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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 [For the entire collection see Zbl 0683.00009.]
Let Y be a normal projective variety over an algebraically closed field k and L an ample line bundle on Y. The author looks for conditions on (Y,L) such that
(*) each normal projective variety X containing Y as an ample Cartier divisor such that the normal bundle of Y in X is equal to L is isomorphic to the projective cone over (Y,L) and Y is embedded in X as the infinite section.
(*) can be considered as a rigidity property of (Y,L) with respect to deformations and the first criterion of the author for (Y,L) to satisfy (*) is in terms of Schlessinger's deformation theory: let S be the graded coordinate algebra of (Y,L) and \(S_ 0\) the localization at the irrelevant maximal ideal. A sufficient condition is that \(depth(S_ 0)\geq 3\) and S has no deformation of negative weight. This is applied to show e.g. that the singular Kummer varieties of dimension \(\geq 3\) satisfy (*) with respect to any ample L.
Then the author considers the case where Y is a \({\mathbb{P}}^ n\)-bundle over a smooth projective curve B of positive genus, \(char(k)=0\). If X is a singular normal projective variety containing Y as an ample Cartier divisor and if L is the normal bundle of Y in X then X is isomorphic to the projective cone over (Y,L). This result allows the author to give a complete description of all normal projective varieties whose hyperplane sections are \({\mathbb{P}}^ n\)-bundles over a curve.
The results of the paper are extensions of previous results of the author. infinitesimal deformations of negative weights; ample Cartier divisor; singular Kummer varieties Infinitesimal methods in algebraic geometry, Formal methods and deformations in algebraic geometry, Embeddings in algebraic geometry, Singularities in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Infinitesimal deformations of negative weights and hyperplane sections | 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 review some basics on the theory of generic singularities in algebraic geometry. We state certain approaches towards the explicit local equations of such singularities. The main point of view is the constructive approach in commutative algebra and algebraic geometry. Singularities in algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc. Local defining ideals of ordinary 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 \(G\) be a complex, semisimple Lie group, and \(N\) the nilpotent variety in the Lie algebra of \(G\). The hyperkähler metrics associated to minimal singularities in the nilnotent variety are studied in this paper. Kronheimer's 4-dimensional ALE (Asymptotically Locally Euclidean) spaces are naturally realized within the context of coadjoint orbits and are identified as certain moduli spaces of SU(2)-invariant instantons on \(\mathbb{R}^k\setminus\{0\}\) with appropriate boundary conditions. The author also shows that the hyperkähler metrics on the resolution of the \(D_2\) singularity arise within coadjoint orbits. The author further gives the description of the hyperkähler metric on the orbit of highest root vectors. hyperkähler metrics; nilnotent variety; coadjoint orbits [6] R. Bielawski, 'On the hyperkähler metrics associated to singularities of nilpotent varieties', Ann. Glob. Anal. Geom. 14 (1996), 177-191. Special Riemannian manifolds (Einstein, Sasakian, etc.), Classical groups (algebro-geometric aspects), Differential geometry of homogeneous manifolds, Singularities in algebraic geometry, Other algebraic groups (geometric aspects) On the hyperkähler metrics associated to singularities of nilpotent 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 aim of this note, says the author in the introduction, is to introduce some recent applications of perverse sheaves to the study of complex hypersurface singularities. Thus it is largely a survey article, but the proof of one result is given here for the first time.
The author first briefly recalls some general facts about Milnor fibres, in particular the vanishing theorem of Kato and Matsumoto for the reduced cohomology \(\widetilde{H}^j(F_0,{\mathbb C})\) of the Milnor fibre \(F_0\), outside the range \(n-s\leq j\leq n\) where \(s\) is the dimension of the stratified singular locus. This theorem is later deduced quickly from general theorems about \({\mathcal D}\)-modules and perverse sheaves. In between, he states and proves the theorem of the author and \textit{P. Nang} [Math. Z. 249, No. 3, 493--511 (2005); addendum ibid. 250, No. 3, 729 (2005; Zbl 1066.14005); \texttt{math.AG/0410383}], generalising earlier results of Nang and the author and of Dimca. This bounds the number of Jordan blocks for a given eigenvalue of monodromy in terms of Betti numbers of complex links. Again the proof uses perverse sheaves and is quite short.
The ideas and main applications of \({\mathcal D}\)-modules and perverse sheaves in the context of singularity theory are briskly but clearly outlined here, and the paper is written in an approachable way, assuming the minimum of technical knowledge. \({\mathcal D}\)-module; perverse sheaf; Milnor fibre; monodromy Takeuchi K.: Perverse sheaves and Milnor fibers over singular varieties. Adv. Stud. Pure Math. 46, 211--222 (2007) Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Singularities in algebraic geometry, Sheaves of differential operators and their modules, \(D\)-modules, Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs, Milnor fibration; relations with knot theory Perverse sheaves and Milnor fibers over 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 Let \(f(x_0, \dots, x_n) = 0\) be a germ of analytic hypersurface in \(\mathbb{C}^{ n + 1}\) with isolated singularity at the origin. The finite dimensional \(\mathbb{C}\)-algebra \(A = \mathbb{C} \{z_0, \dots, z_n\}/(f, \partial f/z_0, \dots, \partial f/z_n)\) is called the moduli algebra of the singularity: by a result of \textit{J. N. Mather} and \textit{S. Yau} [Invent. Math. 69, 243-251 (1992; Zbl 0499.32008)] two isolated singularities are analytically isomorphic if and only if their moduli algebras are isomorphic. To \(A\) there is associated the Lie algebra \(L\) of its derivations. If \(f\) is quasihomogeneous, then both \(A\) and \(L\) are naturally graded. In the present paper it is proven that in such a case all homogeneous pieces of negative degree of \(L\) are zero. This leads to a criterion characterizing quasihomogeneous singularities. isolated singularity; hypersurface singularity; germ of analytic hypersurface; moduli algebra; quasihomogeneous singularities Chen, H.; Xu, Y. -J.; Yau, S.: Nonexistence of negative weight derivations of moduli algebras of weighted homogeneous singularities. J. algebra 172, 243-254 (1995) Complex surface and hypersurface singularities, Invariants of analytic local rings, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Nonexistence of negative weight derivation of moduli algebras of weighted homogeneous 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 Given a smooth irreducible complex projective surface \(X\), the Hilbert scheme \(X^{[n]}\) of \(n\) points on \(X\) can be seen as a smooth resolution of the \(n\)-th symmetric product of \(X\). Many topological properties of \(X^{[n]}\) are known. The Betti numbers have been calculated by \textit{L. Göttsche} [Math. Ann. 286, No. 1--3, 193--207 (1990; Zbl 0679.14007)] and only depend on the Betti numbers of \(X\). This result has been clarified by \textit{H. Nakajima} [Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)] by constructing a representation of a Heisenberg algebra built from the rational cohomology of the surface on the direct sum \(\mathbb H := \bigoplus \mathrm H^*(X^{[n]}, \mathbb Q)\).
In the paper at hand, the author extends these results to Voisin's Hilbert schemes [\textit{C. Voisin}, Ann. Inst. Fourier 50, No. 2, 689--722 (2000; Zbl 0954.14002)] associated to compact almost-complex four-manifolds. He is able to prove both Göttsche's formula and the defining commutation relations of Nakajima's operators in this context. One main ingredient of the proof is Le Poitier's decomposition theorem for semi-small maps (following the decomposition theorem by \textit{A. A. Beilinson, J. Bernstein} and \textit{P. Deligne} [Faisceaux pervers. Astérisque 100, 172 p. (1982; Zbl 0536.14011)] without using any characteristic \(p\)-methods or étale cohomology), which is included together with a proof as it is otherwise unpublished.
Finally, tautological bundles are defined in this almost-complex setting. Hilbert scheme; Voison's Hilbert scheme; almost-complex four-manifolds; Göttsche formula; Nakajima operators J Grivaux, Topological properties of punctual Hilbert schemes of almost-complex fourfolds, to appear in Manuscripta Math. Almost complex manifolds, Parametrization (Chow and Hilbert schemes), \(4\)-folds Topological properties of Hilbert schemes of almost-complex fourfolds. 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 Basset's upper bound for the double points of an algebraic surface is here extended to algebraic hypersurfaces in \(\mathbb{P}_r(\mathbb{C})\) [see \textit{A. B. Basset}, Nature 73, 246 (1906; JFM 37.0646.03)]. number of double points; hypersurfaces Singularities of surfaces or higher-dimensional varieties, Hypersurfaces and algebraic geometry, Singularities in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry The inequality of A. B. Basset for hypersurfaces in a hyperspace | 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 \(Y:=\{algebraic\) surfaces of degree d in \({\mathbb{P}}_ 3\}\) and \(\Sigma_ d:=\{S\in Y| \quad S\quad is\quad\) smooth and Pic(S) is not generated by the hyperplane bundle\(\}\). Previously, the author proved the explicit Noether-Lefschetz theorem [J. Differ. Geom. 20, 279-289 (1984; Zbl 0559.14009)]: For \(d\geq 3\), every component of \(\Sigma_ d\) has codimension \(\geq d-3\) in Y. Here the author gives a new and short proof of this result as a consequence of some vanishing theorem for Koszul cohomology on \({\mathbb{P}}_ n\), the proof of which is given in this paper. explicit Noether-Lefschetz theorem; vanishing theorem for Koszul cohomology M. Green, A new proof of the explicit Noether-Lefschetz theorem, J. Differential Geom. 27 (1988), 155-159. Zbl0674.14005 MR918461 Structure of families (Picard-Lefschetz, monodromy, etc.), Picard groups, Parametrization (Chow and Hilbert schemes), Étale and other Grothendieck topologies and (co)homologies A new proof of the explicit Noether-Lefschetz theorem | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In a previous paper [\textit{A. Granja} and \textit{M. C. Martínez}, Commun. Algebra 26, No. 7, 2241-2263 (1998; Zbl 0930.14007)] the authors extended Zariski's equisingularity theory to the case of plane curves over a non algebraically closed field of any characteristic.
In the present paper they apply their definition to the following situation: Let \(R\) be an \((n+2)\)-dimensional local noetherian regular ring, \(p\) a height 2 prime ideal of \(R\) such that \(R/p\) is a regular ring and \(f\) a non zero element of \(p\); let \((\underline t)=(t_1, \dots ,t_n)\) be a system of \(p\)-transversal parameters (i.e. the classes of \(t_1, \dots ,t_n\) in \(R/p\) form a system of regular parameters); denote by \(\overline f\) the class of \(f\) in \(R/(\underline {t})R\) and by \(R_{(\underline t)}\) the ring \(R/(\underline t)R\). Then the hypersurface \(V= \text{Spec}(R/fR)\) is said to be equisingular at the maximal ideal \(M(R)\) of \(R\) along \(p\) if there is a system of \(p\)-transversal parameters \((\underline {t'}) = (t'_1, \cdots, t'_n)\) such that the plane curves \(V_{p} = \text{Spec}(R_{p}/fR_{p})\) (the generic section of \(V\) at \(p\)) and \(V_{(\underline t)} = \text{Spec}(R_{(\underline t)}/{\overline f} R_{(\underline t)})\) (the \(p\)-transversal section of \(V\) relative to \((\underline {t'})\)) are equisingular. The aim of the paper is to prove in this situation some natural algebraic properties which hold in the classical case: independence of the choice of \(p\)-transversal parameters, good behaviour by monoidal dilatations, \(\dots\). In particular the authors characterize the singular locus and prove an equisingularity criterion in codimension \(1\), which is similar to the one of \textit{S. S. Abhyankar} [Am. J. Math. 90, 342-345 (1968; Zbl 0159.22601)]. The main results are obtained under the assumption that the prime factorization of \(f\) in \(R\) agrees with that of \(f\) regarded as an element of the completion of \(R\). equisingularity; plane singular curve Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings, Plane and space curves Algebraic aspects of Zariski's equisingularity in codimension 1 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is a sequel to the author's famous paper [Ann. Inst. Fourier 54, No. 3, 499--630 (2004; Zbl 1062.14014)] in which he introduced the theory of geometric orbifolds, that is pairs \((X, \Delta)\) where \(X\) is a normal complex variety (most of the time \(X\) is a compact Kähler or projective manifold) and \(\Delta\) is an effective \(\mathbb Q\)-divisor \(\Delta=\sum (1-\frac{1}{m(D)}) D\) on \(X\) with \(m(D) \geq 1\) being rational or \(\infty\) (most of the time \(m(D)\) is an integer). This definition is basically the same as the definition of a log pair in the minimal model program MMP, but the orbifold divisor \(\Delta\) plays a complete different role. While in the MMP the divisor \(\Delta\) is related to the singularities of (some birational model of) \(X\), the orbifold structure is typically defined in terms of some fibration \(f: Y \rightarrow X\) mapping onto \(X\). Given such a fibration and a prime Weil divisor \(D \subset X\), one defines the multiplicity \(m(D)\) (or more precisely \(m(D, f)\)) as the infimum of the multiplicities \(a_i\) appearing in the decomposition \(f^* D = \sum a_i E_i + R\) where the \(E_i\) map surjectively onto \(D\) and \(R\) does not. In his aforementioned paper the author showed, among many other things, that if one endows the base of a fibration with such an orbifold structure, one obtains a much closer relation between the geometric properties of the total space \(Y\) on the one side and the general fibre \(F\) and the orbifold base \((X, \Delta)\) on the other side. For example we always get an exact sequence of orbifold fundamental groups, up to taking an appropriate bimeromorphic model of the fibration. Another crucial example is given by what the author calls the core fibration: an orbifold is said to be of general type if the canonical divisor \(K_X+\Delta\) is big. A compact Kähler manifold \(Y\) is special if it does not admit any (meromorphic) fibration such that the orbifold base is of general type. The author proved that every compact Kähler manifold \(Y\) admits a unique almost holomorphic fibration \(Y \dashrightarrow X\) such that the base \((X, \Delta)\) is of orbifold general type and the general fibre \(F\) is special. This core fibration is a significant refinement of classical fibrations like the Iitaka fibration which typically has not the property that its base is of general type (in the standard sense). \newline In the paper under review the author puts his theory on a larger basis by discussing thoroughly the category of geometric orbifolds, the most important points being the appropriate definition of orbifold morphisms and the core fibration in the orbifold setting. He also introduces various notions of rational connectedness by orbifold curves, a very challenging topic with many problems that are open even for surfaces. The final section covers the conjectural relation of special orbifolds with properties from hyperbolic and arithmetic geometry and surveys some of the progress made since the appearance of the first paper. The numerous technical additions and improvements made in this paper should be very helpful for any mathematician interested in this ambitious research program. birational classification; special variety; special orbifold; core fibration; orbifold rational curve Campana, F., \textit{orbifoldes spéciales et classification biméromorphe des variétés Kählériennes compactes}, J. Inst. Math. Jussieu, 10, 809-934, (2011) \(n\)-folds (\(n>4\)), Minimal model program (Mori theory, extremal rays), Parametrization (Chow and Hilbert schemes), Rational and birational maps, Ramification problems in algebraic geometry, Rational points, Coverings of curves, fundamental group, Transcendental methods of algebraic geometry (complex-analytic aspects), Compact Kähler manifolds: generalizations, classification, Hyperbolic and Kobayashi hyperbolic manifolds, Elliptic curves over global fields Orbifoldes géométriques spéciales et classification biméromorphe des variétés Kählériennes compactes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article under review is devoted to the study of analytic families of two-dimensional vector fields; the set of singular points of a field from the family can be either zero- or one-dimensional. It is assumed that the singularities of any vector field have a non-vanishing first jet, that is, at each singular point the matrix of the linear part of the field is different from zero. A field is called elementary if the first jet of its singularities has a non-zero eigenvalue, otherwise it is called nilpotent.
The aim of the author is to prove a desingularization theorem for nilpotent families of vector fields. To be more precise, he proves that any nilpotent family can be transformed to an elementary one by a finite sequence of successive blowing-ups which are monomial transformations of suitable coordinates; at the first stage the blowing-up is applied to the versal unfolding induced by the initial family. Some applications to problems of singular perturbations, limit cycles and finite cyclicity are discussed. planar vector fields; nilpotent singularities; analytic unfolding; blowing-up; strict transforms; desingularization; canard problem; foliations; power geometry; normal forms; limit periodic sets; limit cycles Panazzolo, D., Desingularization of nilpotent singularities in families of planar vector fields, Mem. Am. Math. Soc., 158, (2002) Singularities of vector fields, topological aspects, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Singular perturbations, turning point theory, WKB methods for ordinary differential equations, Control problems involving ordinary differential equations, Modifications; resolution of singularities (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects) Desingularization of nilpotent singularities in families of planar vector fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review is a sequel of works [\textit{A.Campillo}, et al., Duke Math. J. 117, No. 1, 125--156 (2003; Zbl 1028.32013); \textit{F. Delgado}, et al., Proc. Edinb. Math. Soc., II. Ser. 46, No. 2, 501--509 (2003; Zbl 1083.14502); \textit{A. Campillo}, et al., Int. J. Math. 14, No. 1, 47--54 (2003; Zbl 1056.14035)] where the authors computed two-dimensional Poincaré series of divisorial multi-index filtrations of the local ring of holomorphic function germs at the origin of the complex plane and of the filtration defined by orders of a function on branches of a plane curve germ.
Here the authors develop an equivariant version of these results. To be more precise, they study the corresponding filtration on subspaces of germs of functions equivariant with respect to 1-dimensional representations \(G\rightarrow \mathbb C^*\) of the finite group \(G\) acting on the complex plane and compute three examples of Poincaré series for divisorial multi-index filtrations on the germ \((\mathbb C^2,0)\) endowed with the action of groups \(\mathbb Z_3, \mathbb Z_5\) and \(\mathbb D_2^*\) corresponding to quotient surface singularities \(A_2, A_4\) and \(D_4,\) respectively. The authors underline that the aim of their investigations is to define an equivariant version of the monodromy zeta function. equivariant functions; divisorial filtrations; Poincaré series; plane curves Campillo, A., Delgado, F., Gusein-Zade, S.: On Poincaré series of filtrations on equivariant functions of two variables. Mosc. Math. J. 7(2), 243-255 (2007) Singularities in algebraic geometry, Filtered associative rings; filtrational and graded techniques, Actions of groups and semigroups; invariant theory (associative rings and algebras), Valuations and their generalizations for commutative rings, Singularities of curves, local rings On Poincaré series of filtrations on equivariant functions of 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 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 We construct an action of the Neron-Severi part of the Looijenga-Lunts-Verbitsky Lie algebra on the Chow ring of the Hilbert scheme of points on a \(K3\) surface. This yields a simplification of \textit{D. Maulik} and \textit{A. Neguţ}'s [``Lehn's formula in Chow and Conjectures of Beauville and Voisin'', Preprint, \url{arXiv:1904.05262}] proof that the cycle class map is injective on the subring generated by divisor classes as conjectured by Beauville (see [\textit{A. Beauville} and \textit{C. Voisin}, J. Algebr. Geom. 13, No. 3, 417--426 (2004; Zbl 1069.14006)]). The key step in the construction is an explicit formula for Lefschetz duals in terms of Nakajima operators. Our results also lead to a formula for the monodromy action on Hilbert schemes in terms of Nakajima operators. \(K3\) surfaces; Chow groups; Hilbert schemes of points; Lie algebras \(K3\) surfaces and Enriques surfaces, (Equivariant) Chow groups and rings; motives, Parametrization (Chow and Hilbert schemes) A Lie algebra action on the Chow ring of the Hilbert scheme of points of a \(K3\) 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 Let \(X\) be a normal, proper, integral variety defined over an algebraically closed field. The Weil divisor rank \(\rho_w(X)\) of \(X\) is defined to be the minimum dimension of the \(\mathbb{Q}\)-vector space \(B^1(\tilde{X})=(\text{CH}^1(\tilde{X})/\text{CH}^1(\tilde{X})_{\text{hom}})\otimes\mathbb{Q}\) over all projective, normal modifications \(\pi:\tilde{X}\rightarrow X\) of \(X\). Let \(\{D_i\}_{i\in I}\) be a collection of pairwise-disjoint, reduced, codimension-one, connected subvarieties of \(X\) such that \(\sharp I\geq\rho_w(X)+1\). The main result of this paper shows that there is a smooth, projective curve and a surjective morphism \(f:X\rightarrow C\) with connected fibers such that for any \(i\in I\), the divisor \(D_i\) is contained in a fiber of \(f\). Furthermore, there is a set \(\Sigma\subset I\) so that \(\sharp I\setminus\Sigma\leq\rho_w(X)-2\) and for each \(i\in\Sigma\), \(D_i\) is equal to a fiber of \(f\). On the other hand the authors prove a strong counterexample in the affine case: if \(X\) is a quasi-affine variety over a countable, algebraically-closed field \(k\), then there is a countable family \(\{D_i\}_{i\in I}\) of pairwise-disjoint divisors which cover the \(k\)-points of \(X\), so that for any non-constant morphism from \(X\) to a curve, at most finitely many are contained in the fibers thereof. disjoint divisors; Chow group; regular functions; Hodge index theorem Parametrization (Chow and Hilbert schemes), Algebraic cycles, Structure of families (Picard-Lefschetz, monodromy, etc.) Families of disjoint divisors on 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 view (or aspect) graph of a surface in real 3-space is a graph designed to measure the visual complexity of the surface. It describes the number of topologically distinct views (or aspects) of a surface. Consider projections (parallel or central) of a surface \(M\) in 3-space. A view of \(M\), relative to a given projection, is the branch locus (aka bifurcation set) of the projection together with the projections of the double curves of \(M\). For a general projection of a surface \(M\) satisfying appropriate restrictions, the view has only specific types of isolated singularities (e.g. nodes and cusps when \(M\) is a non-singular algebraic surface), while for certain projections, called degenerate, the singularities are worse. If \(\mathcal V\) denotes the space of projections and \(\mathcal B\) is the subset of degenerate projections, then the view graph is the graph whose nodes are the connected components of \(\mathcal V \setminus \mathcal B\), and whose edges connect nodes separated by a codimension 1 component of \(\mathcal B\). In the present paper the author defines a piecewise smooth surface to be a finite union of \(C^\infty\) smooth surfaces, any two of which intersect transversally if at all, and no more than three intersect in a point. (Thus in the algebraic case the union has the ``generic surface singularities'' of classical algebraic geometry, namely, a finite number of double curves, whose only singularities are triple points, triple also for the surface.) The principal results of the paper under review appear to be
(1) a proof that the view graph of a piecewise smooth \(C^\infty\) surface is well-defined
(2) presentation of a polynomial time symbolic algorithm for determining the view graph of a piecewise smooth algebraic surface.
Intermediate results classifying singularities and bounding the cardinality of the view graph are of independent interest. view graph; aspect graph; piecewise smooth surface; singularity theory; generic surface singularities J.H. Rieger, On the complexity and computation of view graphs of piecewise smooth algebraic surfaces, Philos. Trans. Roy. Soc. London Ser. A 354 (1996), 1899--1940 Computational aspects of algebraic surfaces, Singularities of surfaces or higher-dimensional varieties, Analysis of algorithms and problem complexity, Singularities in algebraic geometry, Computing methodologies for image processing On the complexity and computation of view graphs of piecewise smooth algebraic surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The concept of Kähler different was first introduced by Kähler. In this paper we are mainly interested in the geometric applications of Kähler differents. The results of this paper will be subsequently applied to study the birational geometry of regular schemes. Consider a morphism \(f: X\to Y\) of schemes such that the sheaf \(\Omega_{X/Y}\) of relative differentials of \(X\) over \(Y\) is coherent. The 0-th Fitting ideal (sheaf) \(d_{X/Y}\) of \(\Omega_{X/Y}\) is called the Kähler different of \(X\) over \(Y\). Now suppose that \(X\) and \(Y\) are regular Noetherian integral schemes and \(f\) is dominant and unramified at the generic point. Then the Kähler different \(d_{X/Y}\) is an invertible sheaf, which defines an effective divisor \(E_f\) of \(X\). The support of \(E_f\) consists precisely of ramification points of \(f\). Furthermore, if \(f\) is a birational morphism, the support of \(E_f\) coincides with the exceptional locus of \(f\).
Definition. The divisor \(E_f\) is the ramification divisor of the morphism \(f\). If \(f\) is a birational morphism then \(E_f\) is also called the exceptional divisor of the morphism \(f\).
The ramification divisor \(E_f\) can be determined locally. Consider a dominating pair \(A\supset B\) of regular local rings such that \(A\) is a discrete valuation ring and the fraction field \(Q(A)\) of \(A\) is a finite separable extension of the fraction field \(Q(B)\) of \(B\). Define the ramification index \(r(AB)\) of \(A\) over \(B\) to be the multiplicity of the Kähler different \(d(A/B)\) of \(A\) over \(B\). Then the multiplicity of \(E_f\) at a point \(W\) of codimension 1 of \(X\) is \(r({\mathcal O}_W/{\mathcal O}_{f(W)})\). In algebraic number theory we consider the case that both \(A\) and \(B\) are discrete valuation rings. Since for Dedekind domains the concept of Dedekind different coincides with that of Kähler different, the ramification index can also be defined using Dedekind different. The main theorem of ramification theory of algebraic number theory (due to Dedekind) then states:
Let \(A\supset B\) be a dominating pair of discrete valuation rings as above. Let \(e(AB)\) be the reduced ramification index of \(A\) over \(B\). Then \(r(AB)\geq e(AB)-1\). Equality holds if and only if \(e(AB)\) is not a multiple of \(\text{ch} B/N\) and \(A/M\) is separable over \(B/N\) \((M,N\) are the maximal ideals of \(A,B\) respectively).
For a general pair \(A\supset B\) with Krull \(\dim B=n\geq 1\) we define the reduced ramification index \(e(AB)\) to be the maximum of the multiplicities of the products \(\prod^n_{\lambda=1}b_\lambda\) of all minimal bases \(\{b_\lambda\}\) of the maximal ideal \(N\) of \(B\); then \(e(AB)\geq n\). Our main result is the following generalization of the above theorem:
Let \(A\supset B\) be a dominating pair of regular local rings such that \(A\) is a discrete valuation ring and \(Q(A)\) is a finite separable extension of \(Q(B)\). Then \(r(AB)\geq e(AB)-1\geq \text{Krull}\dim B-1\). If \(Q(A)=Q(B)\) then \(r(AB)=\text{Krull}\dim B-1\) if and only if \(A\) is the \(N\)-adic valuation ring of \(B\). Kähler differents; sheaf of relative differentials; ramification points; birational morphism; exceptional locus; Dedekind different; reduced ramification index; dominating pair of regular local rings Zhao Hua Luo, Ramification divisor of regular schemes, Algebraic geometry and algebraic number theory (Tianjin, 1989 -- 1990) Nankai Ser. Pure Appl. Math. Theoret. Phys., vol. 3, World Sci. Publ., River Edge, NJ, 1992, pp. 77 -- 91. Regular local rings, Modules of differentials, Singularities in algebraic geometry, Schemes and morphisms Ramification divisor 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 [For the entire collection see Zbl 0667.00008.]
The goal of this article is to understand better the structure of the Hilbert scheme \(H(p)=Hilb^ p({\mathbb{P}}^ n_ k)\) of projective subschemes of \({\mathbb{P}}^ n_ k\) with Hilbert polynomial p by considering linkage not just of individual subschemes but instead of entire flat families of them, in effect a study of linkage behavior under very general deformation.
For S locally noetherian, an S-point of D(p,q) is a sequence of closed embeddings XYP of flat S-schemes such that for any \(s\in S\) the schemes \(X_ s\) and \(Y_ s\) have Hilbert polynomials p and q, respectively. Further D(p;\textbf{f})\({}_{CM}\) is the open subscheme of D(p,q) for which the fibers \(X_ s\) are Cohen-Macaulay and equidimensional and \(Y_ s\) are complete intersections of multidegree \textbf{f}\(=f_ 1,...,f_ r\) for all \(s\in S\). The main result is that linkage of families X and \(X'\) with respect to a family of complete intersections Y defines an isomorphism D(p;\textbf{f})\({}_{CM}\to D(p';{\mathbf{f}})_{CM}\). If U is a subset of \(H(p)_{CM}\), all the members of U are contained in complete intersections of the same type, and \(U'\) is the set of linked subschemes in \(H(p')_{CM}\), then, under various additional hypotheses, properties of U (openness, irreducibility, smoothness of H(p) along it) can be carried over to \(U'\). As a corollary, if \(X\in H(p)_{CM}\) is non- obstructed, linked to \(X'\), and certain cohomological conditions hold on X and its ``generizations'', then \(X'\) is also non-obstructed. The author also gives a number of concrete examples, and methods for constructing these.
If, for example X is a (locally Cohen-Macaulay) curve in \({\mathbb{P}}^ 3\) with \(H^ 1(N_ X)=0\), then by linking geometrically by Y to \(X'\) (subject to certain constraints on the degrees for Y) and then again by suitable \(Y'\) to \(X''\), one obtains an obstructed curve that is doubly linked to X. Further results concern invariance of the cotangent sheaf \(A^ 2_ X\) and the obstruction space \(A^ 2(XY)\) under geometric linkage. There is also a subgroup C(XY) of \(A^ 2(XY)\) that is still invariant under appropriate conditions, but easier to compute and with useful applications to calculation of \(H^ 1(N_ X)\) for generic complete intersection curves in \({\mathbb{P}}^ 3\). For related results on invariance of cotangent modules the reader may consult \textit{R.-O. Buchweitz} [Thesis, Paris VII (1981)], \textit{R.-O. Buchweitz} and \textit{B. Ulrich} [``Homological properties invariant under linkage'' (preprint 1983)], \textit{B. Ulrich} [Math. Z. 196, 463-484 (1987; Zbl 0657.13023)], and for results on linkage of curves in \({\mathbb{P}}^ 3\), the sequence of papers by Bolondi and Migliore. liaison of families; obstructed curve; Hilbert scheme; Hilbert polynomial; linkage behavior under very general deformation; linkage of families Kleppe, J. O.: Liaison of families of subschemes in pn. Lecture notes in math. 1389 (1989) Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Formal methods and deformations in algebraic geometry, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Liaison of families of subschemes in \({\mathbb{P}}^ n\) | 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 present paper is devoted to the classification of symplectic automorphisms of some hyper-Kähler manifolds. The result presented here is a proof that all finite groups of symplectic automorphisms of manifolds of \(K3^{[n]}\) type are contained in Conway's group \(Co_1\). symplectic automorphisms; manifolds of \(K3^{[n]}\) type Mongardi, G., \textit{towards a classification of symplectic automorphisms on manifolds of K3[\textit{n}] type}, Math. Z., 282, 651-662, (2016) Automorphisms of surfaces and higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), \(K3\) surfaces and Enriques surfaces Towards a classification of symplectic automorphisms on manifolds of \(K3^{[n]}\) type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We show that any toroidal DM stack \(X\) with finite diagonalizable inertia possesses a maximal toroidal coarsening \(X_{\operatorname{tcs}}\) such that the morphism \(X\to X_{\operatorname{tcs}}\) is logarithmically smooth.
Further, we use torification results of the first two authors [J. Algebra 472, 279--338 (2017; Zbl 1376.14051)] to construct a destackification functor, a variant of the main result of \textit{D. Bergh} [Compos. Math. 153, No. 6, 1257--1315 (2017; Zbl 1372.14002)], on the category of such toroidal stacks \(X\). Namely, we associate to \(X\) a sequence of blowings up of toroidal stacks \(\widetilde{\mathcal F}_X\:Y\to X\) such that \(Y_{\operatorname{tcs}}\) coincides with the usual coarse moduli space \(Y_{\operatorname{cs}}\). In particular, this provides a toroidal resolution of the algebraic~space~\(X_{\operatorname{cs}}\). Both \(X_{\operatorname{tcs}}\) and \(\widetilde{\mathcal F}_X\) are functorial with respect to strict inertia preserving morphisms \(X'\to X\).
Finally, we use coarsening morphisms to introduce a class of nonrepresentable birational modifications of toroidal stacks called Kummer blowings up.
These modifications, as well as our version of destackification, are used in our work on functorial toroidal resolution of singularities. algebraic stacks; toroidal geometry; logarithmic schemes; birational geometry; resolution of singularities Generalizations (algebraic spaces, stacks), Rational and birational maps, Global theory and resolution of singularities (algebro-geometric aspects) Toroidal orbifolds, destackification, and Kummer blowings up | 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 0741.00019.]
This is a survey on the birational classification of the complex projective varieties of dimension \(r\). Having reviewed a few facts on surfaces just to help the reader understand the results for 3-folds, the author makes it clear that the first problem in generalizing those classical results to higher dimensions is to find a reasonable class of varieties where there is a contraction theorem. This leads to define classes of singular varieties (canonical and terminal singularities). The classification program thus starts by fixing a category \({\mathcal C}\) of varieties where to play the classification game, which is the usual category of smooth surfaces if \(r=2\) (projective varieties with \(\mathbb{Q}\)- factorial terminal singularities). The minimal model program consists either in describing \(X\in{\mathcal C}\) as a reasonable fibration (so as to be helped by the classification in lower dimension) or in replacing \(X\in{\mathcal C}\) with a simpler \(X'\in{\mathcal C}\) birational to \(X\) (a minimal model, i.e. \(K_{X'}\) nef). If \(X\) is not a minimal model, there is an extremal ray \(R\), giving rise to a contraction morphism \(\text{cont}_ R:X\to Y\). If \(\dim Y<\dim X\), \(X\) is reasonably well- known (\(-K_ X\) is ample on the fibres of the fibration \(\text{cont}_ R\), \(X\) is uniruled) and the game ends at this point. If \(\text{cont}_ R\) is birational and contracts a divisor, the play goes on by replacing \(X\) by \(X'=Y\). If \(\text{cont}_ R\) is birational but does not contract a divisor, we cannot go on as above since \(Y\notin{\mathcal C}\). To get around the trouble the conjecture on the existence of flips has to be true. In that case the play goes on by replacing \(X\) by \(X^ +\in{\mathcal C}\) birational to \(Y\). --- The game will be over and the minimal model program completed once we have shown that this case cannot occur infinitely many times (conjecture on the termination of flips).
The author recalls what is known about these and other conjectures for 3- folds and 4-folds. He also collects some applications of this program for 3-folds: the factorization result for birational morphisms of nonsingular surfaces is generalized for the 3-folds in \({\mathcal C}\) by considering divisorial contractions and flips; the dichotomy between \(\chi(X)\geq 0\) and \(\chi(X)=-\infty\); the existence of canonical models. Some comments are added on the proofs of the results for 3-folds and some related directions of research are listed. canonical singularities; birational classification; contraction theorem; terminal singularities; extremal ray; existence of flips J. Kollár and S. Mori,Birational Geometry of Algebraic Varieties, Cambridge Univ. Press, to appear. \(3\)-folds, Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps Birational classification of algebraic 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 A class of non-isolated singularities is studied, which still have some nice properties as isolated hypersurface singularities, the class of almost free divisors. This class contains discriminants of finitely determined map germs, bifurcation sets of certain unfoldings of hypersurfaces, certain central arrangements of hyperplanes. To these singularities a so-called singular Milnor fibration is associated, which has similar properties as the Milnor fibration for isolated singularities. Especially, there is a ``singular'' Milnor number and it can be computed as the length of a certain determinantal module, higher multiplicities similar to Teissier's \(\mu^*\) are considered and used to deduce topological properties of these singularities. hyperplane arrangements; isolated hypersurface singularities; almost free divisors; Milnor fibration; Milnor number J. Damon, ''Higher multiplicities and almost free divisors and complete intersections,'' preprint (1992). Deformations of complex singularities; vanishing cycles, Singularities in algebraic geometry, Complete intersections, Holomorphic maps on manifolds Higher multiplicities and almost free divisors and complete 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 The holomorphy conjecture roughly states that Igusa's zeta function associated to a hypersurface and a character is holomorphic on \(\mathbb{C}\) whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article, we prove the holomorphy conjecture for surface singularities that are nondegenerate over \(\mathbb{C}\) with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volumes (which appear in the formula of Varchenko for the zeta function of monodromy) of the faces in a simplex in arbitrary dimension. We then study some specific character sums that show up when dealing with false poles. In contrast to the context of the trivial character, we here need to show fakeness of certain candidate poles other than those contributed by \(B_1\)-facets. Singularities in algebraic geometry, Other character sums and Gauss sums, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) The holomorphy conjecture for nondegenerate 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 prove that every quasi-log canonical pair has only Du Bois singularities. Note that our arguments are free from the minimal model program. Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Vanishing theorems in algebraic geometry, Minimal model program (Mori theory, extremal rays) Quasi-log canonical pairs are Du Bois | 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 Unfortunately, there is a mistake in our paper [ibid. 239, 322--345 (2020; Zbl 1440.13072), Lemma 3.10] which invalidates [loc. cit., Theorem 3.12]. We show that the theorem still holds if the ring is assumed to be Gorenstein. Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Multiplicity theory and related topics, Singularities in algebraic geometry Erratum to: ``Continuity of Hilbert-Kunz multiplicity and F-signature'' | 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 0653.00009.]
\textit{B. Teissier} [Lect. Notes Math. 961, 314-491 (1982; Zbl 0585.14008)] gave an algebraic characterization of the Whitney-conditions. In a similar spirit the author describes a necessary algebraic condition for a stratification to fulfill a Lipschitz-analogue of Whitney-conditions. tangent cone; Whitney-conditions; stratification T. MOSTOWSKI , Tangent Cones and Lipschitz Stratifications, Singularities, Banach Center Publications (S. Łojasiewicz, ed.), Vol. XX, PWN, Warszawa, 1988 , pp. 303-322. MR 92f:32008 | Zbl 0662.32012 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Singularities in algebraic geometry Tangent cones and Lipschitz stratifications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f:\mathbb C^n \rightarrow \mathbb C\) be a quasihomogeneous polynomial of degree \(d\), non-degenerated with respect to its Newton diagram in the sense of Varchenko, and \(P(t) = \sum_s \dim(A_s)t^s\) be the Poincaré series of the graded ring \(A = \mathbb C[x]/f = \bigoplus_s A_s\). Let \(\zeta^*_f(t)\) be the Saito dual of the zeta function of the monodromy of the singularity defined by \(f\). It is proved that
\[
P(t) = \zeta^*_f(t) \prod_m(1 - t^m)^{-\chi(Y_m)}.
\]
\(\chi(Y_m)\) is the Euler characteristic of \(Y_m\), the set of orbits of the \(\mathbb C^*\)-action on \(V(f) \setminus \{0\}\) for which the isotropy group is the cyclic group of order \(m\). Ebeling, W.; Gusein-Zade, S. M., Poincaré series and zeta function of the monodromy of a quasihomogeneous singularity, Math. Res. Lett., 9, 509, (2002) Singularities in algebraic geometry, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) Poincaré series and zeta function of the monodromy of a quasihomogeneous 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 Let \(E\) be an elliptic curve and let \(R\) be a real three dimensional root system. Let \(W\) be the Weyl group associated to \(R\), and put \(W_+=W\cap\text{SL}_3(\mathbb{C}).\) Then \(W_+\) acts naturally on \(E\otimes Q(R)\) and the quotient \(E\otimes Q(R)/W_+\) is singular with two natural crepant resolutions. One is the result of a Jung process of desingularization of singularities, the other the equivariant Hilbert scheme.
The author compares these resolutions case by case by writing up explicit equations, and in the Hilbert scheme case, an explicit example is given and a McKay correspondence is achieved. Finally, this correspondence results in a family of vector bundles on \(E\) parameterized by the \(W_+\)-Hilbert scheme. This article gives an interesting comparison of the two types of crepant resolutions in the dimension three case. crepant resolution; singularities; equivariant Hilbert scheme Global theory and resolution of singularities (algebro-geometric aspects), Geometric invariant theory, Local deformation theory, Artin approximation, etc. Resolution of non-abelian three-dimensional 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 a celebrated preprint from 1968, \textit{J. F. Nash jun.} [cf. Duke Math. J. 81, No. 1, 31--38 (1995; Zbl 0880.14010)] introduced the space of arcs on an isolated singularity of a surface \(X\), and he asked if there is a bijection between the irreducible components of the space of arcs on \(X\) and the exceptional components of a minimal resolution of singularities of \(X\). This problem was extended recently by \textit{S. Ishii and J. Kollár} [Duke Math. J. 120, 601--620 (2003; Zbl 1052.14011)]
to any dimension, they also gave a counterexample in dimension 4. But in dimension two there is no counterexample and until recently there were very few positive answers to this problem by \textit{A. J. Reguera} [Manuscr. Math. 88, 321--333 (1995; Zbl 0867.14012)] for minimal singularities, \textit{A. Reguera} and \textit{M. Lejeune-Jalabert} [Rev. Mat. Iberoam. 19, No.2, 581--612 (2003; Zbl 1058.14006)] for sandwich singularities, \textit{C. Plénat} [Ann. Inst. Fourier 55, No.3, 805--823 (2005; Zbl 1080.14021)] for the singularity \(D_n\) and the reviewer for a very large class of singularities depending only in the dual graph of the exceptional set in the minimal resolution of singularities.
In the paper under review, the author introduces a map from the set of fat arcs to the set of valuations. A fat arc is an arc which does not factor through any proper closed subvarieties. This map is a generalization of the Nash map and the map defined by \textit{L. Ein, R. Lazarsfeld} and \textit{M. Mustata} [Compos. Math. 140, No.5, 1229--1244 (2004; Zbl 1060.14004)]. This paper gives an affirmative answer for a non normal toric variety. Another interesting example is the arc determined by a conjugacy class of a finite group \(G\) which gives the quotient variety \(X=\mathbb C^n/G\). The restriction of the map defined by the author onto a subset of these arcs coincides with the McKay correspondence. singularity; toric variety S. Ishii, ''Arcs, valuations and the Nash map,'' J. Reine Angew. Math., vol. 588, pp. 71-92, 2005. Singularities in algebraic geometry, Valuations and their generalizations for commutative rings, Toric varieties, Newton polyhedra, Okounkov bodies, Divisors, linear systems, invertible sheaves Arcs, valuations and the Nash map | 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 Denote by \(H_{d,g,r}\) the Hilbert scheme parametrizing projective smooth irreducible complex curves of degree \(d\) and genus \(g\) in \({\mathbb P^r}\). A natural question concerning \(H_{d,g,r}\), which goes back to [\textit{F. Severi}, Vorlesungen über algebraische Geometrie, Teubner (1921; JFM 48.0687.01)], is whether it is irreducible under the assumption \(d\geq g+r\).
In more recent years \textit{J. Harris} [see \textit{L. Ein}, Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, Part 1, Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], and \textit{C. Keem} [Proc. Am. Math. Soc., 122, No.~2, 349--354 (1994; Zbl 0860.14003)] proved by examples that \(H_{d,g,r}\) may be reducible for \(d\geq g+r\) and \(r\geq 6\). Moreover \textit{L. Ein} [loc. cit.; Ann. Sci. Éc. Norm. Supér., IV. Sér., 19, No.~4, 469--478 (1986; Zbl 0606.14003)], and \textit{C. Keem} and \textit{S. Kim} [J. Algebra, 145, No.~1, 240--248 (1992; Zbl 0783.14002)] proved the irreducibility of \(H_{d,g,r}\) when \(d\geq g+r\), for \(r=3\) and \(r=4\). In their paper, C. Keem and S. Kim also proved the irreduciblity of \(H_{g+2,g,3}\) if \(g\geq 5\), and of \(H_{g+1,g,3}\) if \(g\geq 11\).
Continuing the quoted works of L. Ein, C. Keem and S. Kim, and using Brill-Noether Theory as developed by [\textit{E. Arbarello, M. Cornalba, P. Griffiths} and \textit{J. Harris}, ``Geometry of Algebraic Curves'', Grundlehren der Mathematischen Wissenschaften, 267 (1985; Zbl 0559.14017)], in the paper under review the author further refines the irreducibility range of \(H_{d,g,r}\) for \(3\leq r\leq 4\), proving that \(H_{g,g,3}\) is irreducible if \(g\geq 13\), and that \(H_{g+i,g,4}\) is irreducible for \(2\leq i\leq 3\) if \(g\geq 23-6i\). projective space curve; Brill-Noether Theory; line bundle; normal bundle; linear series \textsc{H. Iliev}, On the irreducibility of the Hilbert scheme of space curves, Proc. Amer. Math. Soc. \textbf{134} (2006), 2823-2832. Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) On the irreducibility of the Hilbert scheme of space 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 ADHM construction establishes a one-to-one correspondence between framed torsion free sheaves on the projective plane and stable framed representations of a quiver with relations in the category of complex vector spaces. This paper studies the geometry of moduli spaces of representations of the same quiver with relations in the abelian category of coherent sheaves on a smooth complex projective curve \(X\). In particular it is proven that this moduli space is virtually smooth and related by relative Beilinson spectral sequence to the curve counting construction via stable pairs of Pandharipande and Thomas. This yields a new conjectural construction for the local Donaldson-Thomas theory of curves as well as a natural higher rank generalization. moduli spaces of sheaves; Donaldson-Thomas invariants; ADHM construction A. Gholampour, M. Kool and B. Young, \textit{Rank} 2 \textit{Donaldson-Thomas invariants of toric} 3\textit{-folds}, talk slides presented by M. Kool at \textit{String-Math} 2014, University of Alberta, Edmonton Canada (2014). Algebraic moduli problems, moduli of vector bundles, Representations of quivers and partially ordered sets, Yang-Mills and other gauge theories in quantum field theory Moduli of ADHM sheaves and the local Donaldson-Thomas theory | 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 complex algebraic variety \(V\subset\mathbb C^n\) the authors define, in analogy to the local case due to H. Whitney, the geometric tangent cone \(C_{g,\infty }(V)\) of \(V\) at infinity (as vectors \(v\in\mathbb C^n\) such that there exist sequences \(\{x_k\}\subset V\), \(\left\| x_k\right\|\rightarrow \infty\) and \(\{t_k\}\subset\mathbb C\) such that \(t_kx_k\rightarrow v\)) and algebraic tangent cone \(C_{a,\infty}(V)\) of \(V\) at infinity (as the set \(\{v\in\mathbb C^n:f^\ast(v)=0\text{ for all }f\in I(V)\}\), where \(f^\ast\) means homogeneous part of \(f\) of the highest degree and \(I(V)\) the ideal defining \(V\)). The main theorem is the equality \(\;C_{g,\infty }(V)=C_{a,\infty }(V)\). They also show how the tangent cone at inifinity can be computed using Gröbner bases. tangent cone at infinity; algebraic variety; Łojasiewicz inequality; Gröbner base Lê, Công-Trình; Ph\textvietạm, Ti\textviet{\'ê}n-S{\textviet{o}}n, On tangent cones at infinity of algebraic varieties, J.~Algebra Appl., 17, 8, 1850143, 10 pp., (2018) Singularities in algebraic geometry, Local complex singularities On tangent cones at infinity of algebraic 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 aim of the article is to give an arithmetical and elementary proof of the classification of the log canonical and log terminal surface singularities.
Let \((X,p)\) be a germ of a normal surface and \(B = \sum b_ i B_ i\) a sum of irreducible Weil divisors, with rational coefficients \(0 \leq b_ i \leq 1\). Let \(f : Y \to X\) be a resolution of \((X,B)\), i.e. a resolution of the singularity \((X,p)\) such that the inverse image of \(f^*(\sum B_ i)\) is a normal crossing divisor. For any Weil divisor \(F\) on \(X\), we denote by \(f_ *^{-1} (F)\) the strict transform of \(F\) on \(Y\) and we note \(E = \sum^ n_{j = 1} E_ j\) the reduced exceptional divisor of \(f : Y \to X\). Then we have the equality: \(K_ Y + f_ *^{ - 1}(B) = f^*(K_ X + B) + \sum a(E_ j,B) E_ j\).
\(a(E_ j,B)\) is the discrepancy of \(E_ j\) with respect to \(B\) and \(a_ j = a_ l(E_ j, B) = 1 + a(E_ j,B)\) the log discrepancy. Then \((X,B)\) is ``log canonical'' (resp. ``log terminal'') if for all \(j\), \(a_ j \geq 0\) (resp. for all \(j\), \(a_ j > 0)\).
The idea is to consider the minimal resolution \(f : Y \to X\) of the singularity \((X,p)\), and the equality: \(K_ Y + \sum f_ *^{ - 1} (B_ i) + \sum E_ j = f^*(K_ X + \sum B_ i) + \sum a_ j E_ j\). By the adjunction formula for the divisors \(E_ j\), \(1 \leq j \leq n\), we get the following system of \(n\) linear equations in \(n\) variables: \(\sum^ n_{k = 1} a_ k E_ k.E_ j = - c_ j\), where \(c_ j = 2 - 2p_ a(E_ j) - (\sum f^{ - 1}_ *B_ i + \sum_{k \neq j} E_ k).E_ j\). Then we have to solve this system and check the conditions \(a_ k \geq 0\) or \(a > 0\). -- The author studies then the weighted dual graph \(\Gamma\) of the exceptional divisor \(E\). If \(\Gamma'\) is a subgraph of \(\Gamma\), then the discrepancies \(a_ j'\) associated to \(\Gamma'\) are such that \(a_ j \leq a_ j'\). From this result the author can prove that the graph of a log canonical singularity satisfies the following:
-- There is no vertex \(v_ j\) with \(p_ a (E_ j) > 1\) or an edge of weight \(>2\).
-- If the exceptional divisor \(E\) is not irreducible with \(p_ a(E) = 1\), or is not a circle of smooth rational curves, the graph \(\Gamma\) is a tree, all vertices are smooth rational curves and all edges are of weight 1.
The author needs also to look at the determinant \(\Delta\) of the intersection matrix of some subgraphs of \(\Gamma\) and he can give an expression of the discrepancies:
\[
a_ j = (\Delta (\Gamma))^{ - 1} \sum^ n_{k = 1} \Delta (\Gamma-(\text{path from }v_ j \text{ to }v_ k)).
\]
By this results he can then give a complete description of each graph \(\Gamma\) associated to log canonical or log terminal singularities. log canonical surface singularities; log terminal surface singularities; discrepancy Alexeev, V., Classification of log-canonical surface singularities, in ``Flips and Abundance for Algebraic Threefolds--Salt Lake City, Utah, August 1991,'' Asterisque No. 211, Soc. Math. France, 1992, pp. 47-58. Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Classification of log canonical surface singularities: Arithmetical proof | 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 minimal model program for the Hilbert schemes \(\text{Hilb}^n \mathbb{P}^2\) of \(n\)-points on the complex projective plane \(\mathbb{P}^2\) and the correspondence between wall-crossings in effective cones of divisors and wall-crossings in Bridgeland stability manifolds are explored in detail in [\textit{D. Arcara} et al., Adv. Math. 235, 580--626 (2013; Zbl 1267.14023)]. The paper under review studies the relevant problems in ibid. in a broader setting.
The Sklyanin algebra \(S=\text{Skl}(E, \mathscr{L}, \lambda)\) is in general a noncommutative algebra, constructed from the data: an elliptic curve \(E\), a degree \(3\) line bundle \(\mathscr{L}\) on \(E\) and a translation \(\lambda\) on \(E\). When \(\lambda\) is chosen to be the identity, \(S\simeq\mathbb{C}[x_0, x_1, x_2]\), the homogeneous coordinate ring of \(\mathbb{P}^2\). The category of \(\text{qgr-}S\) (see Section 2) corresponds to, in commutative case, the category of coherent sheaves on \(\mathbb{P}^2\). One can introduce Chern classes for the objects in \(\text{qgr-}S\) and hence the notion of slope stability. The moduli space \(\text{Hilb}^n S\) of semistable objects with variants \((\text{rk}, c_1, \chi)=(1, 0, 1-n)\) gives a generic deformation of \(\text{Hilb}^n \mathbb{P}^2\) by [\textit{T. A. Nevins} and \textit{J. T. Stafford}, Adv. Math. 210, No. 2, 405--478 (2007; Zbl 1116.14003)] and [\textit{N. Hitchin}, Mosc. Math. J. 12, No. 3, 567--591 (2012; Zbl 1267.32010)].
In a similar fashion to Bridgeland, Arcara-Bertram's construction for stability conditions on projective surfaces, the authors construct stability conditions \(\sigma=\sigma(s, t)\) on the derived category \(D^b(\text{qgr-}S)\), for \((s, t)\) in the upper-half plane. Let \(\mathfrak{M}_{\sigma}(n)\) be the moduli space of stable objects of invariants \((\text{rk}, c_1, \chi)=(1, 0, 1-n)\) with respect to \(\sigma\). The authors show that \(\mathfrak{M}_{\sigma}(n)\) coincides with \(\text{Hilb}^n S\) for \(s<0\) and \(t\gg 0\). The proof involves an interpretation for the moduli space as a moduli space of quiver representations using G.I.T.
One main theorem is that for general \(\sigma\) not lying on destabilizing walls, \(\mathfrak{M}_{\sigma}(n)\) is a nonsingular, projective variety of dimension \(2n\) so long as it is nonempty, and that for \(\sigma\neq \sigma'\) not on any destabilizing wall, \(\mathfrak{M}_{\sigma}(n)\) and \(\mathfrak{M}_{\sigma'}(n)\) are birationally equivalent.
Fix \(s\in \mathbb{R}\), then for the heart of the bounded \(t\)-structure corresponding to \(s\), one can let the parameter \(t\) decrease. When \(t\) crosses a destabilizing wall in the upper- half plane, there are induced birational transforms between the moduli spaces corresponding to \(\sigma(s, t)\) on the two sides of the wall. By the variation of G.I.T., the authors establish for \(\text{Hilb}^n S\) a one-to-one correspondence between the destabilizing walls in the upper-half plane of stability conditions and the stable base locus walls in the effective cone of divisors. In particular, for \(\text{Hilb}^n \mathbb{P}^2\), an explicit correspondence between the destabilizing walls in the second quadrant and the stable base locus walls is confirmed, as conjectured by Arcara, Bertram, Coskun and Huizenga in [Zbl 1267.14023]. birational geometry; stability condition; geometric invariant theory Algebraic moduli problems, moduli of vector bundles, Minimal model program (Mori theory, extremal rays), Parametrization (Chow and Hilbert schemes) The minimal model program for deformations of Hilbert schemes of points on the projective 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 For all integers \(r\ge 3\), \(d\ge r\), \(g\ge 0\) Let \(\mathcal{H}^{L}_{d,g,r}\) denote the union of the irreducible component Hilbert scheme of \(\mathbb {P}^r\) whose general element is a smooth, connected and linearly normal curve \(C\) with degree \(d\) and genus \(g\) (over an algebraically closed field with characteristic zero). The paper always works in the Brill-Noether range \(\rho(d,g,r) =g-(r+1)(g-d+r)\ge 0\). Since \(C\) is linearly normal, \(h^1(C,\mathcal{O}_C(1)) =g-d+r\). In this range the wide-open Modified Assertion of Severi asks if \(\mathcal{H}^{L}_{d,g,r}\). The author had previously solved the cases \(g-d+r\le 3\). In the paper under review the author studies the case \(g-d+r=4\). He proves the existence part (with a few exceptions) and the irreducibility part for \(3\le r\le 8\). For the latter part he construct explicit families on certain surfaces. Hilbert scheme; algebraic curves; linearly normal; special linear series Parametrization (Chow and Hilbert schemes), Plane and space curves On the Hilbert scheme of linearly normal curves in \(\mathbb{P}^r\) with small index of speciality | 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 for certain projective varieties (e.g., smooth complete intersections in projective space), there are only finitely many components of the Hilbert scheme parametrizing irreducible, smooth, projective subvarieties of low codimension and not of general type. This generalizes a result of \textit{M. Schneider} [Int. J. Math. 3, 397--399 (1992; Zbl 0762.14022)]. We give similar results concerning subvarieties with globally generated tangent bundle. Hilbert schemes; Noether-Lefschetz theorem C. CILIBERTO - V. DI GENNARO, Boundedness for low codimensional subvarieties, to Appear in Proceedings of Conference Held at the ESI of Wien in October 2000 (Editor Prof. Popov). Zbl1066.14007 MR2090667 Parametrization (Chow and Hilbert schemes), Low codimension problems in algebraic geometry, Algebraic moduli problems, moduli of vector bundles Boundedness for low codimensional subvarieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0727.00022.]
This paper deals with a generalization of mutations of exceptional sequences of vector bundles and helices studied by \textit{J. M. Drezet} and \textit{J. Le Potier} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 193- 243 (1985; Zbl 0586.14007)] and \textit{A. L. Gorodentsev} and \textit{A. N. Rudakov} [Duke Math. J. 54, 115-130 (1987; Zbl 0646.14014)] to arbitrary triangulated categories. There are also connections to the theory of tilting modules introduced by \textit{S. Brenner} and \textit{M. C. R. Butler} in Representation theory II, Proc. 2nd int. Conf., Ottawa 1979, Lect. Notes Math. 832, 103-169 (1980; Zbl 0446.16031) which are studied in representation theory of algebras.
Let \({\mathcal C}\) be a triangulated \(k\)-category defined over some field \(k\). An object \(E\) of \({\mathcal C}\) is called an exceptional object if \(\Hom(E,E)=k\) and \(\Hom(E,E[i])=0\) for \(i\neq 0\). A sequence \(\sigma=(E_ 0,E_ 1,\ldots,E_ n)\) of exceptional objects is called an exceptional sequence if \(\Hom(E_ i,E_ j[r])=0\) for all \(i>j\) and all \(r\in\mathbb{Z}\). If moreover \(\Hom(E_ i,E_ j[r])=0\) for all \(i,j\) and \(r\neq 0\) then \(\sigma\) is called a strongly exceptional sequence. The author proves that if \({\mathcal C}\) is generated by the elements of a strongly exceptional sequence then \({\mathcal C}\) is equivalent to the derivative category \(D^ b\hbox{mod}(A)\) of right modules over the algebra \(A=\hbox{End}(\oplus E_ i)\). Provided \({\mathcal C}=D^ b(\hbox{coh}(X))\), where \(\hbox{coh}(X)\) is the category of coherent sheaves over a projective algebraic variety \(X\), several examples are studied. Moreover, under the additional assumption that the anticanonical class of \(X\) is very ample, it is shown that for an exceptional sequence \(\sigma=(E_ 0,E_ 1,\ldots,E_ n)\) the following conditions are equivalent: (i) the \(E_ i\) generate \(D^ b(\hbox{coh}(X))\) as a triangulated category, (ii) \(\sigma\) is the foundation of a helix.
Finally, the author studies from a purely algebraic point of view the problem when mutations of a strongly exceptional sequence remain strongly exceptional. The result are certain homological conditions for the algebra \(A=\hbox{End}(\oplus E_ i)\). Algebras satisfying these conditions generalize Koszul algebras.
Note that the main results of this article are already published in Math. USSR, Izv. 34, No. 1, 23-43 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 1, 25-44 (1989; Zbl 0692.18002). derived category; representations of quivers; mutations of exceptional sequences of vector bundles; helices; tilting modules; exceptional object; Koszul algebra A.~I. Bondal. Helices, representations of quivers and Koszul algebras. In \(Helices and vector bundles\), volume 148 of \(London Math. Soc. Lecture Note Ser.\), pages 75-95. Cambridge Univ. Press, Cambridge, 1990. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Representations of quivers and partially ordered sets, Derived categories, triangulated categories Helices, representations of quivers and Koszul algebras | 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 Zariski stratification of a hypersurface \(V\) in a smooth complex variety \(W\) is a partition of \(V\) into smooth locally closed subvarieties \(Y_i\) such that, locally at \(x\in Y_i\), \(V\) is equisingular and equimultiple along \(Y_i\), and the \(Y_i\) are maximal with respect to these properties.
The question addressed in this paper is stated in the article of \textit{J. Lipman} [In: Prog. Math. 181, 485-505 (2000; Zbl 0970.14011)]: given a Zariski stratum \(Y\subset V\) and a point \(x\in Y\), is \(V\) equiresolvable along \(Y\) locally at~\(x\)?
Equisingularity in the sense of Zariski is defined inductively: two smooth points are equisingular, and \(x\in V\) and \(y\in V\) are equisingular if \(\beta(x)\) and \(\beta(y)\) are equisingular as points of the reduced discriminant locus \(\Sigma_\beta\), where \(\beta\) is a generic smooth projection of relative dimension~\(1\).
The meaning of the term ``equiresolution'' is less settled. Following Lipman [loc. cit.] we say that \(\text{ER}(x,V\subset W, Y)\) holds at \(x\in Y\) if there is a smooth morphism \(\pi:W\to Z\) such that \(\pi|_V\) is a family of reduced hypersurfaces and an embedded resolution \(f:W'\to W\) of \(V\) defining a family of embedded resolutions of the fibres of~\(\pi\), and \(f^{-1}(Y)\) is a union of exceptional divisors. This last condition amounts in practice to requiring that \(f^{-1}(Y)\) be of pure codimension~\(1\).
Note that we do not require \(f\) to be a sequence of blowups. Indeed, an example of \textit{I.~Luengo} [Math. Ann. 267, 487-494 (1984; Zbl 0539.14007)] shows that we cannot expect to achieve equiresolution under such restrictions.
The main technical theorem of the paper is that if \(\beta:W\to W_1\) is a suitably general smooth morphism of relative dimension~\(1\) and \(\Sigma_\beta\) is the reduced discriminant then \(\text{ER}(x,\Sigma_\beta\subset W_1, \beta(Y))\) implies \(\text{ER}(x,V\subset W, Y)\). Applying this to a Zariski stratum~\(Y\) gives an affirmative answer to the question above.
In view of Luengo's example a Hironaka-style approach to constructing~\(f\), by blowing up, cannot succeed directly. Instead the author turns to procedures similar to those of Jung and Abhyankar, among others, using an abelian Galois cover to separate \(V\to W_1\) into a union of sections (locally) and then carefully blowing up the locus of intersection of these sections. There remains a problem of resolving toric quotient singularities in a compatible way, which is solved by explicit calculations. This procedure is in its essentials not very different from the method described e.g. by \textit{K.~Paranjape} [In: Lond. Math. Soc. Lect. Note Ser. 264, 347-358 (1999; Zbl 0958.14007)]; but it is far from clear a priori how to make such an approach succeed. By doing so the paper greatly clarifies both how equiresolution ought to be thought of and when it can be expected to be attainable. resolution of singularities; equisingularity Orlando Villamayor U., On equiresolution and a question of Zariski, Acta Math. 185 (2000), no. 1, 123 -- 159. Equisingularity (topological and analytic), Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) On equiresolution and a question of Zariski | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Cluster algebras weres invented by \textit{S. Fomin} and \textit{A. Zelevinsky} [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)] in order to study total positivity in algebraic groups and canonical bases in quantum groups. Cluster categories [\textit{A. B. Buan} et al., Adv. Math. 204, No. 2, 572--618 (2006; Zbl 1127.16011)] are certain categories of representations of finite dimensional algebras which were introduced to ``categorify'' cluster algebras. The Caldero--Chapoton map was introduced in [\textit{P. Caldero} and \textit{F. Chapoton}, Comment. Math. Helv. 81, No. 3, 595--616 (2006; Zbl 1119.16013)] to formalize the connection between the cluster algebras and the cluster categories. Indeed, using the Caldero--Chapoton map, \textit{P. Caldero} and \textit{B. Keller} [Ann. Sci. Éc. Norm. Supér. (4) 39, No. 6, 983--1009 (2006; Zbl 1115.18301)] established a bijection between the indecomposable rigid objects of a cluster category and the cluster variables of the corresponding cluster algebra.
Let \(Q\) be an acyclic quiver with vertex set \(Q_0=\{1,2,\dots, n\}\). Let \(\mathbb{C}Q\) be the path algebra of \(Q\) and denote by \(P_i\) the indecomposable projective \(\mathbb{C} Q\)-module with the simple top \(S_i\) corresponding to \(i\in Q_0\) and \(I_i\) the indecomposable injective \(\mathbb{C} Q\)-module with the simple socle \(S_i\). The cluster category associated to \(Q\) is the orbit category \(\mathcal{C}(Q):=\mathcal{D}^{b}(Q)/[1]\circ\tau^{-1}\) where \(\mathcal{D}^b(Q)\) is the bounded derived category of \(\mathrm{mod} \mathbb{C} Q\) with the shift functor \([1]\) and the AR-translation \(\tau\). Let \(\mathbb{Q}(x_1,\dots, x_n)\) be a transcendental extension of the rational number field \(\mathbb{Q}\).
The Caldero-Chapton map of an acyclic quiver \(Q\) is \(X_?^Q: \text{obj} (\mathcal{C}(Q))\rightarrow \mathbb{Q}(x_1, \dots, x_n)\) defined by the following rules [Zbl 1119.16013]:
(1) If \(M\) is an indecomposable \(\mathbb{C}Q\)-module, then
\[
X_M^Q=\sum_{\underline{e}}\chi(\mathrm{Gr}_{\underline{e}}(M)) \prod_{i_0 \in Q_0}x_i^{-\langle \underline{e}, s_i \rangle -\langle s_i, \underline{\dim}M-\underline{e} \rangle}.
\]
(2) If \(M=TP_i\) is the shift of the projective module associated to \(i\in Q_0\), then \(X_M^Q=x_i.\)
(3) For any two objects \(M, N\) of \(\mathcal{C}_Q\), we have \(X_{M\oplus N}^Q=X_M^QX_N^Q\).
Here, we denote by \(\langle -,-\rangle\) the Euler form on \(\mathbb{C}Q\)-module and \(\mathrm{Gr}_{\underline{e}}(M)\) is the \(\underline{e}\)-Grassmannian of \(M\). Note that the indecomposable \(\mathbb{C}Q\)-modules and \(TP_i\) for \(i\in Q_0\) exhaust the indecomposable objects of the cluster category \(\mathcal{C}(Q)\). For any object \(M \in \mathcal{C}(Q),\) \(X_M^Q\) will be called the generalized cluster variable for \(M\).
Let \(R=(r_{ij})\) be a matrix with \(r_{ij}=\dim_{\mathbb{C}}\mathrm{Ext}^1(S_i,S_j)\) for any \(i, j\in Q_0\). For \(v=(v_1, \dots, v_n),\) we set \(x_v=x_1^{v_1} \dots x_n^{v_n}\). The Caldero-Chapton map can be reformulated by the following rules \textit{F. Xu} [Trans. Am. Math. Soc. 362, No. 2, 753--776 (2010; Zbl 1200.16021)]:
(1) \( X_{\tau P}=X_{P[1]}=x^{\underline{\mathrm{dim}}{P/\mathrm{rad} P}}, X_{\tau^{-1}I}=X_{I[-1]}=x^{\underline{\mathrm{dim}} \mathrm{soc}I}\) for any projective \(\mathbb{C} Q\)-module \(P\) and any injective \(\mathbb{C} Q\)-module \(I\);
(2)
\[
X_{M}=\sum_{\underline{e}}\chi(\mathrm{Gr}_{\underline{e}}(M))x^{\underline{e} R+(\underline{\mathrm{dim}}M-\underline{e})R^{\mathrm{tr}}- \underline{\mathrm{dim}}M }.
\]
Let \(\mathcal{AH}(Q)\) be the subalgebra of \(\mathbb{Q}(x_1, \dots, x_n)\) generated by all \(X_M, X_{\tau P}\), where \(M, P \in \text{mod}-\mathbb{C}{Q}\) and \(P\) is projective. Let \(\mathcal{EH} (Q)\) be the subalgebra of \(\mathcal{AH}(Q)\) generated by all \(X_M\), where \(M\in \text{ind}-\mathbb{C}(Q)\) and \(\text{Ext}_{\mathcal{C}}^1(M,M)=0.\)
In the paper under review, the authors obtain a \(\mathbb{Z}\)-basis of \(\mathcal{AH}(Q)\) and proved that \(\mathcal{EH}(Q)\) coincides with \(\mathcal{AH}(Q)\) for the quiver \(Q=\widetilde{D}_4.\) Moreover, the authors prove that coefficients of Laurent expansions of the basis elements are non-negative integers. \(\mathbb Z\)-basis; cluster algebra; cluster category; \(\tilde D_4\) Ding, M; Xu, F, A \(\mathbb {Z}\)\(\mathbb{Z}\)-basis for the cluster algebra of type \(\widetilde {D}_{4}\)D~4, Algebra Colloq., 19, 591-610, (2012) Special varieties, Representations of quivers and partially ordered sets, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers A \(\mathbb Z\)-basis for the cluster algebra of type \(\tilde D_4\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is an expository article that discusses in detail the contents of [Ann. Inst. Fourier 54, No. 3, 499--630 (2004; Zbl 1062.14014)] and [J. Inst. Math. Jussieu 10, No. 4, 809--934 (2011; Zbl 1236.14039)]. The author considers geometric orbifolds (pairs) \((Y,\Delta =\sum (1-1/m(D))D)\) where \(Y\) is a normal connected complex projective variety, the sum is finite, \(D\subset Y\) are prime divisors, and \(m(D)\in \mathbb Q\cap [1,+\infty[\cup \{+\infty \}\). Any fibration (algebraic fiber space) \(f:X\to Y\), gives a geometric orbifold \((Y,\Delta =\sum (1-1/m(D))D)\) where the \(m(D)\) depend on the multiplicities of fibers over codimension \(1\) points. The Kodaira dimension of \(Y/\Delta\) is \(\kappa (Y/\Delta )= \kappa (Y,K_Y+\Delta )\) and the Kodaira dimension of \(f\) is the minimum Kodaira dimension of orbifolds \(Y'/\Delta '\) arising form fibrations \(f':X'\to Y'\) birational to \(f:X\to Y\). A fibration \(f:X\to Y\) has (base) general type if \(\kappa (Y/\Delta ) =\dim Y\), \(X\) is special if it admits no fibrations \(f:X\to Y\) of general type and the fibration \(f\) is special if so is its (orbifold) general fiber. It turns out that if \(f:X\to Y\) is a sufficiently prepared (neat) special fibration whose base \((Y,\Delta )\) is special, then \(X\) is special. Special surfaces can be explicitly described and it is shown that \(X\) is special if two generic points can be joined by a chain of special submanifolds. The main result is that if \(X\) is not special, then there is a uniquely determined special fibration of base general type \(X\dasharrow C(X)\) (the core of \(X\)). Connections with rationally connected fibrations, the log minimal model program, hyperbolicity and arithmetic are also explored. birational classification; orbifolds; general type; core F Campana, Special orbifolds and birational classification: a survey (editors C Faber, G van der Geer, E Looijenga), EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich (2011) 123 Rational and birational maps, Parametrization (Chow and Hilbert schemes), Ramification problems in algebraic geometry, Minimal model program (Mori theory, extremal rays), Rational points, \(n\)-folds (\(n>4\)), Rationally connected varieties Special orbifolds and birational classification: a survey | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study the connection between the existence of finite dihedral covers of the projective plane ramified along an algebraic curve \( C\), infinite dihedral covers, and pencils of curves containing \( C\). Galois cover; degeneration of curves Artal, E.; Cogolludo, J. I.; Tokunaga, H. O., Pencils and infinite dihedral covers of \(\mathbb{P}\)2, Proc. Amer. Math. Soc., 136, 1, 21-29, (2008) Coverings of curves, fundamental group, Pencils, nets, webs in algebraic geometry, Singularities in algebraic geometry Pencils and infinite dihedral covers of \(\mathbb{P}^2\) | 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 Using the concept of characteristic polyhedron [\textit{H. Hironaka}, J. Math. Kyoto Univ. 7, 251--293 (1967; Zbl 0159.50502)], the authors continue to develop a constructive approach to the resolution of singularities in three-dimensional space in a more general context similar to [\textit{V. Cossart} and \textit{O. Piltant}, Math. Ann. 361, No. 1--2, 157--167 (2015; Zbl 1308.14008)]. The key idea is to determine the Hironaka polyhedron without passing to the completion of the local ring of a singularity. Thus, they prove that this is possible in a number of special situations, including the cases of local Henselian \(G\)-rings and singularities whose defining ideals satisfy certain numerical conditions on their standard bases. singularities; embedded resolutions; polyhedra; Hironaka's characteristic polyhedron; excellent rings; strong normalization; Hilbert-Samuel function; Hironaka schemes; standard bases Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) Characteristic polyhedra of singularities without completion. 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 This is a survey on recent research directly related to the other papers in the same volume.
After introducing the mapping class group and the Torelli group, the moduli space \({\mathcal M}_{g,n}\) of genus \(g,n\)-pointed, smooth curves is constructed as an analytic orbifold. The Deligne-Mumford-Knudsen compactification \(\overline{{\mathcal M}_{g,n}}\) of \({\mathcal M}_{g, n}\) is then regarded as the quotient of a smooth complete variety by a finite group. The \(k\)-th Chow group of \(\overline{{\mathcal M}_{g,n}}\) is then definable as the invariant part of the \(k\)-th Chow group of this smooth variety.
Next the authors recall the definition of the tautological classes and the results on the stability of the homology of the mapping class group as well as Mumford's conjecture. Finally the authors discuss the Witten conjecture, proved by Kontsevich, and its generalization to moduli spaces of stable maps.
The paper also contains an overview on general methods for finding complete subvarieties of \({\mathcal M}_g\). mapping class group; Torelli group; moduli space; Deligne-Mumford-Knudsen compactification; Chow group; tautological classes; Witten conjecture Carel Faber and Eduard Looijenga, Remarks on moduli of curves, Moduli of curves and abelian varieties, Aspects Math., E33, Friedr. Vieweg, Braunschweig, 1999, pp. 23 -- 45. Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, (Equivariant) Chow groups and rings; motives, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Remarks on moduli of 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 Let \(X\) be a normal complex projective threefold with only quotient singularities in codimension two. Such a threefold admits two different approaches to define its second Chern class: the class \(\widetilde c_ 2 (X)\) defined in the terms of the \(Q\)-tangent bundle of the locus of orbifold points on \(X\), and the class \(c_ 2 (X)\), defined in terms of a crepant in codimension two resolution of the singularities of \(X\). In this paper the authors prove the following main theorem: Let \(X\) be a 3- fold with canonical singularities with numerically trivial canonical class \(K_ X\), and with numerically trivial \(\widetilde c_ 2 (X)\). Then \(X\) is isomorphic to a quotient of an abelian threefold by a finite group acting freely in codimension one. This result is a generalization of a similar result for smooth varieties, but is perhaps unexpected because \(X\) is not assumed a priori to have quotient singularities. The key step of the proof is to establish that such \(X\) has only quotient singularities (this way proving the correctness of the notion \(\widetilde c_ 2 (X))\). The main corollary of the theorem shows that the triviality of \(c_ 2 (X)\) is stronger than that of \(\widetilde c_ 2 (X)\):
Let \(X\) be as above, but \(c_ 2 (X) = 0\). Then \(X\) is a quotient of an abelian threefold by a finite group acting freely in codimension two. In particular, these results are a generalization of the structure theorem for smooth Calabi-Yau threefolds with trivial \(c_ 2\). threefold with only quotient singularities; second Chern class; quotient of an abelian threefold; smooth Calabi-Yau threefolds Shepherd-Barron N.I., Wilson P.M.H.: Singular threefolds with numerically trivial first and second Chern classes. J. Alg. Geom. 3, 265--281 (1994) \(3\)-folds, Homogeneous spaces and generalizations, Characteristic classes and numbers in differential topology, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Singular threefolds with numerically trivial first and second Chern 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 \textit{E. Witten} [Commun. Math. Phys. 121, No. 3, 351--399 (1989; Zbl 0667.57005)] interpreted the Jones invariants of links in \(S^3\) (such as the Jones polynomial) in terms of his topological quantum field theory using Chern-Simons theory (see \textit{K. Marathe} [in: The mathematics of knots. Theory and application. Banagl, Markus (ed.) et al., Berlin: Springer. Contributions in Mathematical and Computational Sciences 1, 199--256 (2011; Zbl 1221.57023)] for a recent survey). This theory depends on a rank \(N\) and level \(k\), the quantum Hilbert space being identified with level \(k\) highest weight representations of the Lie algebra corresponding to \(\text{SU}(N)\) and acted on by the group \(\text{SL}(2,\mathbb Z)\) via matrices \(S\) and \(T\). Recently \textit{M. Aganagic} and \textit{Sh. Shakirov} proposed a refinement of the \(\text{SU}(N)\) Chern-Simons theory for links in 3-manifolds having \(S^1\) symmetry [String-Math 2011, Proc. Symp. Pure Math. 85, Amer. Math. Soc., Providence RI 2012, 3--31 (2012)] in which the matrices \(S\) and \(T\) are replaced by matrices used by \textit{I. Cherednik} [Invent. Math. 122, No. 1, 119--145 (1995; Zbl 0854.22021)] and \textit{A. A. Kirillov, jun.} [J. Am. Math. Soc. 9, No. 4, 1135--1169 (1996; Zbl 0861.05065)]. In the refined theory, the Hilbert space is identified with the MacDonald polynomials of type \(\text{SU}(N)\) with parameters \(q,t\) satsifying \(q^k t^N = 1\).
In the paper under review, the author computes the limit of the matrix \(S\) as \(N \to \infty\) for the refined theory. Starting with the explicit form of \(S\) given by \textit{M. Aganagic} and \textit{Sh. Shakirov} [``Knot homology from refined Chern-Simons theory'', Preprint, 2011, \url{arXiv:1105.5117}], he replaces \(t^N\) by a variable \(u\) to obtain a stable version expressed in terms of the modified MacDonald polynomials of \textit{A. Garsia} et al. [Sémin. Lothar. Comb. 42, B42m, 45 p. (1999; Zbl 0920.05071)]. To compute the kernel function, he starts with the Cherednik-MacDonald-Mehta identity in the form used by Garsia, Haiman and Tesler [loc. cit.] and using the relation between MacDonald polynomials and Hilbert schemes of \(n\) points in \(X=\mathbb C^2\) due to \textit{M. Haiman} [J. Am. Math. Soc. 14, No. 4, 941--1006 (2001; Zbl 1009.14001), Invent. Math. 149, No. 2, 371--407 (2002; Zbl 1053.14005)] he expresses two of the terms as power series in \(u\): the coefficient of \(u^n\) in each case is an equivariant Euler characteristics of certain sheaves on the Hilbert scheme \(X^{[n]}\) for the group action of \(\mathbb C^* \times \mathbb C^*\) - in one case the structure sheaf \({\mathcal O}_{X^{[n]}}\) and in the other case the tensor product of arbitrary Schur functors \(s_{\lambda}\) and \(s_{\mu}\) applied to a universal sheaf. Hilbert scheme of points in the plane; Chern-Simons theory Hiraku Nakajima, Refined Chern-Simons theory and Hilbert schemes of points on the plane, Perspectives in representation theory, Contemp. Math., vol. 610, Amer. Math. Soc., Providence, RI, 2014, pp. 305 -- 331. Parametrization (Chow and Hilbert schemes), Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Refined Chern-Simons theory and Hilbert schemes of points on 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 We study pure Yang-Mills theory on \(\Sigma \times S^{2}\), where \(\Sigma\) is a compact Riemann surface, and invariance is assumed under rotations of \(S^{2}\). It is well known that the self-duality equations in this setup reduce to vortex equations on \(\Sigma\). If the Yang-Mills gauge group is \(\operatorname{SU}(2)\), the Bogomolny vortex equations of the Abelian Higgs model are obtained. For larger gauge groups, one generally finds vortex equations involving several matrix-valued Higgs fields. Here we focus on Yang-Mills theory with gauge group \(\operatorname{SU}(N)/\mathbb{Z}_N\) and a special reduction which yields only one non-Abelian Higgs field. One of the new features of this reduction is the fact that while the instanton number of the theory in four dimensions is generally fractional with denominator \(N\), we still obtain an integral vortex number in the reduced theory. We clarify the relation between these two topological charges at a bundle geometric level. Another striking feature is the emergence of nontrivial lower and upper bounds for the energy of the reduced theory on \(\Sigma\). These bounds are proportional to the area of \(\Sigma\). We give special solutions of the theory on \(\Sigma\) by embedding solutions of the Abelian Higgs model into the non-Abelian theory, and we relate our work to the language of quiver bundles, which has recently proved fruitful in the study of dimensional reduction of Yang-Mills theory.{
\copyright 2011 American Institute of Physics} Manton, NS; Rink, NA, Geometry and energy of non-abelian vortices, J. Math. Phys., 52, 043511, (2011) Yang-Mills and other gauge theories in mechanics of particles and systems, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Representations of quivers and partially ordered sets Geometry and energy of non-Abelian vortices | 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 variety. A line bundle \(L\) on \(X\) is said to be \(3\)-very ample if the restriction map \(H^0(L) \to H^0(L|Z)\) is surjective for every degree \(4\) zero-dimensional subscheme \(Z\subset X\). The main result proved here is that if \(L\) is \(3\)-very ample and it has another technical assumption, then the secant variety \(\Sigma (X,L)\) of \(X\) of the image of \(X\) by the complete linear system \(|L|\) is normal. This technical assumption is satified if \(L\cong \omega _X\otimes A^{\otimes 2(n+1)}\otimes B\), \(n:= \dim (X)\), with \(A\) very ample and \(B\) nef. If \(X\) is a smooth curve of genus \(g\), it is sufficient to assume that \(\deg (L)\geq 2g+3\). She also proves the normality of the secant variety of the canonical embedding of all smooth curves with Clifford index \(\geq 3\). These are very nice results and they also fill a gap in the published proofs by other authors.
Among the papers used for the proofs or generalized by the present one or whose quotations are now fixed I mention: [\textit{A. Bertram}, J. Differ. Geom. 35, No. 2, 429--469 (1992; Zbl 0787.14014)], [\textit{J. Sidman} and \textit{P. Vermeire}, Algebra Number Theory 3, No. 4, 445--465 (2009; Zbl 1169.13304); Abel Symposia 6, 155--174 (2011; Zbl 1251.14043)], [\textit{P. Vermeire}, Compos. Math. 125, No. 3, 263--282 (2001; Zbl 1056.14016); J. Algebra 319, No. 3, 1264--1270 (2008; Zbl 1132.14027); Proc. Am. Math. Soc. 140, No. 8, 2639--2646 (2012; Zbl 1279.14068)]. secant variety; normality of secant variety; \(k\)-very ample line bundle; Hilbert scheme; positivity; vector bundle Ullery, B.: On the normality of secant varieties. Adv. Math. \textbf{288}, 631-647 (2016). arXiv:1408.0865v2 Projective techniques in algebraic geometry, Parametrization (Chow and Hilbert schemes), Plane and space curves, Questions of classical algebraic geometry On the normality of secant 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 [For the entire collection see Zbl 0614.00006.]
Let \(H(d,g)_ S\) be the open subscheme of the Hilbert scheme of curves of degree \(d\) and arithmetic genus g in \({\mathbb{P}}^ 3\) parametrizing smooth irreducible curves. The first author who pointed out the existence of irreducible non reduced components of \(H(d,g)_ S\), was \textit{D. Mumford} [Am. J. Math. 84, 642-648 (1962; Zbl 0114.131)] who found a non reduced component of \(H(14,24)_ S\), the general curve of which lies on a smooth cubic surface in \({\mathbb{P}}^ 3\). Mumford's example has been widely generalized by the author of the present paper in his thesis (``The Hilbert-flag scheme, its properties and its connection with the Hilbert scheme. Applications to curves in 3-space'', Preprint no. 5-1981, Univ. Oslo). Among other things it turns out from his analysis that if \(W\subseteq H(d,g)_ S\) is a closed irreducible subset whose general point corresponds to a curve C lying on a smooth cubic surface, W is maximal under this condition and \(d>9\), then W irreducible, non reduced component of \(H(d,g)_ S\) yields \(g\geq 3d-18\) and \(H^ 1({\mathcal J}_ C(3))\neq 0\) (the latter inequality implying that \(g\leq (d^ 2-4)/8.\)
The author conjectures that these necessary conditions are also sufficient for W to be a non reduced component of \(H(d,g)_ S\), and he proves this conjecture in the ranges \(7+(d-2)^ 2/8<g\leq (d^ 2-4)/8\), \(d\geq 18\) and \(-1+(d^ 2-4)/8<g\leq (d^ 2-4)/8\), 17\(\geq d\geq 14\). The proof consists in an interesting analysis of the tangent and obstruction space to the so called Hilbert-flag scheme (parametrizing pairs (curve, surface), the first contained in the latter) in particular for curves lying on surfaces of degree \(s\leq 4.\) space curves; Hilbert scheme; degree; arithmetic genus; obstruction space; Hilbert-flag scheme J. O. Kleppe, Nonreduced components of the Hilbert scheme of smooth space curves. In Space curves (Rocca di Papa, 1985), volume 1266 of Lecture Notes in Math. (Springer, Berlin, 1987), pp. 181-207. Zbl0631.14022 MR908714 Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Projective techniques in algebraic geometry Non-reduced components of the Hilbert scheme of smooth space 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 Hilbert in his famous paper in 1890 used the projective resolution of a graded module \(M\) over \(k[X_ 1, \dots, X_ n]\) to create its Hilbert function. The authors, in this paper, consider the converse problem. That is, given a Hilbert function of a graded module \(M\), they consider all possible projective resolutions with this Hilbert function. They consider only cyclic modules. They are successful only in the case of rings of dimension two. If \(R = k[x,y]\) is a two-dimensional graded regular ring and if \(R/I\) is a graded module of finite length with minimal resolution, they find, in theorem 1, some conditions on degrees of generators and relations. The converse of this is dealt with in theorem 2 as a consequence of which we can get, for a given Hilbert series of a cyclic module, resolutions with the smallest or biggest possible graded Betti numbers. It is illustrated with a concrete example. In the higher dimensional case the situation is not so satisfactory. If the dimension of \(R = k[x_ 1, \dots, x_ n]\) is at least three, there exist a Hilbert series for a cyclic module of finite length and two incomparable smallest sets of graded Betti numbers for that series. Betti number; Hilbert polynomial; projective resolution; graded module; Hilbert function; Hilbert series Hara Charalambous and E. Graham Evans Jr., Resolutions with a given Hilbert function, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 19 -- 26. Resolutions; derived functors (category-theoretic aspects), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Parametrization (Chow and Hilbert schemes), Deformations and infinitesimal methods in commutative ring theory Resolutions with a given Hilbert function | 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 focuses on families of Artinian or one-dimensional quotients of a polynomial ring \(R\). Let \(H\) be a Hilbert function and let GradAlg\(^H (R)\) be the scheme parametrizing all graded quotients of \(R\) with Hilbert function \(H\). Let \(B \rightarrow A\) be a graded surjection of quotients of \(R\), with Hilbert functions \(H_B\) and \(H_A\) respectively. If \(\dim A = 0\) or 1 and making some additional assumptions on both \(A\) and \(B\), the author gives close connections between GradAlg\(^{H_B}(R)\) and GradAlg\(^{H_A}(R)\). These connections involve for instance smoothness and dimension of these parameter schemes. In a more general setting he describes the dual of the tangent and obstruction space of graded deformations. He then applies this machinery to the case of level algebras of Cohen-Macaulay type 2 (this is a natural first case after the Gorenstein algebras). As a result, he proves a conjecture of Iarrobino on the existence of at least two irreducible components of GradAlg\(^H(R)\) when \(H = (1,3,6,10,14,10,6,2)\), such that the general elements of these components are level. The work is very technical, but many examples are given to illustrate the methods. Similar parameter schemes have been studied in the past, but generally those papers have considered the reduced scheme structure, while GradAlg\(^H (R)\) may be non-reduced. parametrization; Artinian algebra; level algebra; Gorenstein algebra; licci; Hilbert scheme; duality; algebra (co)homology; canonical module; normal module Kleppe, J.O., Families of Artinian and one-dimensional algebras, J. algebra, 311, 665-701, (2007) Parametrization (Chow and Hilbert schemes), Commutative Artinian rings and modules, finite-dimensional algebras, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Deformations and infinitesimal methods in commutative ring theory, Linkage, complete intersections and determinantal ideals Families of Artinian and one-dimensional algebras | 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 \(N\) be a free abelian group of rank \(r,\) let \(K\) be a rational polyhedral cone inside \(N \otimes {\mathbb R}\) satisfying the standard conditions, and let \(R = {\mathbb C}[K\cap N]\) be the semigroup ring. Set \(Z_j = \sum_{i=1}^d \langle m_j, e_i \rangle \exp(2\pi\sqrt{-1}a_i) x_i,\) \(j =1, \ldots, r,\) where \(\{e_1, \ldots, e_d\}\) is a set of lattice points of degree \(1\) that lie in the cone \(K,\) \(m_1, \ldots, m_r\) is a basis of the vector space \(\text{ Hom}(N, {\mathbb Z}) \otimes {\mathbb C},\) and the element \(x_i \in R\) corresponds to \(e_i, i = 1, \ldots, d,\) respectively. It is proved that \(Z_1, \ldots, Z_r\) is a regular sequence in the ring \(R\) as well as in its open part \(R^{\text{open}}.\) In particular this implies the result by \textit{ {}M. Hochster} [Ann. Math. (2) 96, 318-337 (1972; Zbl 0233.14010)] that \(R\) is a Cohen-Macaulay ring. In the note the author also describes an analog of the Poincaré duality and discusses how his results relate to the mirror symmetry and string cohomology. toroidal singularity; polyhedral cones; string cohomology; Cohen-Macaulay rings; mirror symmetry; Calabi-Yau hypersurfaces Borisov, LA, String cohomology of a toroidal singularity, J. Algebraic Geom., 9, 289-300, (2000) Toric varieties, Newton polyhedra, Okounkov bodies, Calabi-Yau manifolds (algebro-geometric aspects), Singularities in algebraic geometry, String and superstring theories; other extended objects (e.g., branes) in quantum field theory String cohomology of a toroidal 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 In this paper the authors introduce new quiver gauge theories which can be associated with Sasaki-Einstein manifolds \(S^3/\Gamma\) where \(\Gamma\) is a finite subgroup of \(\mathrm{SU}(2)\cong S^3\) with its standard metric. These new quiver gauge theories are put in comparison on the one hand with quiver gauge theories associated to Calabi-Yau cones \(C(S^3/\Gamma)\) (same as singular multi-Eguchi-Hanson or ALE Gibbons-Hawking spaces of all the \(ADE\)-types) and on the other hand with quiver gauge theories on the corresponding coset spaces of \(\mathbb {CP}^1\).
The idea is that a Sasaki-Einstein \(3\)-manifold \(S^3/\Gamma\) interpolates between two distinct singular Kähler manifolds: the \(4\)-orbifold \(C(S^3/\Gamma)\) and the corresponding \(2\)-orbifold coset of \(\mathbb {CP}^1\), allowing one to clarify the dynamical relations between the quiver bundles underlying the field theory for \(C(S^3/\Gamma)\) and the quiver bundles underlying the field theory for the corresponding coset of \(\mathbb {CP}^1\). The authors use Nahm equations to describe the moduli space of vacua of the quiver gauge theory associated to \(C(S^3/\Gamma )\) as spherically symmetric instantons over these cones and relate them to the Nakajima quiver varieties and to the moduli spaces of spherically symmetric solutions of hypothetical non-abelian generalizations of two imensional affine Toda field theories.
Apparently the overall aim of these investigations is to understand better and generalize the recently discovered Alday-Gaiotto-Tachikawa relation between \(4\) dimensional gauge theories and \(2\) dimensional conformal field theories. Lechtenfeld, O.; Popov, AD; Szabo, RJ, Sasakian quiver gauge theories and instantons on Calabi-Yau cones, Adv. Theor. Math. Phys., 20, 821, (2016) Yang-Mills and other gauge theories in quantum field theory, Representations of quivers and partially ordered sets, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Calabi-Yau manifolds (algebro-geometric aspects), Special Riemannian manifolds (Einstein, Sasakian, etc.), Topology and geometry of orbifolds Sasakian quiver gauge theories and instantons on Calabi-Yau cones | 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 expounds the theory of reduction of singularities of algebraic surfaces along the lines of \textit{O. Zariski}'s papers [Ann. Math., II. Ser. 40, 639-689 (1939; Zbl 0021.25303) and Ann. Math., II. Ser. 43, 583-593 (1942; Zbl 0063.08389)]. In line with Zariski's ideas and techniques the paper gives a detailed account of uniformization of valuations and desingularization by blowing up and normalization (allowing, thanks to a new argument, a non-algebraically closed field). resolution of singularities; uniformization; desingularization; non-algebraically closed field V. Cossart, Uniformisation et désingularisation des surfaces d'apr` es Zariski, in Res- olution of singularities (Obergurgl, 1997), 239-258, Progr. Math., 181, Birkhäuser, Basel. Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Other nonalgebraically closed ground fields in algebraic geometry Uniformization and desingularization of surfaces after Zariski | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\alpha_ 1,\alpha_ 2,...,\alpha_ n\) be pairwise coprime positive integers. \(\Sigma (\alpha_ 1,\alpha_ 2,...,\alpha_ n)\) be the Seifert homology 3-sphere associated to \((\alpha_ 1,\alpha_ 2,...,\alpha_ n)\), and \(V(\alpha_ 1,\alpha_ 2,...,\alpha_ n)\) be the 2-dimensional smooth affine variety in \({\mathbb{C}}^ n\) which is the Brieskorn complete intersection of weight \((\alpha_ 1,\alpha_ 2,...,\alpha_ n)\). It is known that \(\Sigma (\alpha_ 1,\alpha_ 2,...,\alpha_ n)\) is diffeomorphic to \(V(\alpha_ 1,\alpha_ 2,...,\alpha_ n)\cap S_ r^{2n-1}\), where \(S_ r^{2n-1}\) is the (2n-1)-sphere centered at \(0\in {\mathbb{C}}^ n\) with radius r sufficiently large [\textit{H. Hamm}, Math. Ann. 197, 44-56 (1972; Zbl 0239.14003)]. Let \(\sigma (\alpha_ 1,\alpha_ 2,...,\alpha_ n)\) denote the signature of \(V(\alpha_ 1,\alpha_ 2,...,\alpha_ n)\). Concerning the \(\lambda\)-invariant \(\lambda (\alpha_ 1,\alpha_ 2,...,\alpha_ n)\) of \(\Sigma (\alpha_ 1,\alpha_ 2,...,\alpha_ n)\), we prove the following theorem which extends Theorem 2.10 of \textit{Fintushel} and \textit{Stern} [Instanton homology of Seifert fibered homology three spheres (preprint)]: Theorem 1. \(\lambda (\alpha_ 1,\alpha_ 2,...,\alpha_ n)=(1/8)\sigma (\alpha_ 1,\alpha_ 2,...,\alpha_ n)\). More explicitly: \(\lambda (\alpha_ 1,\alpha_ 2,...,\alpha_ n)=(- 1/8)+(1/24\alpha)(1+\sum^{n}_{j=1}(\alpha /\alpha_ j)^ 2-(n- 2)\alpha^ 2)-(1/2)\sum^{n}_{j=1}s(\alpha /\alpha_ j,\alpha_ j),\) where \(\alpha =\prod^{n}_{j=1}\alpha_ j\) and \(s(\quad,\quad)\) denotes the Dedekind sum. Related results are obtained, too. lambda invariant; Brieskorn variety; signature; Seifert fibered homology three spheres Fukuhara S., Matsumoto Y., Sakamoto K. (1990). Casson's invariant of Seifert homology 3-spheres. Math. Ann. 287:275--285 Topology of general 3-manifolds, Algebraic topology on manifolds and differential topology, Singularities in algebraic geometry, Local complex singularities Casson's invariant of Seifert homology 3-spheres | 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 One of many definitions of equisingularity of a smooth family of smooth map-germs \(f_{t}:(\mathbb{R}^{n},0)\rightarrow(\mathbb{R}^{p},0)\) is the bilipschitz triviality. The family \(f_{t},\) \(t\in I,\) is bilipschitz trivial iff there exist \(t_{0}\in I,\) \(\lambda>0\) and a map \(\varphi(x,t)=(\varphi _{t}(x),t)\) such that for all \(t\in I\)
(1) \(\varphi_{t}\) is \(\lambda\)-lipschitz and \(\varphi_{t}^{-1}\) is \(1/\lambda\)-lipschitz,
(2)\(f_{t}\circ\varphi_{t}=f_{t_{0}}.\)
The main theorem is a condition on a map-germ \(\theta:(\mathbb{R} ^{n},0)\rightarrow(\mathbb{R}^{p},0)\) for the deformation \(f_{t}=f+t\theta\) to be bilipschitz trivial, where \(f:(\mathbb{R}^{n},0)\rightarrow(\mathbb{R} ^{p},0)\) is a nondegenerate polynomial map. The condition is expressed in terms of some Newton polyhedron. singularity; analytic map-germ; deformation of singularity Fernandes, A., Soares Jr., C. H.: On the bi-Lipschitz triviality of families of real maps. In: Real and Complex Singularities. Contemp. Math., vol. 354, pp. 95--103. Amer. Math. Soc., Providence (2004) Real algebraic and real-analytic geometry, Singularities in algebraic geometry, Equisingularity (topological and analytic) On the bilipschitz triviality of families of real maps | 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 Lê, D. T.: Un application d'un théoréme d'a'campo a l'equisingularité. Indag. math. 35, 403-409 (1973) Singularities in algebraic geometry, Structure of families (Picard-Lefschetz, monodromy, etc.), Complex singularities Une application d'un théorème d'A'Campo à l'équisingularité | 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 0527.00017.]
This paper is concerned with the Milnor lattices of hypersurface singularities and Dynkin diagrams with respect to geometric (weakly distinguished and distinguished) bases of these lattices. It gives a summary of the author's results on the calculation of these invariants for some special classes of singularities, namely the singularities of Arnold's lists and the elliptic hypersurface singularities. These results are partly contained in the author's thesis [Math. Ann. 255, 463-498 (1981; Zbl 0438.32004)] and partly new. The paper starts with a general discussion of the notions of weakly distinguished and distinguished bases and the action of the braid group and symmetric group on the sets of these bases. Conjecture 2.3, stated in this context, has now been proven [cf. \textit{S. P. Humphries}, ''On weakly distinguished bases and free generating sets of free groups'', Q. J. Math., Oxf. II. Ser. (to appear)]. Then some general results on the discriminant quadratic forms of the Milnor lattices and on weakly distinguished bases are given. More details and further results can be found in the author's joint paper with \textit{C. T. C. Wall} [''Kodaira singularities and an extension of Arnold's strange duality'', Compos. Math. (to appear)] and for the general elliptic hypersurface case in the author's paper: ''The Milnor lattices of the elliptic hypersurface singularities'' (in preparation). Finally the author discusses a normal form for Dynkin diagrams with respect to distinguished bases of Arnold's bimodular singularities. These diagrams were used in a study of the deformation theory of these singularities [\textit{D. Balkenborg, R. Bauer} and \textit{F.-J. Bilitewski} (Diplomarbeit, Bonn 1984)]. discriminant quadratic form; weakly distinguished basis; Milnor lattices of hypersurface singularities; Dynkin diagrams; elliptic hypersurface singularities; action of the braid group; bimodular singularities W. Ebeling, ''Milnor Lattices and Geometric Bases of Some Special Singularities,'' in Noeuds, tresses et singularit és, Ed. by C. Weber (Enseign. Math. Univ. Genève, Genève, 1983), Monogr. Enseign. Math. 31, pp. 129--146; Enseign. Math. 29, 263--280 (1983). Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Braid groups; Artin groups, Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Complex singularities, Quadratic forms over global rings and fields Milnor lattices and geometric bases of some special 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 symmetric quiver \((Q,\sigma)\) is a finite quiver without oriented cycles \(Q=(Q_0,Q_1)\) equipped with a contravariant involution \(\sigma\) on \(Q_0\sqcup Q_1\). The involution allows us to define a nondegenerate bilinear form \(\langle-,-\rangle_V\) on a representation \(V\) of \(Q\). We shall say that \(V\) is orthogonal if \(\langle-,-\rangle_V\) is symmetric and symplectic if \(\langle-,-\rangle_V\) is skew-symmetric. Moreover, we define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. So we prove that if \((Q,\sigma)\) is a symmetric quiver of tame type then the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type \(c^V\) and, when the matrix defining \(c^V\) is skew-symmetric, by the Pfaffians \(pf^V\). To prove it, moreover, we describe the symplectic and orthogonal generic decomposition of a symmetric dimension vector. symmetric quivers of tame type; representations of quivers; rings of semi-invariants; actions of products of classical groups; Coxeter functors; Pfaffians; Schur modules; generic decompositions; bilinear forms Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Quadratic and bilinear forms, inner products, Geometric invariant theory, Vector and tensor algebra, theory of invariants Semi-invariants of symmetric quivers of tame type. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider finite-dimensional modules over tame path algebras and study the `building blocks' of their degenerations. These are the minimal disjoint degenerations to the direct sum of two indecomposables \(U\), \(V\). For the building blocks, we derive a reduction theorem that holds if \(U\) or \(V\) are non-regular. Subsequently we focus on minimal degenerations to the direct sum of a preprojective and a preinjective indecomposable and use the reduction theorem to introduce a new method to analyze the codimensions of this sort of degenerations. We show that the codimensions are bounded. By means of computer calculations we determine this bound explicitly. The codimension is always one. Finally, this leads to a minimality preserving periodicity theorem, which reduces the classification of the building blocks to a finite problem. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets On minimal disjoint degenerations with preprojective and preinjective direct summands over tame path algebras | 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 scheme of finite type over a field \(k\). Let \(D_1, \dots, D_h\) be effective divisors having simple normal crossings on \(X\). A simple constructive proof (working over fields of any characteristic) is given for the existence of a finite sequence of blow-ups at non-singular centers principalizing the ideal \(I_{D_1}+\dots+I_{D_h}\), i.e. if \(\pi: X'\to X\) is the composite of these blow-ups, then the ideal \(\sum I_{D_i}\mathcal O_{X'}\) is a principal ideal defining a divisor with simple normal crossings. In particular it is proved that the centers can always be chosen to be the intersection of two of the prime divisors in the support of \(\sum D_i\). Principalization of monomial ideals is one step in the constructive resolution of singularities. Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: \textsc{Singular} 3-1-6--A Computer Algebra System for Polynomial Computations. http://www.singular.uni-kl.de (2012) Global theory and resolution of singularities (algebro-geometric aspects), General commutative ring theory, Rational and birational maps A simple algorithm for principalization of monomial 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 Artin-Schreier coverings \(\sigma:W\to V\) where the function field extension of \(k(W)\) over \(k(V)\) is of degree \(p>0\) are studied first. Let \(V=\mathbb{A}^ 2\), the affine 2-space, \(\sigma\) be étale and then \(W\) be defined by \(\tau^ p-\tau-f(x,t)=0\). There are three cases:
(i) If \(\deg(f)\equiv 0\pmod p\), certain conditions on \(f\)'s non- singularity imply that \(W\) is non-singular.
(ii) Let \(\deg(f)\equiv p-1\pmod p\). Here \(W\) has exactly \(d\) rational double points of type \(A_{p-1}\), given that the homogenization of \(f\) meets the line at infinity transversally. The same properties that hold for sheaves and their invariants in case (ii) hold here.
(iii) Let \(\deg(f)\not\equiv 0, p-1\pmod p\). For a Gorenstein scheme \(W'\) there is a map \(\Psi:W'\to\mathbb{P}^ 2\). There is also a normalization map \(\pi:W''\to\mathbb{P}^ 2\) corresponding to the extension of \(k(W)\) by \(k(\mathbb{P}^ 2)\). The author fills in the exact sequence \(0\to\Psi_ *{\mathcal O}\to\pi_ *{\mathcal O}\to{\mathcal H}\to 0\) of sheaves over \(\mathbb{P}^ 2\) under the conditions of case (ii). The canonical sheaf of the minimal resolution \(N\) of singularities of \(W''\) is also determined. Independent of the cases above, if \(1+e=mp (0\leq e<p)\) and \(e=p-1\), then, with the hypothesis of case (ii) above, one has \(H^ 1(N,{\mathcal O}_ N)=0\) for \(m\geq p-1\). etale Galois coverings; normalization; cohomology; Artin-Schreier coverings; Gorenstein scheme; minimal resolution of singularities Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Étale and other Grothendieck topologies and (co)homologies Etale Galois coverings of degree \(p\) of the affine 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 the paper under review, the authors obtain a maximum principle for a type of elliptic systems and they use it to analyze the Hitchin equation for a special case of Higgs bundle they call cyclic Higgs bundles. They prove several domination results on the pullback metric of the (possibly branched) minimal immersion associated to cyclic Higgs bundles. They also obtain a lower and upper bound of the extrinsic curvature of the image of this minimal immersion. As an application, they give a complete picture for maximal \(\mathrm{Sp}(4,\mathbb{R})\)-representations in the \(2g-3\) Gothen components and the Hitchin components. Higgs bundle; cyclic Higgs bundle; Hitchin component Vector bundles on curves and their moduli, Moduli problems for differential geometric structures, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Representations of quivers and partially ordered sets On cyclic Higgs 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 this intriguing paper, the authors introduce a new operation on the modules of an \(F\)-finite ring of characteristic \(p>0\) which they term the tight interior. It is a true interior operation when \(R/\sqrt{R}\) is \(F\)-finite and is a dual to the tight closure operation developed by Hochster and Huneke. In addition to tight interior the authors introduce dual notions of several common tight closure constructs: co-test elements, co-\(F\)-regularity, co-persistence, co-contraction, co-colon capturing and co-phantom resolutions. Strikingly, in the \(F\)-finite reduced setting the co-test elements are precisely the big test elements and a co-\(F\)-regular ring is precisely a strongly \(F\)-regular ring. They also show that the tight interior of a finite \(R\)-modules \(M\) whose support is equal to the support of \(R\) is the smallest \(\mathcal{C}_M\)-fixed submodule of \(M\) proposed in [\textit{M. Blickle}, J. Algebr. Geom. 22, No. 1, 49--83 (2013; Zbl 1271.13009)] where \(\mathcal{C}_M\) is the full Cartier algebra on \(M\). Hence, showing that this submodule does indeed exist. They also improve a result of the reviewer [Trans. Am. Math. Soc. 350, No. 10, 4041--4051 (1998; Zbl 0913.13005)] showing that if \(R\) is a reduced equicharacteristic Noetherian ring with minimal primes \(\{\mathfrak{p}_i\}_{i=1}^{n}\) such that each \(R/\mathfrak{p}_i\) is weakly \(F\)-regular, then the test ideal is the conductor which is precisely the sum of the annihilators of the minimal primes and further show in the characteristic \(p\) setting that if the normalization of \(R\) is strongly \(F\)-regular then the test ideal is the big test ideal. tight closure; \(F\)-finite; big test ideal; \(F\)-regular; Cartier algebra; conductor Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Integral closure of commutative rings and ideals, Étale and flat extensions; Henselization; Artin approximation, Singularities in algebraic geometry, Multiplier ideals A dual to tight closure theory | 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 simplest version of the main result of this paper is the following. Let \(f\) be a polynomial on a finite dimensional vector space \(W\). Then the Fourier transform of the absolute value of \(f\) is smooth on a dense open subset of \(W^*\). This result holds uniformly over all local fields \(F\) of characteristic \(0\): If the inputs are defined over some field \(K\) (i.e., \(W = W_K \otimes_K F\) for some \(K\)-vector space \(W_K\) and \(f\) is obtained from a polynomial on \(W_K\) with coefficients in \(K\)), then the authors obtain a dense open sub-variety \(L \subset W_K^*\) such that the Fourier transform is smooth on \(L(F)\), for every \(F\) as above and for every embedding of \(K\) into \(F\).
The full main result strengthens the above in several ways. In particular, the absolute value of a polynomial can be replaced by a more general distribution on \(W\), e.g., by the direct image \(\phi_*(|\omega|)\) of the measure induced by a regular top differential form \(\omega\) on a smooth algebraic variety \(X\), under a proper map \(\phi: X \to W\). Moreover, the result about the Fourier transform is more precise, stating that its wave front set is contained in an isotropic algebraic sub-variety of \(T^*(W^*) = W \times W^*\) (which again can be given uniformly in \(F\)).
Several of the weaker versions of the main result have been known before. What is new is a uniform (in \(F\)) description of the wave front set of Fourier transforms. Moreover, this description is specified explicitly in terms of a desingularization. (Desingularization is a key ingredient to the proof, in contrast to older proofs which used \(D\)-modules in the archimedean case and model theory in the non-archimedean case.) wave front set; distribution; Fourier transform; local field; desingularization 2 A. Aizenbud and V. Drinfeld, 'The wave front set of the fourier transform of algebraic measures', \textit{Israel J. Math.}207 (2015) 527-580 (English). Local ground fields in algebraic geometry, Measures on groups and semigroups, etc., Global theory and resolution of singularities (algebro-geometric aspects), Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups The wave front set of the Fourier transform of algebraic measures | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let G be a simple complex Lie group of type A, D, E. The author gives a construction of the hypersurface singularity of the same type in terms of conjugacy classes of G which is different from the one found by Grothendieck-Brieskorn [see \textit{F. Brieskorn}, ``Singular elements of semisimple algebraic groups'' in Actes Congr. Internat. Math. 1970, vol. 2, 279-284 (1971; Zbl 0223.22012)]. Namely, let X be the closed orbit of G under the adjoint representation in the projective space attached to the Lie algebra \({\mathfrak g}\) of G. Consider a regular nilpotent element \(y\in {\mathfrak g}\) and let H(y) be the hyperplane orthogonal to y with respect to the Killing form. Then \(X\cap H(y)\) has exactly one singularity which is simple of the desired type. Its dimension is quite high, for example equal to 57 in the case of \(E_ 8\). By varying y in \({\mathfrak g}\) one can realize a versal deformation of this singularity.
These results are generalized to algebraic groups over fields of any characteristic and applied to get interesting deformations in characteristic two and three. conjugacy classes of complex Lie group; hypersurface singularity; versal deformation Knop, F.: Ein neuer Zusammenhang zwischen einfachen Gruppen und einfachen Singularitäten. Invent. Math. \textbf{90}, 579-604 (1987) Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Simple groups, General properties and structure of complex Lie groups, Linear algebraic groups over arbitrary fields, Deformations of singularities Ein neuer Zusammenhang zwischen einfachen Gruppen und einfachen Singularitäten. (A new connection between simple groups 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 Consider the action of an algebraic group \(G\) on an irreducible algebraic variety \(X\) defined over a field \(k\). \textit{M. Rosenlicht} [Am. J. Math. 78, 401--443 (1956; Zbl 0073.37601)] showed that orbits in general position in \(X\) can be separated by rational invariants. We prove a dynamical analogue of this theorem, where \(G\) is replaced by a semigroup of dominant rational maps \(X\dashrightarrow X\). Our semigroup \(G\) is not required to have the structure of an algebraic variety and can be of arbitrary cardinality. This generalizes earlier work of \textit{E. Amerik} and \textit{F. Campana} [Pure Appl. Math. Q. 4, No. 2, 509--545 (2008; Zbl 1143.14035)], where \(k=\mathbb{C}\) and the semigroup \(G\) is assumed to be generated by a single endomorphism. Bell, Jason; Ghioca, Dragos; Reichstein, Zinovy, On a dynamical version of a theorem of Rosenlicht, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 17, 1, 187-204, (2017) Rational and birational maps, Parametrization (Chow and Hilbert schemes), Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables On a dynamical version of a theorem of Rosenlicht | 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 Consider the periodic Toda lattices corresponding to extended Dynkin diagrams [cf. \textit{M. A. Ol'shanetsky} and \textit{A. M. Peremolov}, Invent. Math. 54, 261-269 (1979; Zbl 0419.58008)]. The level variety is generically an open part of an Abelian variety. The main theme of this article is that the divisor at infinity is entirely specified by the Dynkin diagram. For example, the components are canonically in bijection with the vertices and the intersection of these components can be described in combinatoric terms. The case of 3-particle Toda lattices is given in great details. Dynkin diagram; Abelian variety; divisor at infinity; 3-particle Toda lattices Adler, M. and van Moerbeke, P., The Toda Lattice, Dynkin Diagrams, Singularities and Abelian Varieties, Invent. Math., 1991, vol. 103, no. 2, pp. 223--278. Abelian varieties and schemes, Singularities in algebraic geometry The Toda lattice, Dynkin diagrams, singularities and Abelian 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 authors work on the problem of determining which zero dimensional schemes deform to a set of distinct points, i.e. are smoothable. This is a fundamental problem in the theory of Hilbert schemes of points -- and active and exciting area of research in algebraic geometry and commutative algebra.
The authors define a syzygetic invariant which gives insight to the problem mentioned above. Using their invariant the authors are able to deduce several interesting examples including: families which are not smoothable; Hilbert schemes of points which have components intersecting away from the smoothable component; and information about the Hilbert scheme of nine points in five dimensional affine space.
The paper is expertly written and contains necessary background information on Hilbert schemes, inverse systems, and regularity for homogeneous ideals. Several enlightening examples are given including computations of their introduced \(\kappa\)-vector. Theorems establishing necessary, as well as both necessary and sufficient conditions for certain classes of schemes of regularity two to be smoothable are given. Artinian algebras; smoothability; syzygies; deformation theory; punctual Hilbert schemes; Hilbert schemes of points Erman D., Velasco M.: syzygetic approach to the smoothability of 0-schemes of regularity two. Adv. Math. 224(3), 1143--1166 (2010) Parametrization (Chow and Hilbert schemes), Syzygies, resolutions, complexes and commutative rings, Deformations and infinitesimal methods in commutative ring theory, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Commutative Artinian rings and modules, finite-dimensional algebras A syzygetic approach to the smoothability of zero-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 \(Q\) be a finite connected quiver. Consider the vector space \(\mathbb{A}^{Q_1}\) of its representations with dimension vector \((1,1,\dots,1)\). Each rational weight \(\theta\) on the quiver \(Q\) defines open subsets \((\mathbb{A}^{Q_1})^{SS}_{\theta}\) and \((\mathbb{A}^{Q_1})^S_{\theta}\) of \(\theta\)-semistable and \(\theta\)-stable representations. The isomorphism classes of representations are orbits of the action of a torus \(G\). By \textit{A. D.~King} [Q. J. Math., Oxf. II. Ser. 45, No. 180, 515--530 (1994; Zbl 0837.16005)], there exists a categorical quotient \((\mathbb{A}^{Q_1})^{SS}_{\theta} \to (\mathbb{A}^{Q_1})^{SS}_{\theta} /\!/ G.\) The quotient space \(\mathcal{M}_{\theta}(Q):= (\mathbb{A}^{Q_1})^{SS}_{\theta} /\!/ G\) is a semiprojective toric variety [cf. \textit{L.~Hille}, Toric quiver varieties. Algebras and Modules, II. Eighth international conference on representations of algebras, Geiranger, Norway, August 4--10, 1996. Providence, RI: American Mathematical Society. CMS Conf. Proc. 24, 311--325 (1998; Zbl 0937.14039)]. Moreover, if the quiver \(Q\) is acyclic, the variety \(\mathcal{M}_{\theta}(Q)\) is projective. By a result of A.D.~King, for a generic \(\theta\) the variety \(\mathcal{M}_{\theta}(Q)\) is the fine moduli space of \(\theta\)-stable representations of \(Q\).
The main result of the article under review states that every projective toric variety \(X\) may be realized as the fine moduli space for stable representations of an appropriate bound quiver i.e., quiver with relations. To this end with a list \(\mathcal{L}=\{\mathcal{O}_X, L_1,\dots,L_r\}\) of line bundles on \(X\) the author associate a quiver \(Q\), called the complete quiver of sections for \(\mathcal{L}\), the ideal of relations \(R\) in the path algebra \(\mathbb{K}Q\), and a smooth projective toric variety \(|\mathcal{L}|\) called the multilinear series of \(Q\). The variety \(|\mathcal{L}|\) can be defined combinatorially, by geometric invariant theory, or via representation theory. If the line bundle \(L_1,\dots,L_r\) are basepoint-free, then there is an induced morphism \(\varphi_{|\mathcal{L}|} : X \to |\mathcal{L}|\) defined on the level of the total coordinate rings. When \(r=1\), the morphism \(\varphi_{|\mathcal{L}|}\) coincides with the morphism from \(X\) to the linear series \(|L_1|\). The authors give conditions under which the morphism \(\varphi_{|\mathcal{L}|}\) is a closed embedding, and show that the image of \(\varphi_{|\mathcal{L}|}\) is presented as a geometric quotient. Moreover, it is proved that for every collection \(L_1,\dots,L_{r-2}\) of basepoint-free line bundles on \(X\) such that the subsemigroup of \(\text{Pic}(X)\) generated by \(L_1,\dots,L_{r-2}\) contains an ample line bundle, there exist line bundles \(L_{r-1}\) and \(L_r\) such that the list \((\mathcal{O}_X,L_1,\dots,L_r)\) defines the closed embedding \(\varphi_{|\mathcal{L}|}\) whose image is of the form \(\mathcal{M}_{\theta}(Q,R)\). toric variety; quiver; representation; geometric quotient; fine moduli space A. Craw, \(Quiver Representations in Toric Geometry\), 2008, available at: https://arxiv.org/pdf/0807.2191v1.pdf Toric varieties, Newton polyhedra, Okounkov bodies, Algebraic moduli problems, moduli of vector bundles, Geometric invariant theory, Representations of quivers and partially ordered sets Projective toric varieties as fine moduli spaces of 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 The articles of this volume will be reviewed individually. Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings of conferences of miscellaneous specific interest, Festschriften, Singularities in algebraic geometry, Hypersurfaces and algebraic geometry, Configurations and arrangements of linear subspaces, Relations with arrangements of hyperplanes, Knots and links (in high dimensions) [For the low-dimensional case, see 57M25] Topology of algebraic varieties and singularities. Invited papers of the conference in honor of Anatoly Libgober's 60th birthday, Jaca, Spain, June 22--26, 2009 | 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 show a partial answer to a question of \textit{C. Huneke} and \textit{G. J. Leuschke} [Proc. Am. Math. Soc. 131, No. 10, 3003--3007 (2003; Zbl 1021.13011)]: Let \(R\) be a standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay representation type, and assume that \(R\) has an isolated singularity. Is \(R\) then necessarily of graded finite Cohen-Macaulay representation type? In particular, this question has an affirmative answer for standard graded non-Gorenstein rings as well as for standard graded Gorenstein rings of minimal multiplicity. Along the way, we obtain a partial classification of graded Cohen-Macaulay rings of graded countable Cohen-Macaulay type. Linkage, complete intersections and determinantal ideals, Singularities in algebraic geometry Non-Gorenstein isolated singularities of graded countable Cohen-Macaulay type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0657.00005.]
This paper is a survey of progress on the problem of constructing minimal models for algebraic varieties of arbitrary dimension. The key conjecture is that every nonsingular projective algebraic variety X over an algebraically closed field of characteristic zero has either a minimal model (one with only terminal singularities and numerically effective canonical bundle) or a relative anticanonical model (a morphism \(h: X\to Z\) such that \(h_*{\mathcal O}_ X={\mathcal O}_ Z\) and such that the anticanonical bundle is relatively ample for h). The author describes two theorems, the contraction theorem and the cone theorem, and outlines two outstanding conjectures, the flip conjecture and the termination conjecture, which together would imply the model conjecture. The termination conjecture holds in dimensions \(\leq 4,\) and both conjectures hold for toric varieties.
As the author points out in a note added in proof, the model conjecture has now been proved in dimension 3 by \textit{S. Mori}. constructing minimal models; relative anticanonical model; contraction theorem; cone theorem; flip conjecture; termination conjecture; toric varieties Shokurov, V.V.: Numerical geometry of algebraic varieties. Proc. ICM 1986 (to appear) Minimal model program (Mori theory, extremal rays), Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves Numerical geometry of algebraic 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 explore the connections between three classes of theories: \(A_r\) quiver matrix models, \(d=2\) conformal \(A_r\) Toda field theories, and \(d = 4\; \mathcal{N} = 2\) supersymmetric conformal \(A_r\) quiver gauge theories. In particular, we analyze the quiver matrix models recently introduced by Dijkgraaf and Vafa (unpublished) and make detailed comparisons with the corresponding quantities in the Toda field theories and the \(\mathcal{N} = 2\) quiver gauge theories. We also make a speculative proposal for how the matrix models should be modified in order for them to reproduce the instanton partition functions in quiver gauge theories in five dimensions.{
\copyright 2010 American Institute of Physics} Schiappa, R.; Wyllard, N., An A\_{}\{r\} threesome: matrix models, 2D CFTs and 4d N\ =\ 2 gauge theories, J. Math. Phys., 51, 082304, (2010) Model quantum field theories, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Supersymmetric field theories in quantum mechanics, Yang-Mills and other gauge theories in quantum field theory, Representations of quivers and partially ordered sets, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) An \(A_r\) threesome: matrix models, \(2d\) conformal field theories, and \(4d\; \mathcal{N} = 2\) gauge theories | 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 \(O\) be a nilpotent orbit of a complex semisimple Lie algebra \(\mathfrak{g}\) and let \(\pi: X \to \bar{O}\) be the finite covering associated with the universal covering of \(O\). In [\textit{Y. Namikawa}, in: Handbook of moduli. Volume III. Somerville, MA: International Press; Beijing: Higher Education Press. 1--38 (2013; Zbl 1322.14013)], he has explicitly constructed a \(\mathbb{Q}\)-factorial terminalization \(\widetilde{X}\) of \(X\) when \(\mathfrak{g}\) is classical.
In this paper, the author counts different \(\mathbb{Q}\)-factorial terminalizations of \(X\). He also constructs the universal Poisson deformation of \(\widetilde{X}\) over \(H^2(\widetilde{X}, \mathbb{C})\) and looks at the action of the Weyl group \(W(X)\) on \(H^2(\widetilde{X}, \mathbb{C})\), leading toward an explicit geometric description of \(W(X)\). birational geometry; nilpotent orbit of a complex Lie algebra; Poisson deformation Global theory and resolution of singularities (algebro-geometric aspects), Coadjoint orbits; nilpotent varieties Birational geometry for the covering of a nilpotent orbit closure. 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 Let $C$ be an integral proper complex curve with compactified Jacobian $J$. Letting $C^{[n]}$ denote the Hilbert scheme of length $n$ subschemes of $C$, the Abel-Jacobi morphism $\varphi: C^{[n]} \to J$ sends a closed subscheme $Z$ to ${\mathcal I}_Z \otimes {\mathcal O}(x)^{\otimes n}$, where $x \in C$ is a nonsingular point. When $C$ has at worst planar singularities, both $C^{[n]}$ and $J$ are integral schemes with local complete intersection singularities according to \textit{A. B. Altman} et al. [in: Real and compl. Singul., Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, 1--12 (1977; Zbl 0415.14014)] and \textit{J. Briancon} et al. [Ann. Sci. Éc. Norm. Supér. (4) 14, 1--25 (1981; Zbl 0463.14001)]. Furthermore $\varphi$ has the structure of a $\mathbb P^{n-g}$-bundle for $n \geq 2g-1$ by work of \textit{A. B. Altman} and \textit{S. L. Kleiman} [Adv. Math. 35, 50--112 (1980; Zbl 0427.14015)] so that the rational homology group $H_* (C^{[n]})$ is determined by $H_* (J)$. Recent work of \textit{D. Maulik} and \textit{Z. Yun} [J. Reine Angew. Math. 694, 27--48 (2014; Zbl 1304.14036)] and \textit{L. Migliorini} and \textit{V. Shende} [J. Eur. Math. Soc. (JEMS) 15, No. 6, 2353--2367 (2013; Zbl 1303.14019)] endows $H^* (J)$ with a certain perverse filtration $P$ for which $H^* (C^{[n]})$ can be recovered from the $P$-graded space $\text{gr}_*^P H^* (J)$. \par Motivated by these results and a suggestion of Richard Thomas, the author shows how $H_* (C^{[n]})$ can be recovered from a filtration on $H_* (J)$ using a method not reliant on perverse sheaves. Taking an approach inspired by work of \textit{H. Nakajima} [Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)], he defines two pairs of creation and annihilation operators acting on $V(C)=\bigoplus_{n \geq 0} H_* (C^{[n]})$. The first pair $\mu_{\pm} [\text{pt}]$ corresponds to adding or removing a nonsingular point in $C$. The second pair $\mu_{\pm}[C]$ come from the respective projections $p,q$ from the flag Hilbert scheme $C^{[n,n+1]}$ to $C^{[n]}$ and $C^{[n+1]}$, namely $q_* p^{!}$ and $p_* q^{!}$ for appropriate Gysin maps $p^!$ and $q^!$. The main theorem states that the subalgebra of $\text{End} (V(C))$ generated by $\mu_{\pm} [\text{pt}], \mu_{pm}[C]$ is isomorphic to the Weyl algebra $\mathbb Q [x_1, x_2, \partial_1, \partial_2]$ and that the natural map $W \otimes \mathbb Q [\mu_+ [\text{pt}], \mu_+ [C]] \to V(C)$ is an isomorphism, where $W$ is the intersection of the kernels of $\mu_- [\text{pt}]$ and $\mu_- [C]$; moreover the Abel-Jacobi pushforward map $\varphi_*: V(C) \to H_* (J)$ induces an isomorphism $W \cong H_* (J)$. Dual variations for cohomology groups recover and strengthen the results of Maulik-Yun [Zbl 1304.14036] and Migliorini-Shende [Zbl 1303.14019]. locally planar curves; Hilbert scheme; compactified Jacobian; Weyl algebra Parametrization (Chow and Hilbert schemes), Jacobians, Prym varieties, Singularities of curves, local rings Homology of Hilbert schemes of points on a locally planar curve | 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 Nagoya Math. J. 100, 97-126 (1985; Zbl 0561.14008) and 104, 85-115 (1986; Zbl 0592.14014) the reviewer stated the following theorem:
``A morphism of noetherian (commutative) rings is regular if and only if it is a filtered inductive limit of standard smooth morphisms'' (that is a special type of finite type smooth morphisms). -- This is a positive answer to a question of \textit{M. Raynaud} [Colloque d'algèbre commutative, Rennes 1972, Publ. Sem. Math. Univ. Rennes, No. 13 (1972; Zbl 0252.13010)] which has many consequences: The Bass-Quillen conjecture holds in the equicharacteristic case [cf. the reviewer, Nagoya Math. J. 113, 121-128 (1989; Zbl 0663.13006)], excellent henselian local rings have the property of Artin approximation -- a positive answer to a conjecture of \textit{M. Artin} [Actes Congr. Intern. Math. 1970, No. 1, 419-423 (1971; Zbl 0232.14003)]\(\dots\) Another form of this theorem stated by \textit{M. Cipu} and the reviewer [Ann. Univ. Ferrara, Nuova Ser., Sez. VII, 30, 63-76 (1984; Zbl 0581.14006)] says that if \(A \to A'\) is a regular morphism of noetherian rings, then every \(A\)-algebra of finite type \(B\), every \(A\)-morphism from \(B\) to \(A'\) can be factorized through a \(B\)-algebra of finite type \(B'\), which is standard smooth over \(A\) and which is ``as smooth as possible'' over \(B\) (that is except the singular locus of \(B\) over \(A)\). This new form relies completely on the quoted theorem and it is a positive answer to a conjecture of \textit{M. Artin} [see Contemp. Math. 13, 223-227 (1982; Zbl 0528.13021)].
Unfortunately, lemma (9.5) of the reviewer's first cited paper (in Nagoya Math. J. 100) does not hold in the case of condition \(\text{(iii}_ 2)\) as the author notices and repairs in the present paper. Mostly the paper rewrites in a nice way the original proof giving good names to concepts like containerizer, standardizer. The reviewer enjoys this new presentation.
Also the present paper introduces a new concept -- the so-called ``residual smoothing''. This is a difficult notion which hides many technical details and makes the reading hard. The reviewer has to confess that perhaps it was more difficult for him to understand this notion than for the author to understand and to repair the original proof. After reading a preliminary version of this paper, the reviewer gave a much easier version [Nagoya Math. J. 118, 45-53 (1990; Zbl 0685.14009)] of his original proof (loc. cit.). Meanwhile \textit{M. André} [``Cinq exposés sur la desingularization'' (Preprint 1991)] has also given an interesting version. Also a preprint of \textit{M. Spivakovsky} should be mentioned [``Smoothing of ring homomorphisms, approximation theorems and the Bass- Quillen conjecture'' (Preprint 1992)], following a different approach. Néron desingularization; smoothification; Bass-Quillen conjecture; Artin approximation; residual smoothing Ogoma, T., General Néron desingularization based on the idea of Popescu, J. Algebra, 167, 57-84, (1994) Étale and flat extensions; Henselization; Artin approximation, Regular local rings, Global theory and resolution of singularities (algebro-geometric aspects), Excellent rings General Néron desingularization based on the idea of Popescu | 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 complex quasi-projective variety and \(\text{Hilb}^n (X)\) its Hilbert scheme of zero dimensional subschemes of length \(n\). The author expresses the virtual Hodge polynomials of \(\text{Hilb}^n (X)\) -- defined by cohomology with compact support -- in terms of those of \(X\) and the Hilbert scheme of subschemes of length \(n\) supported at a point of \(X\). -- The proof proceeds by comparison with the \(n\)-fold symmetric product of \(X\) and related spaces and uses a lemma on point Hilbert schemes from \(L\). Göttsche's 1991 Bonn thesis [see \textit{L. Göttsche}, ``Hilbertschemata nulldimensionaler Unterschemata glatter Varietäten'', Bonner Math. Schr. 243 (1991; Zbl 0846.14002)]. The key properties of the virtual Hodge polynomial used in the proof are its additivity over stratifications and multiplicativity for fibrations. The results extend those found for the Poincaré and Hodge polynomials of surfaces in a paper by \textit{L. Göttsche} [Math. Ann. 286, No. 1-3, 193-207 (1990; Zbl 0679.14007)]. Hilbert scheme; virtual Hodge polynomials; symmetric product J. Cheah, ''On the Cohomology of Hilbert Schemes of Points,'' J. Algebr. Geom. 5, 479--511 (1996). Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the cohomology of Hilbert schemes 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 Let \(Q\) be a quiver with no oriented cycles. For dimension vectors \(\alpha\) and \(\beta\), define \(N(\beta,\alpha)\) as the number of \(\beta\)-dimensional subrepresentations of a general \(\alpha\)-dimensional representation of \(Q\). If \(\langle\beta,\alpha-\beta\rangle=0\) (here \(\langle\cdot,\cdot\rangle\) denotes the Ringel bilinear form), then \(N(\beta,\alpha)\) is finite. Denote by \(M(\beta,\alpha)\) the dimension of the space of semi-invariant polynomial functions with weight \(\langle\beta,\cdot\rangle\) on the space of \((\alpha-\beta)\)-dimensional representations of \(Q\) (note that any non-zero semi-invariant on \(\text{Rep}(Q,\alpha-\beta)\) has weight of such special form).
The main result of this paper is that \(M(\beta,\alpha)=N(\beta,\alpha)\) when \(\langle\beta,\alpha-\beta\rangle=0\). The proof is that the number \(M(\beta,\alpha)\) can be expressed via a Littlewood-Richardson calculation, which is then compared by the authors with the expression of \(N(\beta,\alpha)\) given by \textit{W. Crawley-Boevey} [Bull. Lond. Math. Soc. 28, No. 4, 363-366 (1996; Zbl 0863.16008)] using intersection theory.
Applying results of \textit{A. Schofield} [J. Lond. Math. Soc., II. Ser. 43, No. 3, 383-395 (1991; Zbl 0779.16005)], a basis of the corresponding space of semi-invariants is obtained. The result is generalized to covariants as follows: the cohomology class of the variety of \(\beta\)-dimensional subrepresentations of an \(\alpha\)-dimensional representation in general position can be expressed in terms of multiplicities in isotypic components of the coordinate ring of \(\text{Rep}(Q,\beta-\alpha)\). semi-invariants of quivers; Schubert classes; Littlewood-Richardson coefficients; representations of quivers; Ringel bilinear forms Derksen, H., Schofield, A., Weyman, J.: On the number of subrepresentations of a general quiver representation. J. Lond. Math. Soc. (2) 76(1), 135--147 (2007) Representations of quivers and partially ordered sets, Actions of groups on commutative rings; invariant theory, Grassmannians, Schubert varieties, flag manifolds On the number of subrepresentations of a general quiver representation. | 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 curve X in \({\mathbb{P}}^ 3\) is said to be \textit{obstructed} if the corresponding point of the Hilbert scheme is singular. A long-standing problem has been to give a geometrical characterization of non- obstructedness. Let \({\mathcal I}_ X\) be the ideal sheaf of X in \({\mathbb{P}}^ 3\). A curve is said to be of \textit{maximal rank} if for each n, at most one of \(h^ 0({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\) and \(h^ 1({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\) is non-zero. A curve is said to have \textit{natural cohomology} if at most one of \(h^ 0({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\), \(h^ 1({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\) and \(h^ 2({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\) is non-zero for each n. Prior to this paper, several examples were known of smooth obstructed space curves, but it was an open question (due to E. Sernesi) whether there exists one which furthermore has maximal rank.
For much of the paper X is assumed to have the property that \(h^ 1({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\) is nonzero for exactly one n. The first set of results go to show that the ``generic'' such X (in a precise sense) is non-obstructed. The second part of the paper gives some useful techniques for constructing obstructed curves, using liaison. The authors apply these techniques in the third part to produce a concrete example of a smooth, obstructed curve with maximal rank.
The authors point out that their examples of a smooth obstructed maximal rank curve has also been constructed independently (and with different techniques) by \textit{C. Walter} (``Some examples of obstructed curves in \({\mathbb{P}}^ 3\)'', Proc., Trieste 1989). They also point out that their example does not have natural cohomology, and indeed Walter asked whether all curves with this property may be non-obstructed. However, recently \textit{M. Martin-Deschamps} and \textit{D. Perrin} have produced a curve (but not a smooth one) which has natural cohomology but is obstructed (``Courbes gauches et modules de Rao''). linkage; Hilbert scheme; maximal rank; obstructed curves; liaison; natural cohomology Bolondi, G.; Kleppe, J. O.; Miro-Roig, R. M.: Maximal rank curves and singular points of the Hilbert scheme. Compos. math. 77, 269-291 (1991) Plane and space curves, Linkage, Parametrization (Chow and Hilbert schemes) Maximal rank curves and singular points of the 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 We consider a mixed function of type \(H(\mathbf{z},\overline{\mathbf{z}})=f(\mathbf{z})\,\overline{g}(\mathbf{z})\) where \(f\) and \(g\) are convenient holomorphic functions which have isolated critical points at the origin, and assume that the intersection \(f=g=0\) is a complete intersection variety with an isolated singularity at the origin and \(H\) satisfies the multiplicity condition. We show that \(H\) has a tubular Milnor fibration at the origin. We also prove that \(H\) has a spherical Milnor fibration, provided the Newton non-degeneracy of the intersection variety \(f=g=0\) and Newton multiplicity condition hold. Finally, we give examples of \(f, g\) such that \(H\) does not satisfy the Newton multiplicity condition and it has or has not Milnor fibration. mixed function; multiplicity condition Milnor fibration; relations with knot theory, Fibrations, degenerations in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry On the Milnor fibration for \(f(\mathbf{z})\,\overline{g}(\mathbf{z})\) | 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 observe that linear relations among Chern-Mather classes of projective varieties are preserved by projective duality. We deduce the existence of an explicit involution on a part of the Chow group of projective space, encoding the effect of duality on Chern-Mather classes. Applications include Plücker formulae, constraints on self-dual varieties, generalizations to singular varieties of classical formulas for the degree of the dual and the dual defect, formulas for the Euclidean distance degree, and computations of Chern-Mather classes and local Euler obstructions for cones. Projective techniques in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Singularities in algebraic geometry Projective duality and a Chern-Mather involution | 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 constructs virtual classes and virtual motives on the Quot schemes of a locally free sheaf on certain smooth quasi-projective 3-folds and solves their enumerative theories, obtaining new examples of higher rank Donaldson-Thomas invariants.
Let \(F\) be a locally free sheaf on a smooth projective 3-fold \(Y\) and denote by \(\mathrm{Quot}_Y(F,n)\) the Quot scheme of \(0\)-dimensional quotients of \(F\). Exploiting the deformation theory of the kernels of the surjections (rather than looking at deformations of the quotients), the authors constructs a perfect obstruction theory in the following two cases:
1. \(H^i(Y, \mathcal{O}_Y)=0\) for \(i> 0\) and \(F\) is an exceptional vector bundle;
2. \(Y\) is Calabi-Yau and \(F\) is a simple rigid vector bundle.
A vector bundle \(F\) is said to be simple if \(\mathrm{Hom}(F,F)=\mathbb{C} \), rigid if \(\mathrm{Ext}^1(F,F)=0\) and exceptional if it is simple and \(\mathrm{Ext}^i(F,F)=0\) for any \(i> 0\). Remarkably, this perfect obstruction theory is point-wise symmetric in both cases, but is globally symmetric only if \(Y\) is Calabi-Yau.
By the machinery of Behrend-Fantechi, the perfect obstruction theory defines a virtual fundamental class \([\mathrm{Quot}_Y(F,n)]^{vir}\in A_0(\mathrm{Quot}_Y(F,n))\). For a fixed pair \((Y,F)\) satisfying the hypotheses needed to construct the virtual cycle, one can consider the generating series \(\mathrm{DT}_F(q)\) counting the degree of the virtual cycle for any \(n\geq 0\). The author proposes the conjectural closed formula
\[
\mathrm{DT}_F(q)=\mathrm{M}((-1)^r q)^{r\int_Y c_3(T_Y\otimes \omega_Y )}
\]
where \(\mathrm{M}\) is the Macmahon function and \(r\) is the rank of \(F\). Surprisingly, these invariants seem to depend only on \(Y\) and the rank of the vector bundle. This formula has been proven in the rank 1 case for any 3-fold by Levine-Pandharipande and Jun Li. In the higher rank case, the author proves the formula in the Calabi-Yau case, by using Behrend's constructible function. In this case, the formula reduces to
\[
\mathrm{DT}_F(q)=\mathrm{M}((-1)^r q)^{r\chi(Y)}
\]
where \(\chi(Y)\) is the topological Euler characteristic of \(Y\).
In the second part of the paper, all the conditions on \((Y,F)\) are dropped and it is constructed a virtual motive \([\mathrm{Quot}_Y(F,n)]_{vir}\in \mathcal{M}_{\mathbb{C}}\) for any locally free sheaf \(F\) on a smooth quasi-projective 3-fold \(Y\), where \(\mathcal{M}_{\mathbb{C}}\) is obtained by formally inverting a square root of the Lefschetz motive \(\mathbb{L}=[\mathbb{A}^1_{\mathbb{C}}]\) in the Grothendieck group of complex varieties. For a fixed pair \((Y,F)\), denote by \(Z_r(Y, t)\) the generating series of the virtual motives. The author proves the formula
\[
Z_r(Y, (-1)^r t)=\mathrm{Exp}\left(\frac{(-1)t^r [Y\times \mathbb{P}^{r-1}]}{(1-(-\mathbb{L}^{-1/2})^rt)(1-(-\mathbb{L}^{1/2})^rt)} \right),
\]
where \(\mathrm{Exp}\) is the motivic exponential, generalizing the computation of Behrend-Bryan-Szendröi in the rank 1 case. virtual classes; (motivic) Donaldson-Thomas invariants; Quot schemes Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Virtual classes and virtual motives of Quot schemes on 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 0667.00008.]
The authors study the deformations of space curves which lie on rational ruled surfaces. Let \(W(m,d,g,n)\) be the subscheme of the Hilbert scheme of projective n-space corresponding to those smooth degree \(d,\) genus \(g\) curves which are m-secant curves lying on a two dimensional rational scroll. Using the properties of ruled surfaces, the authors are able to give an upper bound on the Zariski tangent space at a general point of \(W(m,d,g,n)\). From that they are able to conclude that the closure of \(W(m,d,g,n)\) is a component of the Hilbert scheme, if d, g, n and m satisfy certain inequalities. This gives a generalization of an earlier example of \textit{J. Harris} which shows that \(H(d,g,n)\), the Hilbert scheme of smooth curves of degree \(d\) and genus \(g\) in \({\mathbb{P}}^ n (n\geq 6)\), is reducible even if \(d\geq g+n.\)
The authors also formulate many different conjectures concerning about the irreducibility of \(H(d,g,n)\). For instance they conjecture that if the Brill-Noether number \(\rho(d,g,n)\) is non-negative then the open set of \(H(d,g,n)\) corresponding to linearly normal curves is an irreducible variety. gonality; deformations of space curves; Hilbert scheme; ruled surfaces; Zariski tangent space; Brill-Noether number E. Mezzetti and G. Sacchiero, Gonality and Hilbert schemes of smooth curves, in Algebraic Curves and Projective Geometry, (Trento, 1988), 183--194, Lecture Notes in Math. 1389, Springer, Berlin, 1989. Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) Gonality and Hilbert schemes of smooth 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 It is shown that the minimal free resolution of the homogeneous ideal of \(s\) points in \(\mathbb{P}^4\) has the expected form given by the minimal resolution conjecture of \textit{A. Lorenzini} [J. Algebra 156, No. 1, 5-35 (1993; Zbl 0811.13008)] for almost all \(s\) using a version of ``Horace's method''. points in \(\mathbb{P}^ 4\); minimal free resolution Walter, C., The minimal free resolution of the homogeneous ideal of \textit{s} general points in \(\mathbb{P}^4\), Math. Z., 219, 2, 231-234, (1995) Projective techniques in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The minimal free resolution of the homogeneous ideal of \(s\) general points in \(\mathbb{P}^ 4\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Since A.~Grothendieck discovered that the foundation of algebraic geometry has to be the language of schemes it became rather hard to non-specialists to follow basic ideas of the subject as well as to get an feeling for recent research in algebraic geometry. This is in particular true for mathematicians working on complex analysis, which had and has great influence of algebraic geometry. The book under review is intended for the working mathematician who is unfamiliar with algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites. The book is not conceived as a subsitute for an introduction to algebraic geometry as given by \textit{R.~Hartshorne} [``Algebraic geometry'', Graduate Texts in Math. 52 (1977; Zbl 0367.14001)], or by \textit{I. R. Shafarevich} [``Basic algebraic geometry'', Grundlehren 213 (1974); translation from the Russian (1972; Zbl 0258.14001)], for example. It is the authors' goal to inspire the readers to undertake a more serious study of the subject, i.e. to invite them to algebraic geometry.
The book grew out of a course of the first author at the University of Jyväskylä (Finland) for Ph. D. students and interested mathematicians whose research is quite far removed from algebra. The text is divided into eight chapters (affine algebraic varieties, algebraic foundations, projective varieties, quasi-projective varieties, classical constructions, smoothness, birational geometry, and maps to projective space) with an appendix (sheaves and abstract algebraic varieties). Starting with the affine algebraic varieties as the zero loci of sets of polynomials the basic commutative algebra is introduced and projective and quasi-projective varieties are sketched. All the material is nicely presented. Several constructions are illustrated by instructive pictures. The chapter on classical constructions covers Veronese maps, Segre products, Grassmannians, and the Hilbert polynomial. The chapter on smoothness sketches the Jacobian criterion, smoothness in families and the Bertini theorem. The chapter `Birational geometry' explains the problem of resolution of singularities, blowing up along a subvariety, birational equivalence, and the classification problem. Most of the chapters provide some information about current research. The final chapter introduces line bundles, rational maps, and very ample line bundles from a very concrete point of view. The appendix contains a short introduction to some basic notions of modern algebraic geometry.
Because of the nature of this book as an invitation it was necessary to omit several proofs and sacrifice some rigor. For all details there are references and explanations. Only a few prerequisites are presumed beyond a basic course in linear algebra. So the lectures are also intended to an interested undergraduate student. The reviewer believes that the authors convey in their invitation that the main objects in algebraic geometry and the main research questions about them are as interesting and accessible as ever, even for mathematicians far from the subject. affine variety; projective variety Smith, K.-E.; Kahanpää, L.; Kekäläinen, P.; Traves, W., An invitation to algebraic geometry, (2000), Springer-Verlag New York Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Foundations of algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Affine geometry, Real algebraic and real-analytic geometry An invitation to algebraic 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 \(\mathbb F[x]\) be the space of polynomials in \(d\) variables, let \({\mathfrak G}^N\) be the Grassmannian of \(N\)-dimensional subspaces of \(\mathbb F[x]\) and let \({\mathfrak J}_N\) stand for the family of all ideals in \(\mathbb F[x]\) of codimension \(N\). For a given \(G\in{\mathfrak G}^N\), we let
\[
{\mathfrak J}_G:= \{J\in{\mathfrak J}_N: J\cap G=\{0\}\}.
\]
Is it true, that (with appropriate topology on \({\mathfrak J}_N\)) the set \({\mathfrak J}_G\) is dense in \({\mathfrak J}_N\)? In general the answer is ``no''. What is even more surprising, is that there are ``good ideals'' \(J\in {\mathfrak J}_N\) such that every ``neighborhood'' \({\mathcal U}(J)\subset{\mathfrak J}_N\) has a non-empty intersection with \({\mathfrak J}_G\) for any \(G\in{\mathfrak G}^N\) and there are ``bad'' ideals \(J\in{\mathfrak J}_N\) (for \(d\geq 3\)) such that some ``neighborhoods'' \({\mathcal U}(J)\subset{\mathfrak J}_N\) have an empty intersection with \({\mathfrak J}_G\) for some \(G\in{\mathfrak G}^N\). This contrast illuminates the non-homogeneous nature of \({\mathfrak J}_N\). ideal complements; ideal projections Ideals, maximal ideals, boundaries, Parametrization (Chow and Hilbert schemes), Ideals and subalgebras, General theory of commutative topological algebras On perturbations of ideal complements | 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 describes how a resolution of a given singularity can be computed explicitly, and also describes how this computation can be used for computing an invariant which arises as a geometric analogue to some arithmetic invariant. By the algorithm, the resolution is obtained as a collection of affine charts, each represented by a polynomial (and some additional information that take care of the necessary bookkeeping and make sure that the charts glue together). The paper shows nicely how these local data are used to obtain the global invariant. resolution; singularities; desingularization; algorithm Frühbis-Krüger, Anne: An application of resolution of singularities: computing the topological \({\zeta}\)-function of isolated surface singularities in (C3,0), Singularity theory (2007) Computational aspects in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) An application of resolution of singularities: computing the topological \(\zeta\)-function of isolated surface singularities in \((\mathbb C^3,0)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((X,0)\subseteq (\mathbb C^4,0)\) be the germ of an isolated hypersurface singularity defined by \(f=0\) and \(\rho(X)\) be the rank of the analytic divisor class group \(\text{Cl}(\mathcal O_{X,0}^{an})\). In case that \((X,0)\) is a canonical singularity and \(f\) is non-degenerate with respect to its Newton boundary, \(\rho(X)\) is computed in terms of a suitable set of monomials whose residue classes form a basis for the Milnor algebra \(\mathbb C [x_1, \ldots, x_4]/\left(\frac{\partial f}{\partial x_1}, \cdots \frac{\partial f}{\partial x_4}\right)\). canonical singularity; Newton polyhedron Caibăr, M., On the divisor class group of 3-fold singularities, Int. J. Math., 14, 105, (2003) Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties On the divisor class group of 3-fold 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 study properties of the algebras of
planar quasi-invariants. These algebras are Cohen-Macaulay and
Gorenstein in codimension one. Using the technique of matrix
problems, the authors classify all Cohen-Macaulay modules of rank
one over them and determine their Picard groups. In terms of this
classification, they describe the spectral modules of the planar
rational Calogero-Moser systems. Finally, the authors elaborate
the theory of the algebraic inverse scattering method, computing a
new unexpected explicit example of a deformed Calogero-Moser
system. This paper is organized as follows. Section 1, is an
introduction to the subject and a description of the results
obtained. Section 2, deals with ring-theoretic properties of the
algebra of surface quasi-invariant. Section 3, deals with rank one
Cohen-Macaulay modules over the algebra of planar
quasi-invariants. In this section, the authors classify all
Cohen-Macaualy \(A\)-modules of rank one, specifying those of them,
which are locally free in codimension one. Next, they give an
explicit description of a canonical module of \(A\) (algebra of
planar quasi-invariant) and describe the Picard group \(Pic(A)\)
viewed as a subgroup of the group \(\mathbf{CM}^{lf}_1(A)\) (the set
of the isomorphism classes of Cohen-Macaualy \(A\)-modules of rank
one, which are locally free in codimension one). Section 4, deals
with spectral module of a rational Calogero-Moser system of
dihedral type. In this section, the authors discuss a link between
results on Cohen-Macaulay modules over an algebra planar
quasi-invariants with the theory Calogero-Moser systems. Section
5, discusses the elements of the higher-dimensional Sato theory.
Section 6, concerns the algebraic inverse scattering method in
dimension two. In this section, the authors discuss some examples
of the theory developed in the previous section. The paper is
supported with an appendix concerning the compactified Picard
variety of an affine cuspidal curve. Cohen-Macaulay modules; algebra of planar quasi-invariants; Calogero-Moser systems Singularities in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Pseudodifferential operators and other generalizations of partial differential operators, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Cohen-Macaulay modules over the algebra of planar quasi-invariants and Calogero-Moser systems | 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 According to \textit{P. Deligne} [Lect. Notes Math. 340, 165--196 (1973; Zbl 0266.14010)], the theory of nearby and vanishing cycles is an important tool to study of hypersurface singularities. Nearby and vanishing cycles are derived functors from the derived category of constructible sheaves on \(X\) (or \(X_{\text{ét}}\)) to the derived category of constructible sheaves on the fiber \(X_\theta\). These come equipped with an action of the (étale) fundamental group. For a complex manifold \(X\), the famous Riemann-Hilbert correspondence states that there is a functor from the bounded derived category of regular holonomic \(D_X\)-modules to the bounded derived category of constructible sheaves inducing an equivalence of category which respects the ``six operations''.
By work of \textit{B. Malgrange} [Astérisque 101--102, 243--267 (1983; Zbl 0528.32007)] and \textit{M. Kashiwara} [Lect. Notes Math. 1016, 134--142 (1983; Zbl 0566.32022)] one has a direct construction of nearby and vanishing cycles for regular holonomic \(D_{X}\)-modules without passing through the Riemann-Hilbert correspondence. The key ingredient in this construction is the so-called \(V\)-filtration of a (regular holonomic) \(D_{X}\)-module whose associated graded pieces recover nearby and vanishing cycles. The construction of the \(V\)-filtration itself uses so-called Bernstein-Sato polynomials.
The author of the paper under review is concerned with a partial characteristic \(p\) analog of \(V\)-filtrations. The characteristic \(p\)-version of multiplier ideals are so-called test ideals. In work of \textit{M. Blickle} [J. Algebr. Geom. 22, No. 1, 49--83 (2013; Zbl 1271.13009)] this point of view is further emphasized and the construction of test ideals is extended to modules. The author constructs a V -filtration for Cartier modules and show that this coincides with the test module filtration under some conditions. In particular, this means that the test ideal filtration (say in a polynomial ring) is uniquely determined by discreteness, rationality, the Brian-Skoda property and how the Cartier operator acts on the filtration.
The author uses work of \textit{M. Blickle} and \textit{G. Böckle} [J. Reine Angew. Math. 661, 85--123 (2011; Zbl 1239.13007); ``Cartier crystals'', Preprint, \url{arXiv:1309.1035}] and of \textit{M. Emerton} and \textit{M. Kisin} [The Riemann-Hilbert correspondence for unit \(F\)-crystals. Paris: Société Mathématique de France (2004; Zbl 1056.14025)] to drive an analog of a Riemann-Hilbert correspondence. The paper is good for all researchers in the field of the subject. nearby and vanishing cycles; Riemann-Hilbert correspondence; V-filtration; Cartier modules; test ideals Stäbler, A., \textit{V}-filtrations in positive characteristic and test modules, Trans. Am. Math. Soc., 368, 11, 7777-7808, (2016) Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry \(V\)-filtrations in positive characteristic and test 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 The Abuaf-Ueda flop is a 7-dimensional flop related to \(G_2\) homogeneous spaces. The derived equivalence for this flop was first proved by \textit{K. Ueda} [Manuscr. Math. 159, No. 3--4, 549--559 (2019; Zbl 1440.14096)] using mutations of semi-orthogonal decompositions. In this article, we give an alternative proof for the derived equivalence using tilting bundles. Our proof also shows the existence of a non-commutative crepant resolution of the singularity appearing in the flopping contraction. We also give some results on moduli spaces of finite-length modules over this non-commutative crepant resolution. derived category; non-commutative crepant resolution; flop; tilting bundle Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Noncommutative algebraic geometry On the Abuaf-Ueda flop via non-commutative crepant resolutions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X_F\) be hypersurface in \({\mathbb{P}}^n\) defined by a homogeneous polynomial \(F\) of degree \(d\), and \(G\) the set of all lines in \({\mathbb{P}}^n\). Let \(1\leq m\leq d+1\) and \(Y_{F, m}\) be the set of \((p, L)\in {\mathbb{P}}^n\times G\), where \(L\) and \(X_F\) intersect at \(p\) with the multiplicity \(\geq m\). The author gives the defining equations of the projective variety \(Y_{F, m}\) and states that for a general hypersurface \(X_F\), \(Y_{F, m}\) is smooth of dimension \(2n-m-1\). By the Jacobian ring method developed by Griffiths, the author studies the injectivity of the infinitesimal period map for \(Y_{F, m}\) and is able to describe the Hodge cohomologies of \(Y_{F, m}\). The detailed proofs will appear somewhere. Parametrization (Chow and Hilbert schemes), Transcendental methods, Hodge theory (algebro-geometric aspects) The varieties of intersections of lines and hypersurfaces in projective 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 The present book, published in two volumes, is the English translation of the second revised and expanded edition of the author's famous introductory textbook on algebraic geometry. The Russian edition appeared in 1988 (Zbl 0675.14001), and rewardingly has been translated (and slightly commented on by \textit{M. Reid}. The translator has attempted -- with the author's permission -- to put the text into the language used by the present generation of English-speaking algebraic geometers and, moreover, added some footnotes concerning terminology or further references. In this English translation, the two volumes have now a common index and list of references, in contrast to the Russian original edition.
As for the present revised and expanded edition of the author's textbook itself, a few remarks might be appropriate. The very first edition (in Russian) appeared in 1972 (Zbl 0258.14001). At that time, this textbook was the first and only one which built bridges between the geometric intuition, the classical origins and achievements of algebraic geometry, the modern concepts and methods, and the complex-analytic aspects in algebraic geometry. The English translation of this unique textbook was published in 1974 under the title ``Basic algebraic geometry'' (Zbl 0284.14001). In the meantime, it has become one of the most valuable, recommended and used textbooks on algebraic geometry, together with the subsequent standard texts by \textit{R. Hartshorne}, \textit{D. Mumford}, \textit{Ph. A. Griffiths} and \textit{J. Harris}, and others. The special feature of the author's book, in comparison to the others, has always been provided by the fact that it really conveys the many different aspects of modern algebraic geometry, without particularly focusing on any special approach, and without assuming any advanced prerequisites such as commutative algebra, differential geometry, functions of several complex variables, etc. In this sense, it has proved an extremely useful addition to the other (here and there) more thorough-going textbooks, in particular for beginners, and -- simultaneously -- a highly recommendable introduction to them and to the current research literature. Besides, and in any case, the author's book is still a lovely and fascinating invitation to algebraic geometry.
Now, in the second edition, he maintains his ground-rules and the tried arrangement of the original text. That means, he has left the aims, the character, and the chapters basically intact. However, taking into account the rapid development and the various interconnections of algebraic geometry during the past two decades, he has added -- in an organic manner -- some important topics of current interest as well as some further motivating and instructive examples.
The first volume of the new edition corresponds to chapters I--IV of the first edition. The material is enhanced by the additional treatment of various examples of concrete algebraic varieties such as plane cubic curves, cubic surfaces, grassmannians and determinantal varieties. The study of singularities of algebraic varieties and maps is remarkably deepened and applied to the current topic of degenerations in algebraic families of varieties, in particular to degenerations of quadrics and elliptic curves. Also, the Bertini theorems are now included, as is a discussion of normal singularities of algebraic surfaces. Furthermore, some arithmetic aspects are worked into the text, for example: the zeta function for algebraic varieties over a finite ground field, a version of the Riemann conjecture for elliptic curves, and other applications.
Finally, in order to keep the text as self-contained as possible, the author has added an appendix entitled ``Algebraic supplements'', in which he compiles the basic algebraic facts utilized in the text.
The many instructive exercises (of various degrees of difficulty) and the updated bibliography have been adjusted to the reworked material, some inaccuracies in the original text have been removed, and several proofs of theorems have been ameliorated. Thus the first part of the author's well-tested textbook has undergone an evident enrichment in divers regards. The author has managed, with his inimitable masterly skill, to organically insert more concrete, advanced and topical material, to lucidly present the interrelations, the complexity, and the vividness of algebraic geometry in a comprehensible way, and to make his already outstanding textbook even more useful for both learning and teaching.
[See also the following review]. affine algebra varieties; quasiprojective varieties; intersection theory; birational equivalence; algebraic groups; degenerations; Bertini theorems; singularities; zeta function I.~Shafarevich. \textit{Basic algebraic geometry~1: varieties in projective space}, 2nd edition, Springer, Berlin 1994. Foundations of algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities in algebraic geometry, Curves in algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Basic algebraic geometry. 1: Varieties in projective space. Transl. from the Russian by Miles Reid. | 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 \(\mathbb{X}\) be an affine toric variety, and let \(T \subset \mathbb{T}\) be a subtorus of its defining torus. Let \(H\) denote the toric Hilbert scheme of \(T\)-varieties in \(\mathbb{X}\) having the same Hilbert function as \(X=\overline{Tx}\), where \(x\) is a point in the open \(\mathbb{T}\)-orbit. There is an induced \(\mathbb{T}/T\)-action on \(H\), and \(H_0\) is defined as the closure of \(\mathbb{T}X\). To a toric variety there is an associated fan. The main result announced in the article is that the fan of \(H_0\) is the normal fan of the average over all integral fibres of the cone projection determined by \(T\). toric Hilbert scheme; fan Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) The fan of the main component 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 Let \(f_ 1,...,f_ k\) be polynomials in d complex variables, and let \(\Gamma^+_ 1,...,\Gamma^+_ k\) denote the corresponding Newton polyhedra. The author proves that the equations \(f_ 1=...=f_ k=0\) define an isolated complete intersection singularity at the origin if the coefficients of \(f_ 1,...,f_ k\) satisfy a suitable condition of non- degeneracy with respect to the system \((\Gamma^+_ 1,...,\Gamma^+_ k)\). The geometric genus of this singularity is calculated in terms of the elementary geometry of the system of polyhedra \((\Gamma^+_ 1,...,\Gamma^+_ k)\). The proofs use resolution of singularities via toroidal embeddings. Newton polyhedra; isolated complete intersection singularity; resolution of singularities; toroidal embeddings Morales, M., Polyèdre de Newton et genre géométrique d'une singularité intersection complète, Bull. Soc. Math. France, 112, 325-341, (1984) Local complex singularities, Singularities in algebraic geometry, Implicit function theorems; global Newton methods on manifolds Polyèdre de Newton et genre géométrique d'une singularité intersection complète | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study invariance of multiplicity of complex analytic germs and degree of complex affine sets under outer bi-Lipschitz transformations (outer bi-Lipschitz homeomorphims of germs in the first case and outer bi-Lipschitz homeomorphims at infinity in the second case). We prove that invariance of multiplicity in the local case is equivalent to invariance of degree in the global case. We prove invariance for curves and surfaces. In the way we prove invariance of the tangent cone and relative multiplicities at infinity under outer bi-Lipschitz homeomorphims at infinity, and that the abstract topology of a homogeneous surface germ determines its multiplicity. Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Singularities in algebraic geometry Multiplicity and degree as bi-Lipschitz invariants for complex sets | 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, the author continues the study of the essential dimension of an algebraic stack introduced in [\textit{P. Brosnan} et al., J. Eur. Math. Soc. (JEMS) 13, No. 4, 1079--1112 (2011; Zbl 1234.14003)], focusing on (the stack of) representations of a quiver with a fixed dimension vector. In particular, the author investigates the question of when the essential dimension and the generic essential dimension (the essential dimension of a generic object) agree. This is known to hold in the case of smooth algebraic stacks with reductive automorphism groups, but, as the author shows, also holds in many cases beyond this (and fails to hold in many cases).
The article gives an upper bound on the generic essential dimension for the representations of a quiver when the dimension vector is taken to be a root of the quiver. Conjecturally this upper bound is an equality. The author shows that for quivers of finite type, or those with at least one loop at every vertex, the essential and generic essential dimensions agree for every dimension vector, and moreover shows that this fails for at least one dimension vector for all other quivers. A detailed study is also made of the essential dimensions of the (generalised) Kronecker quivers. essential dimension; genericity property; quiver representations; algebraic stack Representations of quivers and partially ordered sets, Generalizations (algebraic spaces, stacks) Essential dimension and genericity for 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 Given an ideal I in a noetherian ring A, let Ass(I) denote the set of prime ideals associated to I, we show that \(Ass(I^ k)\) stabilizes for k sufficiently large in a set which we call \textit{Ass}(I). Let \(\pi: Z\to Spec(A)\) denote the blowing up at I and assume that \(A\cong \Gamma (O_ Z)\), then \textit{Ass}(I) is the image via \(\pi\) of those primes associated to the sheaf of ideals \(IO_ Z=\pi^{- 1}(I)O_ Z\). - Now, let A be a normal two-dimensional local ring, assume that \textit{Ass}(I) is known, then the only point, to determine whether it belongs or not to \textit{Ass}(I) is the closed point. If P is a height one prime ideal the condition \textit{Ass}(P)\(=\{P\}\) is equivalent to \(P^ k=P^{(k)}\) (symbolic power) for k sufficiently large, which in turn is equivalent to P being invertible. Given now P and Q, height one primes, the embedded components to \(P^ k\) and \(Q^ k\) for k large depend only on the class of P and Q in the divisor class group Cl(A). - \textit{J. Lipman} [Publ. Math., Inst. Haut. Étud. Sci. 36(1969), 195-279 (1970; Zbl 0181.489)] proves that a local normal two dimensional ring A has a rational singularity if and only if Cl(A) is finite.
We shall show that if A has a rational singularity and \(\pi: X\to Spec(A)\) denotes a resolution of singularities, then for each height one prime ideal P the sheaf of ideals \(\pi^{-1}(P)O_ X\) is invertible at X. Using this property we exhibit a reduction of the group Cl(A) by blowing up at height one prime ideals and also the existence of a unique minimal model \(\alpha: Z\to Spec(A)\) for \(\alpha\) proper and birational and Z a locally factorial scheme. This model is dominated by the minimal resolution. - \textit{E. Brieskorn} [Invent. Math. 4, 336-358 (1968; Zbl 0219.14003)] and \textit{J. Lipman} [loc. cit.] show, under certain conditions, that a local two dimensional ring is factorial (U.F.D.) if and only if it is isomorphic to the singularity of \(x^ 2+y^ 3+z^ 5=0\) at the origin.
One can ask whether a normal two dimensional local excellent ring admits a resolution of singularities \(\pi: X\to Spec(A)\) such that for each height one prime ideal P the sheaf of ideals \(\pi^{-1}(P)\cdot O_ X\) is invertible. In section \(9\) we exhibit an example of a non rational singularity which admits no resolution of singularities with that property. The question stands as to whether this property characterizes rational singularities. noetherian ring; height one prime ideal; resolution of singularities; rational singularities Ideals and multiplicative ideal theory in commutative rings, Singularities in algebraic geometry, Commutative Noetherian rings and modules, Local rings and semilocal rings On powers of prime ideals and rational singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f(z_ 0,\dots,z_ n)\) be a polynomial in complex variables, having an isolated singularity at the origin. If the zero set \(f^{-1}(0)\) intersects a small sphere \(S^{2n+1}_ \varepsilon\) in a manifold \(\Sigma\) homeomorphic to a sphere, then \(\Sigma\) is called a spherical algebraic knot. Write \(A_ n\) for the set of isomorphism classes of such knots \(\Sigma\). Replacing \(f\) by \(f+z^ 2_{n+1}+z^ 2_{n+2}\) induces an injective map \(A_ n\to A_{n+2}\). By adapting an example of Malgrange, the authors construct a knot in \(A_{n+1}\) whose monodromy has a Jordan block of size \(n\). Such a knot is not in the image of \(A_{n-3}\). spherical algebraic knot Singularities in algebraic geometry, Knots and links (in high dimensions) [For the low-dimensional case, see 57M25] Spherical algebraic knots increase with dimension | 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 constructs a subvariety of the multigraded Hilbert scheme for each dominant character of the special linear group over a field of positive characteristic. It is shown that this subvariety can be interpreted as lattices endowed with infinitesimal structure in the proper setting. The author relates this via a an equivariant universal homeomorphism to a Demazure resolution of the Schubert variety associated to the character in the affine Grassmannian.
The paper is well written and thorough, including necessary background on several topics, including the affine Grassmannian, Demazure varieties, and multigraded Hilbert schemes. Demazure varieties; multigraded Hilbert schemes; affine Grassmanian Kreidl, M, Demazure resolutions as varieties of lattices with infinitesimal structure, J. Algebra, 324, 541-564, (2010) Parametrization (Chow and Hilbert schemes) Demazure resolutions as varieties of lattices with infinitesimal structure | 0 |
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