;\sum\limits^n_{i=1} c_ig_i=0\right\}\). The main tool is the Hilbert--Burch matrix \(\varphi\) such that the sequence
\[
0\to \overset{n-1}{\underset{i=1}{\oplus}} B(-d-d_i)@> {\varphi}>> B(-d)^n @> {(g_1, \ldots, g_n)}>> B
\]
is exact.
In Chapter 1 the row ideals of \(\varphi\) are used to identify and describe the singular points of \(C\) including the number of branches and the multiplicity of each branch. Chapter 2 deals with the blow--up of a singular point \(p\in C\). A matrix \(\varphi'\) is constructed such that the maximal minors of \(\varphi'\) parametrize the closure of the blow--up at \(p\) in a neighbourhood of \(p\). Chapter 3 studies curves of even degree \(d=2c\). The singular points on or infinitely near \(C\) of multiplicity \(c\) are described. Chapter 4 gives a complete classification of parametrizations of rational plane curves of degree \(d=2c\) by the configuration of multiplicity \(c\) singularities appearing on or infinitely near \(C\). Chapter 5 studies families of curves given by the Hilbert--Burch matrix. This is a basis for a stratification of the parameter space of the family in Chapter 6, each stratum gives rise to a curve with a predetermined configuration of singularities. In Chapter 7 the Jacobian matrix associated to the parametrization is considered. The Jacobian identifies the non--smooth branches of the curve and gives the multiplicity of the branches. In Chapter 8 the degree of the singularities are studied using conductor techniques. In Chapter 9 the results are applied to rational plane quartics. Hilbert-burch matrix; rational curve; singularities; rational plane quartics D. Cox, A. Kustin, C. Polini and B. Ulrich. \textit{A study of singularities on rational curves via syzygies}. MemoirsAMS 222 (2013),Amer. Math. Soc., Providence, RI. ArXiv 1102.5072. Singularities of curves, local rings, Plane and space curves, Families, moduli of curves (algebraic), Computational aspects of algebraic curves, Multiplicity theory and related topics A study of singularities on rational curves via syzygies | 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 Der kompaktifizierte Quotientenraum \(Y_K\) des Produkts von \(n\) oberen Halbebenen nach der Hilbertschen Modulgruppe eines total-reellen algebraischen Zahlkörpers \(K\) vom Grad \(n\) über \(\mathbb{Q}\) ist im Fall \(n=2\) genau dann rational, wenn das geometrische Geschlecht \(p_g=0\) ist. Für \(n=3\) fand \textit{H. G. Grundman} [Math. Ann. 300, 77-88 (1994; Zbl 0807.11026)] drei total-reelle nicht-galoissche kubische Zahlkörper, für die \(p_g=0\) aber \(Y_k\) nicht rational ist. Der Autor untersucht nun den desingularisierten kompaktifizierten Quotientenraum \(X_0({\mathfrak n})\) der Kongruenzuntergruppe \(\Gamma_0({\mathfrak n})\) der Hilbertschen Modulgruppe \(\Gamma\) eines galoisschen total-reellen kubischen Zahlkörpers \(K\) zu einem ganzen Ideal \({\mathfrak n}\), das nicht die Hauptordnung ist. Für \(d_k\geq 13^2\) bestimmt er \(p_g\) mit der Formel von \textit{H. Saito} [J. Math. Kyoto Univ. 24, 285-303 (1984; Zbl 0547.10027)]. Mit Hilfe der Formel von \textit{E. Thomas} für den Defekt einer Spitze [Math. Ann. 264, 397-411 (1983; Zbl 0511.14001)] gewinnt der Verfasser sein Hauptresultat, daß alle \(X_0({\mathfrak n})\) mit \(p_g\leq 1\) vom allgemeinen Typ sind, insbesondere, dass es sowohl für \(p_g=0\) als auch für \(p_g=1\) Quotientenräume vom allgemeinen Typ gibt. Hilbert modular threefolds; geometric genus Modular and Shimura varieties, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Families, moduli, classification: algebraic theory A note on Hilbert modular 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 [For the entire collection see Zbl 0653.00006.]
This is a survey describing the results in and the background for the author's papers in J. Reine Angew. Math. 359, 90-105 and 362, 4-24 (1985; Zbl 0603.14006 and 14007)]. The problem is to find criteria for the infinite dimensionality of the so called \(A_ 0(X)\), the kernel of the degree map from the Chow group of zero cycles of a normal projective surface. Bloch's conjecture; Chow group; zero cycles (Equivariant) Chow groups and rings; motives, Algebraic cycles, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes) Zero cycles on a singular surface: An introduction | 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 \(G(b^+, b^-)\) be the Grassmannian of \(B^+\)-dimensional positive definite subspaces of the inner product space \(\mathbb R^{b^+, b^-}\) of signature \((b^+, b^-)\). This paper concerns the construction of automorphic forms on \(G(b^+, b^-)\) which have singularities along smaller sub-Grassmannians. The main tool used in the paper is the extension of the usual theta correspondence to automorphic forms with singularities developed by \textit{J. Harvey} and \textit{G. Moore} [Nucl. Phys. B 463, 315--368 (1996; Zbl 0912.53056)]. It is used to construct families of holomorphic automorphic forms which can be written as infinite products. This extends the previous results for \(G(2, b^-)\) by the author [\textit{R. E. Borcherds}, Invent. Math. 120, 161--213 (1995; Zbl 0932.11028)], and such automorphic forms provide many new examples of generalized Kac-Moody superalgebras.
The paper gives a common generalization of several well-known correspondences, including the Shimura and Maass-Gritsenko correspondences, to modular forms with poles at cusps. It also contains proofs of some congruences satisfied by the theta functions of positive definite lattices and provides a sufficient condition for a Lorentzian lattice to have a reflection group with a fundamental domain of finite volume. Finally, the paper discusses some examples suggesting that these automorphic forms with singularities are related to Donaldson polynomials and to mirror symmetry for \(K3\) surfaces. automorphic forms; Grassmannians; theta functions; Kac-Moody algebras; correspondences R. E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491-562. Other groups and their modular and automorphic forms (several variables), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Relationship to Lie algebras and finite simple groups, Modular correspondences, etc., Grassmannians, Schubert varieties, flag manifolds Automorphic forms with singularities on Grassmannians | 0 |
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