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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 If \(X\) is a smooth quasiprojective scheme and \(G\) is a finite group acting faithfully on \(X\), the quotient space of orbits \(X/G\) is in general a singular scheme. A kind of more refined variant of a quotient of \(X\) by \(G\) is the \(G\)-Hilbert scheme of \((G,X)\) introduced in \textit{Y. Ito} and \textit{I. Nakamura} [Proc. Japan Acad., Ser. A 72, 135--138 (1996; Zbl 0881.14002)] by replacing the \(G\)-orbits with \(0\)-dimensional \(G\)-invariant subsets of \(X\). Again in general the \(G\)-Hilbert scheme is a singular variety but expectedly ``less'' singular than \(X/G\). The known cases of smooth \(G\)-Hilbert schemes appear as minimal resolutions of Klein singularities and crepant resolutions of the quotients of \({\mathbb{C}}^3\) by a finite subgroup of \(\text{SL}_3({\mathbb{C}})\) [cf. \textit{T. Bridgeland, A. King} and \textit{M. Reid}, J. Am. Math. Soc. 14, 535--554 (2001; Zbl 0966.14028)]. But the question for which finite subgroups \(G\) of \(\text{GL}_n({\mathbb{C}}), n \geq 4\) the \(G\)-Hilberts scheme of \(({\mathbb{C}}^n,G)\) is a crepant resolution of the quotient \({\mathbb{C}}^n/G\) still remains open. The only attempt was given by \textit{D. Dais, C. Haase} and \textit{G. Ziegler} [Tôhoku Math. J., II. Ser. 53, 95--107 (2001; Zbl 1050.14044)], and it is restricted to the 2-dimensional case. In the present paper a special example of an action of a finite subgroup of \(\text{GL}_4({\mathbb{C}})\) on \({\mathbb{C}}^4\)
is described, where the answer to the above question is positive. The group is the cyclic group \(\mu_{15}\) of order \(15\) with a generator \(\varepsilon = \text{exp}(2{\pi}i/15)\), which acts on \({\mathbb{C}}^4\) by weights \((1,2,4,8)\). The quotient \({\mathbb{C}}^4/{\mu}_{15}\) has a Gorenstein canonical singularity at the origin. The main result of the paper (theorem 2.9) affirms that the \(\mu_{15}\)-Hilbert scheme of \({\mathbb{C}}^4\) is smooth and gives a crepant resolution of the singularity of \({\mathbb{C}}^4/\mu_{15}\). quotient singularities; crepant resolutions; toric varieties Sebestean, M.: A smooth four-dimensional G-Hilbert scheme. Serdica math. J. 30, No. 2 -- 3, 283-292 (2004) Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Global theory and resolution of singularities (algebro-geometric aspects), \(4\)-folds A smooth four-dimensional \(G\)-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 This paper studies the geometry of chains of points in the so-called \textsl{Semple tower} \(\mathcal S(Z)\) over a nonsingular algebraic variety \(Z\) (over an algebraically closed field of characteristic zero); in particular it clarifies in dimension 2 the intention of \textit{J. G. Semple} [Proc. Lond. Math. Soc., III. Ser. 4, 24--49 (1954; Zbl 0055.14505)] to parametrize chains of infinitely near points on \(Z\).
Before describing the results we give an idea of the construction of \(\mathcal S(Z)\). Let \(d:= \dim Z\). One produces inductively pairs \((Z_n,F_n)\), consisting of a nonsingular variety \(Z_n\) and a (rank \(d\))-subbundle \(F_n\) of its tangent bundle \(T_{Z_n}\), starting with \((Z_0=Z,F_0=T_Z)\). Given \((Z_{n-1},F_{n-1})\), we define \(Z_n\) as the projectivized bundle \(\mathbb P F_{n-1}\), equipped with the projection map \(\pi_n:Z_n \to Z_{n-1}\). The tautological line bundle \(\mathcal O_{F_{n-1}}(-1)\) on \(Z_n=\mathbb P F_{n-1}\) is a subbundle of \(\pi_n^* F_{n-1}\), hence also of \(\pi_n^* T_{Z_{n-1}}\). Then \(F_n\) is defined as the pullback of \(\mathcal O_{F_{n-1}}(-1)\) by the tangent map \(d\pi_n: T_{Z_n} \to \pi_n^* T_{Z_{n-1}}\). One easily verifies that \(F_n\) then has the same rank as \(F_{n-1}\).
The author introduces a proximity relation \((z_m \in Z_m) \mapsto (z_n \in Z_n)\) for \(m>n\), parallel to the classical proximity relation among infinitely near points. For the following results if is important to note that, given a reduced curve \(C\subset Z\), the \(n\)th iterated Nash blowing-up \(C_n\) of \(C\) is naturally embedded in \(Z_n\). In arbitrary dimension the author proves a formula, parallel to a classical formula of Enriques, expressing the multiplicity of \(C_n\) at a point \(z_n\) as the (finite) sum of the multiplicities of the curves \((C_m)_{m>n}\) at its proximate points. When \(d=2\) it is shown that (1) the multiplicity sequence of the iterated Nash blowing-ups \(C_n\) of a plane branch \(C\) is equal to the multiplicity sequence of its iterated quadratic transforms, and (2) the iterated Nash blowing-ups of two branches get separated at the same level as their iterated quadratic transforms. The author also gives an example to indicate that these last results do not generalize to \(d>2\), using the branches in 3-space parametrized by \((t^8,t^{10}+t^{13},t^{12}+at^{15})\) for adequate coefficients \(a\). Semple tower; infinitely near points; Nash blowing-ups M. Lejeune-Jalabert, Chains of points in the Semple tower, American Journal of Mathematics 128 (2006), 1283--1311. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of curves, local rings, Plane and space curves, Modifications; resolution of singularities (complex-analytic aspects) Chains of points in the Semple tower | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 generalizes his paper [Rev. Mat. Iberoam. 25, No. 3, 995--1054 (2009; Zbl 1207.14009)] which gave an algorithmic resolution of singularities in characteristic zero. Here he develops a theory of simultaneous resolution completing his precedent results. The algorithm is an adaptation of the algorithm of \textit{S. Encinas} and \textit{O. Villamayor} [Prog. Math. 181, 147--227 (2000; Zbl 0969.14007)] and involves multi-ideals wich are essentially a pair consisting of a sheaf of ideals and a positive integer.
This algorithm is classical if the object are defined over a field but has a real interest if they are defined over an artinian ring. resolution of singularities; multi-ideals; equiresolution; artinian rings Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Computational aspects of higher-dimensional varieties, Commutative Artinian rings and modules, finite-dimensional algebras Resolution algorithms and deformations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 monograph presents a modern theory of resolution of isolated singularities of algebraic curves and surfaces over algebraically closed fields of characteristic zero. The authors develop appropriate algebraic techniques, the valuation theory of rings and fields, ideals of one- and two-dimensional rings, differential modules and ramifications, algebras of formal and convergent power series, and then treat the resolution of singularities of curves and the resolution of isolated normal surface singularities. The emphasis is made on a controlled resolution process, presented by a finite sequence of birational transformations. For curve singularities, it results in successive blowing-ups of the curve embedded into a projective space. In turn the resolution of surface singularities can be performed in two ways. One algorithm follows the Jung idea. It is based on projecting the surface onto a plane with a normal crossing discriminant locus and then parameterizing a complete neighborhood of the singularity by conjugate Puiseux power series. Another algorithm is that of Zariski, and it amounts in a sequence of blowing-ups and normalizations, governed by valuations which can be uniformized. The exposition is self-contained and is supplied by an appendix, covering some classical algebraic geometry and commutative algebra. algebraic curves and algebraic surfaces; resolution of singularities; local rings; valuation theory; Cohen-Macaulay rings; differential modules; analytic algebras Kiyek, K.; Vicente, J. L., Resolution of curve and surface singularities in characteristic zero, (2004), Kluwer Dordrecht Research exposition (monographs, survey articles) pertaining to algebraic geometry, Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Singularities of curves, local rings, Singularities of surfaces or higher-dimensional varieties, Analytic algebras and generalizations, preparation theorems, Sheaves of differential operators and their modules, \(D\)-modules, Global theory and resolution of singularities (algebro-geometric aspects) Resolution of curve and surface singularities in characteristic zero. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let f(x,y,z) be a germ of an analytic function defined in a neighbourhood of the origin and assume that the Newton boundary \(\Gamma\) (f) is non- degenerate and \(V=f^{-1}(0)\) has an isolated singularity at the origin. Let \(\Gamma^*(f)\) be the dual Newton diagram. Let \(\Sigma^*\) be a simplicial subdivision of \(\Gamma^*(f)\). It is well known that there is an associated resolution \(\pi:\tilde V\to V.\) However \(\Sigma^*\) is not unique. The author proves that there is a canonical way to get a simplicial subdivision \(\Sigma^*\) so that the graph of the resolution is obtained by a canonical surgery from \(\Gamma^*(f)\) which is considered as a graph. He also proves that a compact two-face \(\Delta\) of \(\Gamma\) (f) corresponds to an exceptional divisor of genus \(g(\Delta)\) which is equal to the number of the integral points on the interior of \(\Delta\). dual Newton diagram; canonical primitive sequence; germ of an analytic function; simplicial subdivision M. OKA, On the resolution of two-dimensional singularities, Proc. Japan Acad., 60 (1984), 174-177 Modifications; resolution of singularities (complex-analytic aspects), Complex singularities, Germs of analytic sets, local parametrization, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry On the resolution of two-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 The notion of defect was introduced as a measure to what extend the imposition of double points fails to lower the dimension of certain cohomology groups [\textit{C. H. Clemens}, Adv. Math. 47, 107--230 (1983; Zbl 0509.14045)].
In the article under review, a hypersurface \(X\) in \(\mathbb{P}^n\) is said to have defect, if \(h^i(X)\neq h^i(\mathbb{P}^n) \) for some \(i\in\{n,\dots,2n-2\}\), where \(h^i\) stands for the \(i\)-th Betti number in a ``reasonable'' cohomology theory.
Extending the classical situation (projective hypersurfaces over the complex numbers with only ordinary double points), the author allows base fields of arbitrary characteristic and hypersurfaces with more general singularities. Results are given in the spirit of ``defect implies many singularities''.
In Theorem 1.1. an estimate is obtained for the global Tjurina number; the result is non-trivial if all singularities of \(X\) are isolated:
Let \(K\) be a field of characteristic zero. Suppose that \(X\subseteq\mathbb{P}^n_K\), \(n\geq 3\), is a hypersurface with defect in algebraic de Rham, Kähler-de Rham, singular or étale cohomology. Denote by \(\tau (X)\) the global Tjurina number of \(X\). Then \(\tau (X)\geq\frac{\mathrm{deg}(X)-n+1}{n^2+n+1}\).
Moreover, if \(X\) has at most weighted homogeneous singularities, then \(\tau(X)\geq\mathrm{deg}(X)-n+1\).
If the base field has positive characteristic, the list of simple singularities is found in [\textit{G. M. Greuel} and \textit{H. Kröning}, Math. Z. 203, No. 2, 339--354 (1990; Zbl 0715.14001)]; they are known to be absolutely isolated with relatively easy resolutions, especially for the \(A_k\)-type-singularities.
Based on a detailed description of the resolutions, the author states the following for a base field \(K\) of characteristic \(\neq 2\) and if the hypersurface \(X\subseteq\mathbb{P}_K^n\) has defect with respect to étale or rigid cohomology:
Assume \(X\) has only singularities \(x\in X\) which are ordinary multiple points of multiplicity \(m_x\) or of type \(A_{k_x}\), then \(\sum m_x+\sum 2\left\lceil \frac{k_x}{2}\right\rceil\geq\mathrm{deg}(X)\).
Reviewers remark: Local resolution graphs of \(A\)-type singularities coincide in all characteristics. Furthermore, they are weighted homogeneous for \(n\) even. This might indicate a possible extension of the result to the case \(\mathrm{char }(K)=2\). de Rham cohomology; singular hypersurfaces; isolated singularities; factorial hypersurfaces; defect of a projective hypersurface Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Étale and other Grothendieck topologies and (co)homologies, Motivic cohomology; motivic homotopy theory, Singularities of surfaces or higher-dimensional varieties Hypersurfaces with defect | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Author's abstract: Let \(M\) be an analytic manifold over \(\mathbb R\) or \(\mathbb C\), \(\theta\) a 1-dimensional log-canonical (resp. monomial) singular distribution and \(\mathcal I\) a coherent ideal sheaf defined on \(M\). We prove the existence of a resolution of singularities for \(\mathcal I\) that preserves the log-canonicity (resp. monomiality) of the singularities of \(\theta\). Furthermore, we apply this result to provide a resolution of a family of ideal sheaves when the dimension of the parameter space is equal to the dimension of the ambient space minus one. resolution of singularities; singular foliations; log-canonical foliations; monomial foliations Belotto, A, Global resolution of singularities subordinated to a \(1\)-dimensional foliation, J. Algebra, 447, 397-423, (2016) Global theory and resolution of singularities (algebro-geometric aspects), Singularities of holomorphic vector fields and foliations, Foliations in differential topology; geometric theory, Global theory of complex singularities; cohomological properties, Dynamical aspects of holomorphic foliations and vector fields, Dynamics induced by flows and semiflows, Real-analytic and Nash manifolds, Singularities in algebraic geometry Global resolution of singularities subordinated to a 1-dimensional foliation | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 summary of Ohmoto's minicourse delivered at the School on Real and Complex Singularities in Sao Carlos 2012. It contains important new results on Thom polynomial theory and its applications.
For a map \(f\) between complex algebraic or analytic varieties and a singularity type \(\eta\) one considers the \(\eta\)-singularity subset \(\eta(f)\) to be the collection of points of the domain where the map \(f\) has singularity \(\eta\). The classical starting point of global singularity theory is the result claiming that the cohomological fundamental class of \(\overline{\eta(f)}\) can be expressed as a multivariate polynomial depending only on the singularity, if one plugs in the Chern classes of the source and target manifolds. The polynomial is called the Thom polynomial \(tp(\eta)\) of the singularity \(\eta\).
The main addition of the present paper to global singularity theory is the extension of this result from the \textit{fundamental class} represented by \(\overline{\eta(f)}\) to the \textit{Chern-Schwartz-MacPherson (CSM) class} of \(\eta(f)\). The CSM class of a variety is an inhomogeneous deformation of its fundamental class, it is an additive invariant, it is consistent with push-forward maps, and equals the total Chern class of the tangent bundle if the variety is smooth. A version of the CSM class, called Segre-Schwartz-MacPherson (SSM) class is consistent with pullback. In Section 3 Ohmoto reviews the CSM and equivariant CSM theory (invented by himself), and in Section 4 he lays dows the foundations of the theory of SSM Thom polynomials (\(tp^{SM}(\eta)\)). A calculating method (``restriction method'') is used to calculate some terms, and an excursion to multisingularities is included.
The author's first application is the proof of several universal weighted Euler characteristics formulas for singularity loci, some of them reprove and generalize such formulas of the 19th century geometers. These results now follow not by case-by-case arguments but from the general framework of SSM-Thom polynomials.
The last two sections are devoted to another spectacular application: the conceptual treatment of the vanishing topology of finitely determined weighted homogeneous map germs. In particular, the multiplicities of stable map germs within degenerate ones are calculated, as well as the image and discriminant Milnor numbers. These are important singularity theory notions which had been studied before without Thom polynomials. Ohmoto's calculations follow from the universal formulas of SSM-Thom polynomials.
The paper is a very important contribution to global singularity theory. Thom polynomial; Chern-Schwartz-MacPherson class; Segre class; vanishing cohomology; Milnor number ; Ohmoto, School on real and complex singularities. Adv. Stud. Pure Math., 68, 191, (2016) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Singularities of maps and characteristic 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 Hironaka has proposed a combinatorial game, played with Newton polyhedra of singularities, whose affirmative solution would imply the local uniformization theorem over a field of any characteristic. In this paper we give a counterexample to this game. However, our examples does not come from algebraic geometry (the game is a good bit stronger than the local uniformization theorem). Hironaka hard polyhedra game; Newton pollyhedron; resolution of singularities; local uniformization theorem; normal flatness; blowing-up Global theory and resolution of singularities (algebro-geometric aspects), 2-person games, Singularities in algebraic geometry, Polytopes and polyhedra, Permutations, words, matrices A counterexample to Hironaka's ''hard'' polyhedra game | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 closed hypersurface of a smooth variety \(Z\) over a field \(k\) of characteristic zero and \(E=\{ H_1, \ldots, H_m \}\) a collection of smooth hypersurfaces of \(Z\) with normal crossings. The pair \((X,E)\) has \textit{simple normal crossings} (or is snc) at a point \(a \in Z\) if there is a regular system of parameters \((x_1, \ldots, x_p,y_1, \ldots, y_q)\) of \({\mathcal O}_{Z,a}\) such that each irreducible component of \(X\) at \(a\) defined by \(x_i\) and each \(H \in E\) by \(y_j\), for suitable indices \(i, j\). If \((X,E)\) is snc at each point of \(Z\), we say that it is a snc pair. The main result of the paper under review is the following theorem.
Consider a pair \((X,E)\) as above. Then there is sequence
\[
(1) \quad Z=Z_0 \leftarrow Z_1 \leftarrow \cdots {\leftarrow} Z_t
\]
of blowing-ups, having smooth centers \(C_i \subset Z_i\), such that (letting \({\sigma}_i\) denote the \(i\)-th blowing-up, \(X_{i+1}\) the strict transform of \(X_i\) and \(E_{i+1}\) the collection of strict transforms of the hypersurfaces in \(E_i\) together with the exceptional divisor of \(\sigma _i\)) then \(C_i\) has snc with respect to \(E_i\) for all \(i\), \(C_i\) contains no snc singularity of \((X_i,E_i)\) (hence \(\sigma_{j+1}\) is an isomorphism over the set of snc of \((X_i,E_i)\)) and \((X_t,E_t)\) has only snc singularities. The association of the ``partial'' desingularization sequence (1) to \((X,E)\) is functorial with respect to smooth morphisms that preserve the number of irreducible components of \(X\) at each point \(a \in X\). Each \(C_i\) is necessarily contained in \(X_i\).
Two of the authors had obtained somewhat weaker results before. For instance, in [\textit{E. Bierstone} and \textit{P. D. Milman}, Adv. Math. 231, No. 5, 3022--3053 (2012; Zbl 1257.14002)] they found similar results but where, in the notation above, the composite morphism \({\sigma}_t \ldots {\sigma}_1\) is an isomorphism over the set of snc points of \((X,E)\), although the more precise statement of the new theorem on each morphism \(\sigma _i\) is not available.
As in the mentioned previous paper, the present algorithm to obtain the sequence (1) involves the use of inv, the fundamental desingularization function introduced in [\textit{E. Bierstone} and \textit{P. D. Milman}, Invent. Math. 128, No. 2, 207--302 (1997; Zbl 0896.14006)]. With the aid of inv and another numerical function \({\mu}_{H,k}\) (involving orders of certrain ideals along suitable exceptional hypersurfaces, previously introduced in [\textit{E. Bierstone} and \textit{P. D. Milman}, Publ. Res. Inst. Math. Sci. 44, No. 2, 609--639 (2008; Zbl 1151.14012)]), the authors characterize the snc points of each pair \((X_i,E_i)\) that appears in the sequence (1). Using centers where inv reaches a maximum, the authors reach a situation where they have to deal exclusively with the numbers \(\mu\). At this point, they start using another technique, a process they call ``cleaning''.
They give examples examples illustrating how the present method works, as well as one showing that it is different from that of the mentioned Adv. Math paper. resolution of singularities; simple normal crossings; desingularization invariant; cleaning; partial resolution algorithm Bierstone, E.; Silva, S.; Milman, P. D.; Vera Pacheco, F., Desingularization by blowings-up avoiding simple normal crossings, Proceedings of the American Mathematical Society, 142, 4099-4111, (2014) Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry, Local complex singularities, Invariants of analytic local rings Desingularization by blowings-up avoiding simple normal crossings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Building on results of \textit{H. Clemens} and \textit{H. P. Kley} [J. Algebr. Geom. 9, No. 1, 175--200 (2000; Zbl 0973.14028)], we find criteria for a continuous family of curves in a nodal \( K\)-trivial threefold \( Y_0\) to deform to a scheme of finitely many smooth isolated curves in a general deformation \( Y_t\) of \( Y_0\). As an application, we show the existence of smooth isolated curves of bounded genera and unbounded degrees in Calabi-Yau complete intersection threefolds. isolated curves; deformations; Hilbert schemes; Calabi-Yau threefolds; singularities Knutsen, Andreas Leopold, On isolated smooth curves of low genera in Calabi-Yau complete intersection threefolds, Trans. Amer. Math. Soc., 364, 10, 5243-5264, (2012) Formal methods and deformations in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Divisors, linear systems, invertible sheaves, Special algebraic curves and curves of low genus, \(K3\) surfaces and Enriques surfaces, Calabi-Yau manifolds (algebro-geometric aspects) On isolated smooth curves of low genera in Calabi-Yau complete intersection 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 Let \(k\) be an algebraically closed field of characteristic zero. Let \(S\) and \(F\) be two surfaces such that all singularities of \(F\) are of the form \(z^3=x^{3s}-y^{3s}, s\in \mathbb{N}\). Assume that \(S\cap F=rC, r\in \mathbb{N}\) and \(C\) an irreducible non-singular curve. It is proved that \(C\) cannot pass through the singularities of \(F\) if \(3\nmid r\).
Let \(Z\) be a reduced irreducible non-singular \((n-1)\)-dimensional variety such that \(rZ=X\cap F\), where \(X\) is a normal \(n\)-fold, \(F\) is a \((N-1)\)-fold in \(\mathbb{P}^N\), such that \(Z\cap \text{Sing}(X)\neq \emptyset\). The singularities of \(X\) through which \(Z\) passes are studied.
For part I, II, see \textit{M. R. Gonzalez-Dorrego} [Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 108, No. 1, 183--192 (2014; Zbl 1304.14007); in: Singularities in geometry and topology 2011. Proceedings of the 6th Franco-Japanese symposium on singularities, Fukuoka, Japan, September 5--10, 2011. Advanced Studies in Pure Mathematics 66 (2015)]. smooth double subvarieties; singular varieties Gonzalez-Dorrego, M.R.: Smooth double subvarieties on singular varieties, III. Banach Center Publ. 108, 85-93 (2016) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, \(3\)-folds, \(4\)-folds, \(n\)-folds (\(n>4\)), Hypersurfaces and algebraic geometry Smooth double subvarieties on singular varieties. III | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the sixties, J. Nash initiated the study of the space of arcs of an algebraic variety \(X\), centered in some singular point, denoted by \(X_\infty^{\text{Sing}}\). This space has the structure of an (infinite dimensional) algebraic variety. J. Nash proved that \(X^{\text{Sing}}_\infty\) has a finite number of irreducible components, when the characteristic of the base field is zero. The proof of J. Nash is based on the existence of a resolution of singularities of \(X\). Therefore his result can be extended to any algebraic variety \(X\), for which there exists a resolution of singularities \(Y\longrightarrow X\).
The goal of this paper, is to strengthen this result. Roughly speaking, the authors show that if local uniformization holds for \(X\), then the space of arcs of \(X\), centred in some singular point, has a finite number of components. Their result is a bit more general, and is the following:
Theorem: Let \(X\) be a variety defined over a field \(k\), and \(Z\) be a subvariety of \(X\). Assume that for every \(z\in Z\), and every irreducible subvariety \(V\subset Z\), with \(z\in V\), local uniformization holds on \(V\) at \(z\). Then the space of arcs of \(X\) centered in a point of \(Z\), has a finite number of irreducible components. arc spaces; local uniformization Valuations and their generalizations for commutative rings, Local rings and semilocal rings, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Local uniformization and arc 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 reader is assumed to already know the definitions of a valuation (which is sometimes a pseudo-valuation in this paper) of a graded algebra, of a quasi-excellent local domain, what is local uniformization, and some notions of local uniformization defined in a previous paper by the same author. The definitions of the henselization, of the defect of an extension and of the graded algebra associated to a valuation, of a complete and \(1\)-complete sets of key polynomials, are given. In the following, the extensions are assumed to be simple transcendental extensions \(k(x)\) of fields. So their elements can be seen as polynomials. Intuitively, a complete set of key polynomials is a set of polynomials that makes it possible to compute the valuation of any polynomial. It is \(1\)-complete if it makes possible to compute the valuation of any polynomial with valuation in the smallest non-zero isolated subgroup of the valuation group. Such a set always exists. The author proves that in a defectless extension one can find such a set which is either finite or indexed by \(\mathbb{N}\). He shows the links between the defect of an extension and the degrees of some polynomials in a set of key polynomials. He deduces an effective computation of the defect of an extension in several examples. In the first one, he proves that the valuation on \(k\) extends in an unique way to \(k(x)\). In the last section the author extends some results about local uniformization in quasi-excellent equicharacteristic local domains, of a previous paper by himself, for a valuation satisfying some inductive assumption about defect. defect; valued fields; local uniformization; key polynomials Giraud, J.: Étude locale des singularités. Publications Mathématiques d'Orsay, Number~26. Mathématique, Université Paris XI, Orsay, 1972. Cours de 3éme cycle (1971-1972) Valuations and their generalizations for commutative rings, Valued fields, General valuation theory for fields, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Defect of an extension, key polynomials and local uniformization | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities There are three one-parameter families of parabolic singularities, \(P_ 8\), \(X_ 9\), and \(J_{10}\). The author describes the intersection forms of these singularities using Dynkin diagrams and shows that the set of these diagrams completely determines the decompositions of parabolic singularities. Previously \textit{C. T. C. Wall} had determined the decomposition of \(P_ 8\) [Philos. Trans. R. Soc. Lond., A 302, 415--455 (1981; Zbl 0451.14009)]. The work under review is a continuation of results on deformations of parabolic singularities the author has published in Invent. Math. 86, 19--33 (1986; Zbl 0578.32037). one-parameter families of parabolic singularities; Dynkin diagrams Jaworski P.: Decompositions of parabolic singularities. Bull. Sci. Math. (2) 112(2), 143--176 (1988) Singularities in algebraic geometry, Complex singularities, Global theory and resolution of singularities (algebro-geometric aspects), Deformations of complex singularities; vanishing cycles, Deformations of singularities Decompositions of parabolic singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be an algebraic variety over an algebraically closed field \(k\) with singular locus \(\text{Sing}(X)\) and \(S\subseteq X\) a closed subvariety. Let \(T=\text{Spec} k[[t]]\) and \(T_ N=\text{Spec} k[[t]]/t^{N+1}\). An analytic arc is \(\gamma:T\to X\) (resp. \(T_ N\to X)\); \(\gamma\) is called an \(S\text{-arc}\) if \(\gamma(0)\in S\). Using the strong approximation theorem one can prove that an \(N\)-truncated arc can be lifted to an arc if \(N\) is big enough. The coefficients of an \(N\)- truncated \(S\)-arc define a point in some \(k^ M\). The closure of these points (coming from all \(N\)-truncated \(S\)-arcs) is called the Nash variety \(V(X,S,N)\).
The paper gives a foundation of the theory of Nash varieties. They may carry a non-reduced structure which turns out to be useful to characterize the smoothness of \(X\) or \(S\) in terms of \(V(X,S,N)\). Each irreducible component of \(V(X,S,N)\) contains a dense open set the points of which correspond to a family of \(N\)-truncated \(S\)-arcs. It is proved that this family considered as family of irreducible curve singularities is equisingular. Using this fact one may use the corresponding invariants (which are constant in an equisingular family) as invariants of the singularities of \(X\). equisingular family of irreducible curve singularities; truncated \(S\)- arcs; Nash variety A. Nobile, On Nash theory of arc structure of singularities, Ann. Mat. Pura Appl. (4) 160 (1991), 129--146. Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings, Real-analytic and Nash manifolds, Singularities in algebraic geometry On Nash theory of arc structure of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is the first part of a series of papers devoted to the resolution of singularities (and containing joint work with K. Matsuki, as mentioned in 0.6). The goal is presenting a program towards the construction of a resolution algorithm which works for an algebraic variety over a perfect field in positive characteristic. The author's intention is however, to develop a program working in full generality, i.e. including characteristic 0 as well.
The intended parts of the complete work are:
I. Foundation; the language of the idealistic filtration
II. Basic invariants associated to the idealistic filtration and their properties
III. Transformations and modifications of the idealistic filtration
IV. Algorithm in the framework of the idealistic filtration
Part I (under review) establishes the notion and fundamental properties of the \textit{idealistic filtration}, which is considered the main language of this program.
Chapter 0 starts with a crash course on the existing algorithms in characteristic 0 and introduces the author's program as a ``new approach to overcome the main source of troubles in the language of the \textit{idealistic filtration}, which is a refined extension of such classical notions as the idealistic exponent by Hironaka, the presentation by Bierstone-Milman, the basic object by Villamayor, and the marked ideal by Wlodarczyk.''
Section 0.2.3 introduces the idealistic filtration and mentions some of its distinguished features as there are: leading generator systems as substitutes for hypersurfaces of maximal contact, construction of the strand of invariants through enlargements of an idealistic filtration, saturation and a ``new nonsingularity principle''.
In 0.3 (``Algorithm constructed according to the program'') the author refers to the forthcomimg part IV of the paper as far as termination of the algorithm (in case of positive characteristic) is concerned; he mentions that this question is not yet settled.
In 0.5 a brief account of (mainly references to) the history of the problem is given, as well as hints to recent announcements and approaches.
The remaining about 70 pages of the paper (Part I) contain essentially the ``local'' ingredients of the program. Below follows (a part of) the author's outline, taken from 0.8:
``In Chapter 1, we recall some basic facts on the differential operators, especially those in positive characteristic. Both in the description of the preliminaries and in Chapter 1, our purpose is not exhaustively cover all the material, bu only to minimally summarize what is needed to present our program and to fix our notation.''
``Chapter 2 is devoted to establishing the notion of an idealistic filtration and its fundamental properties. The most important ingredient of Chapter 2 is the analysis of the \(\mathcal D\)-saturation and the \(\mathcal R\)-saturation and that of their interaction. In our algorithm, given an idealistic filtration, we always look for its bi-saturation, called the \(\mathcal B\)-saturation, which is both \(\mathcal D\)-saturated and \(\mathcal R\)-saturated and which is minimal among such containing the original idealistic filtration. The existence of the \(\mathcal B\)-saturation is theoretically clear. However, we do not know a priori whether we can reach the \(\mathcal B\)-saturation by a repetition of \(\mathcal D\)-saturations and \(\mathcal R\)-saturations starting from the given idealistic filtration, even after infinitely many times. The main result here is that the \(\mathcal B\)-saturation is actually realized if we take the \(\mathcal D\)-saturation and then the \(\mathcal R\)-saturation of the given one, each just once in this order. In our algorithm, we do not deal with an arbitrary idealistic filtration, but only with those which are generated by finitely many elements with rational levels. We say they are of r.f.g. type (short for `rationally and finitely generated'). It is then a natural and crucial question if the propery of being of r.f.g. type is stable under \(\mathcal D\)-saturation and \(\mathcal R\)-saturation.''
``In Chapter 3, through the analysis of the leading terms of an idealistic filtration (which is \(\mathcal D\)-saturated), we define the notion of a leading generator system, which \dots plays the role of a collective substitute for the notion of a hypersurface of maximal contact.
Chapter 4 is the culmination of part I, establishing the new nonsingularity principle of the center for an idealistic filtration which is \(\mathcal B\)-saturated. Its proof is given via three somewhat technical but important lemmas, which we will use again later in the series of papers.'' resolution of singulairties; positive characteristic Kawanoue, H.: Toward resolution of singularities over a field of positive characteristic. I. Foundation; the language of the idealistic filtration. Publ. Res. Inst. Math. Sci. 43 (2007), no. 3, 819-909. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Toward resolution of singularities over a field of positive characteristic. I. Foundation; the language of the idealistic filtration | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main result of this paper is a proof of the monodromy conjecture for the local Igusa and motivic zeta function of a quasi-ordinary polynomial of arbitrary dimension.
Let \(p\) be a prime number and \(K\) a finite extension of the field \(\mathbb Q_p\) of \(p\)-adic numbers. Let \(R\) be the valuation ring of \(K\) with maximal ideal \(P\) and residue field \(\bar K \cong \mathbb F_q\). We denote by \(| \cdot| \) and \(| dx| \) the standard absolute value on \(K\) and Haar measure on \(K^d\), respectively. The global and local Igusa zeta function of a non-constant polynomial \(h \in K[x_1,\dots,x_d]\) are (the meromorphic continuation to \(\mathbb C\) of)
\[
I(h,K;s)=\int_{R^d} | h(x)| ^s | dx| \quad\text{and} \quad I_0(h,K;s)=\int_{P^d} | h(x)| ^s | dx| ,
\]
respectively, for \(s\in \mathbb C\) with \(\Re(s)>0\). Igusa proved that these are rational functions in \(q^{-s}\) and raised the following conjecture.
Monodromy Conjecture. When \(h\) is defined over a number field \(F\), then for all but a finite number of \(p\)-adic completions \(K\) of \(F\) we have that, if \(s_0\) is a pole of \(I(h,K;s)\) or \(I_0(h,K;s)\), then \(\exp(2\pi i \Re(s_0))\) is an eigenvalue of the local monodromy of \(h\) at some complex point of \(h^{-1}\{0\}\).
\textit{F. Loeser} proved the conjecture when \(d=2\) [Am. J. Math. 110, No.1, 1--21 (1988; Zbl 0644.12007)]; since then several partial results by various authors appeared. Here the authors prove it for the local Igusa zeta function of an arbitrary quasi-ordinary polynomial. In fact they show the analogous conjecture for the finer motivic zeta function of \textit{J. Denef} and \textit{F. Loeser} [J. Algebr. Geom. 7, No.~3, 505--537 (1998; Zbl 0943.14010)], which specializes for all but a finite number of \(K\) to the Igusa zeta function. As another consequence they obtain the analogous monodromy conjecture for the (local) topological zeta function from [J. Am. Math. Soc. 5, No.~4, 705--720 (1992; Zbl 0777.32017)] of such polynomials. The proof uses in an essential way motivic integration on the space of arcs on a smooth algebraic variety.
A polynomial \(h\in \mathbb C[x_1,\dots,x_d]\) with \(h(0)=0\) defines a quasi-ordinary singularity at \(0\) if there is a finite morphism from the germ at \(0\) of \(h^{-1}\{0\}\) to \((\mathbb C^{d-1},0)\) whose discriminant locus is contained in \(y_1y_2 \dots y_{d-1} =0\) for some local coordinates \(y_1,\dots,y_{d-1}\). These singularities behave in many aspects as singularities of plane curves; after the Abhyankar-Jung theorem they admit fractional power series parametrizations and a finite set of characteristic exponents.
In order to compute the motivic zeta function the authors use Newton maps. The key point is that after a Newton map one gets a new quasi-ordinary polynomial with less characteristic exponents, allowing a procedure by induction. As an ingredient the authors develop a formula for the motivic zeta function of a non-degenerate polynomial, based on work of Denef-Hoornaert in the \(p\)-adic case. A technical issue worth mentioning is the lifting problem of arcs under usual Newton maps, forcing the authors to work over \(\mathbb C\{t\}\) instead of \(\mathbb C\), which is possible by the refined motivic integration over \(\mathbb C\{t\}\) from \textit{J. Denef, F. Loeser} [Compos. Math. 131, No.~3, 267--290 (2002; Zbl 1080.14001)]. For the proof of the monodromy conjecture the authors use that transversal sections at generic points are also quasi-ordinary singularities, allowing to proceed by induction on dimension. motivic, Igusa and topological zeta functions; monodromy; quasi-ordinary singularities Artal Bartolo, Enrique; Cassou-Noguès, Pierrette; Luengo, Ignacio; Melle Hernández, Alejandro, Quasi-ordinary power series and their zeta functions, Mem. Amer. Math. Soc., 178, 841, vi+85 pp., (2005) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Zeta functions and \(L\)-functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Quasi-ordinary power series and their zeta functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Nash problem on arc spaces is to ask for the bijectivity between the set of irreducible components of arc spaces passing through the singular locus and the set of the essential divisors of the resolution of the singularities. This problem is negatively answered by \textit{J. Kollár} and the reviewer [Duke Math. J. 120, No. 3, 601--620 (2003; Zbl 1052.14011)] for dimension greater than or equal to 4.
This paper part of the author's approach to solve the Nash problem in the surface case and reduces the problem into a topological problem. He proves that the condition that an essential divisor is not corresponding to the irreducible component of the arc spaces through the singularity depends on the dual weighted graph of the minimal good resolution. arc spaces; Nash problem; minimal resolution Fernández de Bobadilla, J., Nash Problem for surface singularities is a topological problem, Adv. Math., 230, 131-176, (2012) Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Local complex singularities, Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects) Nash problem for surface singularities is a topological problem | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0565.00006.]
For a finite subgroup G of SL(2,\({\mathbb{C}})\) the McKay correspondence is an isomorphism between its Dynkin diagram and its diagram of non trivial irreducible finite dimensional representations. The author gives here an approach to McKay's result via invariant theory by looking at the orbits of G in the projective space of the representation. In this way, he recovers by an explicit computation Gonzalez-Sprinberg's and Verdier's description of the McKay correspondence (obtained also by explicit computation), which assigns to each representation the first Chern class of the corresponding vectorbundle on the minimal desingularization of \({\mathbb{C}}^ 2/G\). Later on, several people gave more theoretical proofs of those facts, which partly apply in characteristic p\(>0\). rational double points; McKay correspondence; invariant theory Knörrer, H.: Group representations and the resolution of rational double points. Contemp. math. 45, 175-222 (1985) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Geometric invariant theory, Group actions on varieties or schemes (quotients) Group representations and the resolution of rational double 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 Given an equicharacteristic complete noetherian local ring \(R\) with algebraically closed residue field \(k\), we first present a combinatorial proof of embedded local uniformization for zero-dimensional valuations of \(R\) whose associated graded ring \(\mathrm{gr}_vR\) with respect to the filtration defined by the valuation is a finitely generated \(k\)-algebra. The main idea here is that some of the birational toric maps which provide embedded pseudo-resolutions for the affine toric variety corresponding to \(\mathrm{gr}_vR\) also provide local uniformizations for \(v\) on \(R\). These valuations are necessarily Abhyankar (for zero-dimensional valuations this means that the value group is \(\mathbb{Z}^r\) with \(r=\dim R\)).
In a second part we show that conversely, given an excellent noetherian equicharacteristic local domain \(R\) with algebraically closed residue field, if the zero-dimensional valuation \(v\) of \(R\) is Abhyankar, there are local domains \(R'\) which are essentially of finite type over \(R\) and dominated by the valuation ring \(R_v\) (\(v\)-modifications of \(R\)) such that the semigroup of values of \(v\) on \(R'\) is finitely generated, and therefore so is the \(k\)-algebra \(\mathrm{gr}_vR'\). Combining the two results and using the fact that Abhyankar valuations behave well under completion gives a proof of local uniformization for rational Abhyankar valuations and, by a specialization argument, for all Abhyankar valuations.
As a by-product we obtain a description of the valuation ring of a rational Abhyankar valuation as an inductive limit indexed by \(\mathbb{N}\) of birational toric maps of regular local rings. One of our main tools, the valuative Cohen theorem, is then used to study the extensions of rational monomial Abhyankar valuations of the ring \(k[[x_1,\dots,x_r]]\) to monogenous integral extensions and the nature of their key polynomials. In the conclusion we place the results in the perspective of local embedded resolution of singularities by a single toric modification after an appropriate re-embedding. toric geometry; valuations; key polynomials Teissier, B.: Overweight deformations of affine toric varieties and local uniformization. In: Campillo, A., Kehlmann, F.-V., Teissier, B. (eds.) Valuation Theory in Interaction. Proceedings of the Second International Conference on Valuation Theory, Segovia-El Escorial, 2011. Congress Reports Series, Sept 2014. European Mathematical Society Publishing House, Zürich, pp. 474-565 (2014) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Overweight deformations of affine toric varieties and local uniformization | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper deals with the classification of symplectic linear quotient singularities. Let us recall that this classification is almost complete, but there is an infinite series of groups for which the problem is still open. In this work, the authors study the remaining infinite series and reduce the problem to finitely many open cases. It is proved that these cases do not admit a symplectic resolution. The main theorem of the paper uses results of Verbitsky. symplectic resolution; symplectic linear quotient singularities; classification; smooth symplectic variety; Verbitsky's result Global theory and resolution of singularities (algebro-geometric aspects), Poisson manifolds; Poisson groupoids and algebroids, Singularities in algebraic geometry, Reflection and Coxeter groups (group-theoretic aspects), Deformations of associative rings, Group actions on varieties or schemes (quotients) Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 gives a theorem about simultaneous resolution of families of embedded schemes over a Dedekind scheme containing a field of characteristic zero. This extends former results of the same author about resolutions of embedded families over fields of characteristic zero. family of embedded schemes; Fitting ideals; normal crossing; Jacobian ideal; resolution of singularities Belotto da Silva, A.: Resolution of singularities in foliated spaces. Ph.D. thesis. Université de Haute-Alsace, France (2013) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Structure of families (Picard-Lefschetz, monodromy, etc.) Resolution in families | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The articles of this volume will be reviewed individually. Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Configurations and arrangements of linear subspaces, Derivations, actions of Lie algebras, Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation), Relations with arrangements of hyperplanes, Other hypergeometric functions and integrals in several variables, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Homology with local coefficients, equivariant cohomology, Proceedings of conferences of miscellaneous specific interest Arrangements of hyperplanes. Proceedings of the 2nd Mathematical Society of Japan-Seasonal Institute, MSJ-SI, Sapporo, Japan, August 1--13, 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 The Lyashko-Looijenga mapping associates to a polynomial (i.e. its coefficients) the set of critical values. This map is holomorphic and maps real points to real points.
Let \(X \subset \mathbb{C}^ \mu\) be the subset of the base of the miniversal deformation of a simple singularity corresponding to polynomials with \(\mu\) distinct critical values. The restriction of the Lyashko-Looijenga map to \(X\) is a finite-sheeted covering. The multiplicity of the Lyashko- Looijenga covering restricted to a connected component of the complement in \(\mathbb{R}^ \mu\) is expressed in terms of invariants of this component.
For \(A_ \mu\)-singularities an algorithm to compute these invariants is described. simple singularities; miniversal deformation; Lyashko-Looijenga mapping Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Singularities in algebraic geometry, Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Deformations of complex singularities; vanishing cycles, Coverings of curves, fundamental group, Local complex singularities On the real preimages of a real point under the Lyashko-Looijenga covering for simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a variety over a field \(k\) of characteristic 0 and let \(X_\infty\) be its space of arcs (see [\textit{J. F. Nash jun.}, Duke Math. J. 81, No. 1, 31--38 (1995; Zbl 0880.14010)]). Let \(P_E\) be the stable point of \(X_\infty\) defined by a divisorial valuation \(\nu_E\) on \(X\). The author investigates the analytic structure of the local ring \(\mathcal O_{X_\infty, P_E}\). In particular, she finds a suitable coordinate representation for the completion of this ring with respect to its maximal ideal, computes generators (a regular system of parameters) of the ring, its embedding dimension, and so on. space of arcs; divisorial valuation; graded algebra; henselization; étale coverings Graded rings, Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc., Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Local complex singularities Coordinates at stable points of the space of arcs | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities See the preview in Zbl 0521.14013. Chern numbers of logarithmic cotangent bundle; quotient surface singularities; minimal resolution; bound for number of rational double points Miyaoka Y.: The maximal number of quotient singularities on surfaces with given numerical invariants. Math. Ann. 268, 159--171 (1984) Singularities of surfaces or higher-dimensional varieties, Enumerative problems (combinatorial problems) in algebraic geometry, Characteristic classes and numbers in differential topology, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) The maximal number of quotient singularities on surfaces with given numerical invariants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be an integral variety over a perfect field \(k\), \(\mathcal L_m(X)\) its jet scheme of level \(m\in \mathbb N\) and \(\mathcal L_{\infty}(X)\) its arc scheme. The general component \(\mathcal G_m(X)\) of \(\mathcal L_m(X)\) is the Zariski closure of \(\mathcal L_m(\mathrm{Reg}(X))\).
If \(X\) is smooth on \(k\) then the geometry and topology of \(\mathcal L_m(X)\) are well understood. In this paper the authors consider the case \(X\) is not smooth and study some properties of the general component \(\mathcal G_m(X)\) by means of a smooth birational model of \(X\).
Indeed, under the further hypothesis that \(X\) is affine embedded in \(\mathbb A^N_k\), the authors prove that a birational model of \(X\) provides a description of \(\mathcal G_m(X)\) that gives rice to an algorithm which computes a Groebner basis of the defining ideal of \(\mathcal G_m(X)\) in \(\mathbb A^N_k\) as a subscheme of \(\mathcal L_m(X)\) (Algorithm~2). The authors also extend to arbitrary integral varieties over perfect fields over arbitrary characteristic another algorithm ''already introduced in the Ph.D. Thesis of Kpognon'' (see also [\textit{K. Kpognon} and \textit{J. Sebag}, Commun. Algebra 45, No. 5, 2195--2221 (2017; Zbl 1376.14018)]) ``for the study of arc scheme associated with integral affine plane curves in characteristic zero'' (Algorithm~1). Several examples and comments to the implementation of the algorithms, which is available in SageMath, are provided in Sections~6 and~7.
The given results are applied for further studies of plane curves, concerning differential operators logarithmic along an affine plane curve and the rationality of a motivic power series that is introduced by the authors and ``which encodes the geometry of all \(\mathcal G_m(X)\)'' (Sections 8 and 9). computational aspects of algebraic geometry; derivation module; jet and arc scheme; singularities in algebraic geometry Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Arcs and motivic integration, Computational aspects of algebraic curves, Computational aspects of higher-dimensional varieties, Effectivity, complexity and computational aspects of algebraic geometry, Local complex singularities Two algorithms for computing the general component of jet scheme and applications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This volume contains the proceedings of the 2019 Lluís A. Santaló Summer School on \(p\)-Adic Analysis, Arithmetic and Singularities, which was held from June 24--28, 2019, at the Universidad Internacional Menéndez Pelayo, Santander, Spain.
The main purpose of the book is to present and analyze different incarnations of the local zeta functions and their multiple connections in mathematics and theoretical physics. Local zeta functions are ubiquitous objects in mathematics and theoretical physics. At the mathematical level, local zeta functions contain geometry and arithmetic information about the set of zeros defined by a finite number of polynomials. In terms of applications in theoretical physics, these functions play a central role in the regularization of Feynman amplitudes and Koba-Nielsen-type string amplitudes, among other applications.
This volume provides a gentle introduction to a very active area of research that lies at the intersection of number theory, p-adic analysis, algebraic geometry, singularity theory, and theoretical physics. Specifically, the book introduces \(p\)-adic analysis, the theory of Archimedean, \(p\)-adic, and motivic zeta functions, singularities of plane curves and their Poincaré series, among other similar topics. It also contains original contributions in the aforementioned areas written by renowned specialists.
This book is an important reference for students and experts who want to delve quickly into the area of local zeta functions and their many connections in mathematics and theoretical physics.
The articles of this volume will be reviewed individually. Proceedings, conferences, collections, etc. pertaining to number theory, Zeta functions and \(L\)-functions, Singularities in algebraic geometry, Arcs and motivic integration, Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Non-Archimedean analysis, Modifications; resolution of singularities (complex-analytic aspects), Statistical mechanics of gases, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Proceedings of conferences of miscellaneous specific interest \(p\)-adic analysis, arithmetic and singularities. UIMP-RSME, Lluís A. Santaló summer school, Universidad Internacional Menéndez Pelayo, Santander, Spain, June 24--28, 2019 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(G\)-Hilbert schemes for finite group schemes \(G\). We find a construction of \(G\)-Hilbert schemes as relative \(G\)-Hilbert schemes over the quotient that does not need the Hilbert scheme of \(n\) points, works under more natural assumptions and gives additional information about the morphism from the \(G\)-Hilbert scheme to the quotient. \(G\)-Hilbert scheme; McKay correspondence Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Hopf algebras and their applications, Ordinary representations and characters Construction of \(G\)-Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\subset\text{SL}(n,\mathbb{C})\) be a finite subgroup and \(X=\mathbb{C}^n/G\) the quotient space. Let \(R\) be a common multiple of the orders of all elements of \(G\) and \(\Gamma=\text{Hom}(\mu_R,G)\). Then, \(\Gamma\) is isomorphic to \(G\), but the isomorphism depends on the choice of an \(R\)-th root of unity \(\varepsilon\). There is a naturally defined grading \(\Gamma=\bigcup_{i=0}^{n-1}\Gamma_i\), were \(\Gamma_i\) is defined as follows. Each element \(g\) of \(G\) has eigenvalues \(\varepsilon^{a_1},\dots,\varepsilon^{a_n}\) with \(a_1+\cdots+a_n\equiv 0\pmod R\) and \(0\leq a_j<R\). The integer \(i=(a_1+\cdots+a_n)/R<n\) is called the age of the element of \(\Gamma\) which maps \(\varepsilon\) to \(g\) and this defines the grading on \(\Gamma\). The elements of \(\Gamma_1\) are called junior elements and their \(G\)-conjugacy classes are the junior conjugacy classes.
Now let \(f\colon Y\to X\) be a resolution of \(X\) and write \(K_Y\equiv f^*K_X+\sum a_E E\) with exceptional prime divisors \(E\). The prime divisor \(E\) is called crepant, if \(a_E=0\). It is proved that there is a canonical one-to-one correspondence between junior conjugacy classes and crepant exceptional divisors. McKay correspondence; Euler number; crepant resolution; junior conjugacy classes; crepant exceptional divisors Ito, Y., Reid, M.: The McKay correspondence for finite subgroups of \({{\mathrm SL}(3, {\mathbf C})}\). In: Higher-Dimensional Complex Varieties (Trento, 1994), pp. 221-240. de Gruyter, Berlin (1996) Homogeneous spaces and generalizations, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) The McKay correspondence for finite subgroups of \(SL(3,\mathbb{C})\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Sei \(U\) ein Keim einer algebraischen Varietät über einem algebraisch abgeschlossenen Körper \(k\). Unter einem Bogen auf \(U\) versteht man eine parametrisierte formelle Kurve. Die Menge \(H\) der Bögen auf einem singulären Keim ist eine algebraische Varietät unendlicher Dimension. \textit{J. F. Nash jun.} [Duke Math. J. 81, No. 1, 31--38 (1995; Zbl 0880.14010)] definierte eine injektive Abbildung der Menge der irreduziblen Komponenten von \(H\) in die Menge der essentiellen Komponenten des exzeptionellen Divisors in einer Auflösung der Singularität und stellte die Frage nach der Surjektivität dieser Abbildung. \textit{S.~Ishii and J.~Kollár} [Duke Math. J. 120, 601--620 (2003; Zbl 1052.14011)] haben gezeigt, dass im allgemeinen dieses Nash-Problem eine negative Antwort besitzt.
In der vorliegenden Arbeit werden zwei hinreichende Bedingungen angegeben, die neue Beweise für eine positive Antwort im Falle minimaler Flächensingularitäten ermöglichen. Plénat, C., À propos du problème des arcs de Nash, Ann. inst. Fourier (Grenoble), 55, 3, 805-823, (2005) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Infinitesimal methods in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects), Singularities of surfaces or higher-dimensional varieties On Nash problem of arc. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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{K. Kato} [in: Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf., Baltimore/MD (USA) 1988, 191--224 (1989; Zbl 0776.14004)] extended the classical notion of toric geometry, the notion of toroidal embedding is replaced by a log structure, as defined by Fontaine and Illusie. Kato has defined log regular schemes and he has proved that fine saturated log regular schemes behave like toric varieties, he had defined fans and has shown that fan subdivisions can be used to solve singularities. The aim of the paper under review is to show that singularities of log-regular schemes can be resolved globally by only one log-blow-up, this is a similar result to the well known result that toroidal embeddings over the complex numbers can be resolved globally by equivariant blow-ups. The paper is written in a very technical language. log-regular scheme; blow-up; resolution of singularities; toric Nizioł, {W}iesława, Toric singularities: log-blow-ups and global resolutions, Journal of Algebraic Geometry, 15, 1-29, (2006) Global theory and resolution of singularities (algebro-geometric aspects), Embeddings in algebraic geometry, Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Toric singularities: log-blow-ups and global 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 A be a finitely generated graded \({\mathbb{C}}\)-algebra. There exist polynomials \(P_ 0,...P_{h-1}\in {\mathbb{Q}}[X]\) such that for large \(n: (*)\quad \dim_{{\mathbb{C}}}A_ n=P_ i(n)\) if \(n\equiv i\) (mod h). The ``index of regularity'' of the Hilbert function of A is the smallest integer a(A) such that (*) holds for all \(n>a(A)\) and all i. If \(Y=Spec(A)\) has an isolated singularity which is Gorenstein then its various plurigenera are computed in terms of a(A) as follows: (1) the \(L^ 2\)-plurigenus of order m is \(\sum_{i\leq ma(A)}\dim_{{\mathbb{C}}}A_ i;\) (2) the Log-plurigenus of order m is \(\sum_{i<ma(A)}\dim_{{\mathbb{C}}}A_ i.\) If, moreover, Y is a generic complete intersection, then the plurigenera are also computed in terms of the number of integral points in certain polyhedra arising out of the Newton polyhedra of the defining equations. The proofs depend upon an analysis of Demazure's construction of some normal cones and an explicit resolution process for a complete intersection singularity developed earlier by the author. index of regularity; Hilbert function; plurigenera; number of integral points; Newton polyhedra; resolution; complete intersection singularity Marcel Morales, Resolution of quasihomogeneous singularities and plurigenera, Compositio Math. 64 (1987), no. 3, 311 -- 327. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Complete intersections Resolution of quasi-homogeneous singularities and plurigenera | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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, Minimal model program (Mori theory, extremal rays), Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Fano varieties, Biographies, obituaries, personalia, bibliographies, History of mathematics in the 20th century, History of mathematics in the 21st century, Proceedings of conferences of miscellaneous specific interest, Festschriften Minimal models and extremal rays. Proceedings of the conference, RIMS, Kyoto, Japan, June 20--24, 2011 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 study constructive embedded resolutions of irreducible quasi-ordinary singularities in \(\mathbb{C}^3\). A surface germ \((V,p)\subset (\mathbb{C}^3,0)\) is a quasi-ordinary singularity if it admits a finite projection \(\pi:(V,p) \to(\mathbb{C}^2,0)\) such that the discriminant locus (i.e., the plane curve over which \(\pi\) ramifies) has only normal crossings. If \(f\) is such a singularity and is irreducible, it admits a parametrization (analogous to the Puiseux series of an irreducible algebroid plane curve) from which certain pairs of numbers, called the characteristic pairs, can be extracted. They are important in the study of the germ. For instance, explicit resolutions of such a singularity have been studied by \textit{J. Lipman} [in: Singularities, Summer Inst., Arcata 1981, Proc. Symp. Pure Math. 40, Part 2, 161-172 (1983; Zbl 0521.14014)]. In this process, which does not lead to embedded resolutions, an important role is played by the characteristic pairs. Recall that, informally, ``embedded resolution'' means a process where along which the singular variety \(V\) one transforms the ambient space where it is defined, so that eventually the strict transform of \(V\) is non-singular and its union with the exceptional divisor is locally defined by simple, ``nice'' equations. Recently, several (closely related) methods to constructively (or canonically) obtain embedded resolutions of general singularities have been proposed. That is, procedures which involve a finite sequence of blowing-ups and which tell us, at each stage of the resolution process, how to choose the center of the transformation.
More precisely, in this note the authors explicitly study what results from the application of the general canonical process devised by \textit{E. Bierstone} and \textit{P. D. Milman} [Invent. Math. 128, No. 2, 207-302 (1997; Zbl 0896.14006)] to an irreducible quasi-ordinary singularity. It turns out that the process depends on the (suitably normalized) characteristic pairs of the singularity only. The description of the algorithm given in the paper is very explicit. The authors apply it to a non-trivial example, and they affirm that the method can be actually implemented by a computer.
The authors plan to apply similar techniques to the problem of simultaneous desingularization of a family of quasi-ordinary singularities in a future work. canonical resolution; quasi-ordinary singularities; characteristic pairs; embedded resolutions C. Ban and L. J. McEwan, Canonical resolution of a quasi-ordinary surface singularity , Canad. J. Math. 52 (2000), 1149--1163. Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Complex surface and hypersurface singularities Canonical resolution of a quasi-ordinary surface 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 \(Q=(Q_0,Q_1)\) be a quiver and \(\mathbf d=(d_i)_{i\in Q_0}\) be a dimension vector. Let \(\mathrm{rep}_Q(\mathbf d)\) be the space of \(\mathbf d\)-dimensional representations of \(Q\) over a fixed algebraically closed field \(k\). The group \(\mathrm{GL}(\mathbf d):=\bigoplus\mathrm{GL}(d_i,k)\) acts on \(\mathrm{rep}_Q(\mathbf d)\) by conjugation, and the orbits correspond to isomorphism classes of \(\mathbf d\)-dimensional representations of \(Q\). For \(N\in\mathrm{rep}_Q(\mathbf d)\) we denote its orbit by \(\mathcal O_N\) and its Zariski closure by \(\overline{\mathcal O}_N\).
Denote by \(kQ\) the path algebra for \(Q\), and let \(\mathrm{Ann}(N)\) be the annihilator ideal of \(N\) in \(kQ\). Suppose \(\mathrm{Ann}(N)\) is non-zero and admissible (that is, \(\mathcal R_Q^r\subseteq\mathrm{Ann}(N)\subseteq\mathcal R_Q^2\) for some \(r\), where \(\mathcal R\) is the ideal of \(kQ\) generated by \(Q_1\)). The main result of the work under review is that if \(\overline{\mathcal O}_N\) is a hypersurface (necessarily singular since \(\mathrm{Ann}(N)\neq 0\)), then \(\overline{\mathcal O}_N\) is a normal variety. The proof uses a result by Serre: that \(\overline{\mathcal O}_N\) is normal if and only if the singular locus is a closed subvariety of \(\overline{\mathcal O}_N\) of codimension at least 2. quiver representations; finite-dimensional representations; normality of orbit closures; hypersurfaces Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry On the normality of orbit closures which are hypersurfaces. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author's approach [\textit{G. Faltings}, J. Reine Angew. Math. 483, 183-196 (1997; Zbl 0871.14012)] to resolve singularities of moduli spaces of Abelian varieties relies on some special features of the level structures considered there. Here the author tries a new approach which is more widely applicable, but gives weaker results. The paper is, as the reviewer supposes, not to be regarded as definitive but is rather in the nature of a progress report; but it introduces important ideas.
The singularities studied are those of the moduli space of degree \(p^g\) isogenies \(\phi: A\to B\) of \(g\)-dimensional Abelian varieties factored into a sequence of isogenies of degree \(p\). This moduli problem is representable over \({\mathbb Z}_p\). The local singularities are described, after Deligne and Pappas, in terms of the Hodge filtration on the universal sequence of Dieudonné modules. More concretely one fixes a discrete valuation ring \(V\) with uniformiser \(\pi\) and a complete flag
\[
N_g=\pi N\subset N_{g-1}\subset\ldots\subset N_0=N\cong V^g
\]
and examines the projective scheme \(X\) over \(V\) parametrising compatible families \(F_i\) of direct summands of \(N_i\) of rank \(a\). By studying suitable arrangements of lattices the author constructs a space \(Y\) which (for \(a=2\) at least) has toroidal singularities, and is equipped with a birational morphism to \(X\). This \(Y\) is an example of what is here referred to as a minimal model (for the Deligne scheme). The arrangements fit into a universal family over a suitable embedding of \(\text{PGL}(d)^r/\text{PGL}(d)\), endowed with a logarithmic structure induced by maps to a scheme with toroidal singularities: Unfortunately these maps, which coincide with those of Lafforgue, appear not to be smooth (contrary to a hope expressed, rather tentatively, by the author in the introduction). Shimura variety; toroidal resolution; minimal model Faltings, G.: Toroidal resolutions for some matrix singularities. In Moduli of Abelian Varieties (Texel Island, 1999), Volume 195 of Progr. Math., pp. 157-184. Birkhäuser, Basel (2001) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Algebraic moduli of abelian varieties, classification, Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties, Homogeneous spaces and generalizations, Toric varieties, Newton polyhedra, Okounkov bodies Toroidal resolutions for some matrix 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 finite subgroup of SL\((n,\mathbb{C})\), then the quotient \(\mathbb{C}^n/G\) has a Gorenstein canonical singularity. If \(n=2\) or \(3\), it is known that there exist crepant resolutions of the quotient singularity. In higher dimension, there are many results which assume existence of crepant resolutions. However, few examples of crepant resolutions are known.
In this paper, we will show several trials to obtain crepant resolutions and give a conjecture on existence of crepant resolutions. quotient singularity; McKay correspondence; Hilbert scheme; Crepant resolution; Gröbner basis; toric variety McKay correspondence, Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Existence of 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 \textit{V. Ginzburg} and \textit{D. Kaledin} [Adv. Math. 186, No. 1, 1--57 (2004; Zbl 1062.53074)] posed the problem of comparing the McKay correspondence, the dual McKay correspondence and the multiplicative McKay correspondence for a finite dimensional \(\mathbb C\)-vector space \(V\), with an action of a finite subgroup \(G\) of SL\((V)\). The vector space \(V\) is assumed to be equipped with a symplectic form, which is preserved by \(G\). Moreover a crepant resolution of singularities \(Y\to V/G\) is fixed. Ginzburg and Kaledin proposed to compute explicitly the Poincaré isomorphism and the Chern character isomorphism (see Problems 1.4 and 1.5 of the above quoted article).
In the paper under review, the author solves the problem in the special case when \(V=\mathbb C^n\otimes \mathbb C^2\), with the action by permutations of the symmetric group \(S_n\) and the canonical symplectic form. In this situation \(Y\) is the Hilbert scheme Hilb\(^n(\mathbb C^2)\). The author gives explicit formulae for the Poincaré and the Chern character isomorphisms and uses them to prove the main theorem of the paper (Theorem 1.2). It says that the McKay correspondence is compatible with the topological filtration of the Grothendieck group \(K(\text{Hilb}^n(\mathbb C^2))\) and with the decreasing filtration of the space of symmetric functions \(\Lambda^n\). It is then proved that the graded McKay correspondence so obtained coincides with both the multiplicative and the dual McKay correspondence. symmetric functions; equivariant cohomology; Macdonald polynomials S. Boissière, On the McKay correspondences for the Hilbert scheme of points on the affine plane, arXiv:math.AG/0410281. Parametrization (Chow and Hilbert schemes), Symmetric functions and generalizations, Symmetric groups, Equivariant homology and cohomology in algebraic topology, Global theory and resolution of singularities (algebro-geometric aspects) On the McKay correspondences for the Hilbert scheme of points on 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 If one has an ample line bundle over a projective variety \(X\) with a group action, one may contract the zero section and then divide the cone by the group-action. These varieties are called cone quotients of \(X\). \textit{Pinkham} showed that analytically every quasihomogeneous normal surface singularity is a cone quotient of a curve and that the resolution graph is star-shaped. This holds also algebraically for any characteristic and the group may be chosen abelian. One gets a bijection between quasihomogeneous normal surface singularities and the data of the resolution in the algebraic category. quasihomogeneous normal surface singularity; resolution graph Runge, B.: Quasihomogeneous singularities. Math. ann 281, 295-313 (1988) Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Quasihomogeneous 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 study the Zariski closures of orbits of representations of quivers of type \(A\), \(D\) or \(E\). With the help of Lusztig's canonical base, we characterize the rationally smooth orbit closures and prove in particular that orbit closures are smooth if and only if they are rationally smooth. quantum groups; representations of quivers; singularities; canonical basis Caldero, P., Schiffler, R.: Rational smoothness of varieties of representations for quivers of Dynkin type. Ann. Inst. Fourier 54(2), 295--315 (2004) Quantum groups (quantized enveloping algebras) and related deformations, Representations of quivers and partially ordered sets, Singularities in algebraic geometry, Group actions on varieties or schemes (quotients) Rational smoothness of varieties of representations for quivers of Dynkin 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 Let \(K\) be a field, and suppose that the characteristic of \(K\) is not equal to \(3\). Let \(R = K[[x,y]]/(x^t +y^3)\) be the complete local ring of the plane curve singularity defined by \(x^t +y^3 =0\). Maximal Cohen-Macaulay \(R\)--modules \(N\) are studied such that \(N/yN\) is isomorphic to a direct sum of copies of \(R/(y,x^i)\), \(R/(y,x^{t-1})\), for \(1 \leq i < \frac{t}{2}\), \( t \not= 3i\). Families are constructed containing all these modules. For the set of maximal Cohen-Macaulay modules with \(\min\{i,t-2i\} =3\) the moduli space of these Cohen-Macaulay modules is constructed. The problem is translated into a moduli problem of matrices under the action of a unipotent and a reductive group. The corresponding theory of \textit{D. Mumford} (for reductive groups) and \textit{G.-M. Greuel} and \textit{G. Pfister} (for unipotent groups) is applied to construct the geometric quotient. Cohen-Macaulay modules; singularities; geometric quotient Pfister G., Comm.in Algebra 27 (6) pp 2555-- (1999) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Singularities of curves, local rings, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) A family of Cohen-Macaulay modules over singularities of type \(X^t+Y^3\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main result of the paper provides an explicit description of an embedded resolution of the hypersurface germ \((\{f(x,y)+z^2=0\},0)\subset (\mathbb C^3,0)\) for an arbitrary \(f:(\mathbb C^2,0)\to (\mathbb C,0)\). The topological data of the exceptional divisors, their intersections and embeddings is codified in a (two-dimensional) ``resolution graph''. embedded resolution; exceptional divisors; hypersurface singularities; Jung strategy; quasi-ordinary singularities; ruled surfaces Ban, C; McEwan, LJ; Némethi, A, The embedded resolution of \(f(x, y)+z^2 (\mathbb{C}^3,0) ### (\mathbb{C},0)\), Studia Scientiarum Mathematicarum Hungarica, 38, 51-71, (2001) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects) The embedded resolution of \(f(x,y)+z^2:(\mathbb C^3,0)\rightarrow (\mathbb C,0)\) | 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 \((X,\mathfrak a)\) be a pair consisting of \(X\), a variety with klt singularities, and \(\mathfrak a=\mathfrak a _1^{q_1}\cdots \mathfrak a _r^{q_r}\), a formal product of ideals \(\mathfrak a_i\subset \mathcal O _X\) with positive real coefficients \(q_i\in \mathbb R _{>0}\). The log canonical threshold \(\text{lct}_X (\mathfrak a )\) is the largest \(t>0\) such that \((X,\mathfrak a^t)\) is log canonical. The log canonical threshold is a subtle invariant measuring the singularities of the pair \((X,\mathfrak a)\). It is of fundamental importance in higher dimensional birational geometry. A conjecture of Shokurov predicts that if we fix \(n=\dim X\) and a set \(A\subset \mathbb R _{>0}\) satisfying the descending chain condition, then the set of all possible log canonical thresholds \(\text{lct}_X (\mathfrak a )\) with \(q_i\in A\) satisfies the ascending chain condition.
In this paper, the authors prove that Shokurov's conjecture holds when \(X\) has bounded singularities. This generalizes a similar result of the authors in the context of smooth varieties (or, more generally, locally complete intersection varieties) [Duke Math. J. 152, No. 1, 93--114 (2010; Zbl 1189.14044)]. log canonical threshold de Fernex, Tommaso; Ein, Lawrence; Mustaţă, Mircea, Log canonical thresholds on varieties with bounded singularities.Classification of algebraic varieties, EMS Ser. Congr. Rep., 221-257, (2011), Eur. Math. Soc., Zürich Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays) Log canonical thresholds on varieties with bounded singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the introduction to the article the author writes: ``This account shall serve as a quick guide to the historical development, the main contributors and the basic notions in the context of resolution of singularities. For detailed information we give references to the literature for each item. The presentation, which is necessarily subjective, does not claim completeness or utmost rigor. It shall merely help the reader to access the field and to find further sources.''
The article is very compressed and appears to be more a dictionary than an account of the history of the resolution of singularities. It is divided into several short sections: Main achievements; Some research problems; Contributors; Dictionary; Surveys; Miscellaneous; Selection of references before 1930; Selection of references after 1930.
Wisely the author preempts much of the criticism that can be raised to his selection of material with remarks as ``We list a selections of \dots'', ``The following list is far from complete \dots'', ``For technical and human reasons not all relevant contributions could be included.'' It is clear that there is no hope of writing a complete overview of a vast area like resolutions of singularities. On the other hand it would have been nice to get a suggestion of how the selection of the material was made. Is it just a result of ``human limitations'' or is it a conscious choice? It would also be of use to have some comments on related topics, like the resolutions of special varieties, that may not be considered, by the experts, as belonging to the field proper. The author has succeeded in writing a short and concise guide to students, and experts from other parts of algebraic geometry. Resolutions of singularities is a vast area of mathematics that has contributed to the development of several parts of algebraic geometry and algebra. The present article will make it considerably easier to access the area, and is a real service to the mathematical community. Among the many useful items of the article is a list of problems that, although quite specialized, will inspire younger mathematicians to work in the area. resolution of singularities; curves; surfaces; threefolds Hauser [Hauser 00] H., Resolution of Singularities (2000) History of algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), History of mathematics in the 19th century, History of mathematics in the 20th century, Singularities in algebraic geometry, Global theory of complex singularities; cohomological properties, Modifications; resolution of singularities (complex-analytic aspects), Singularities of curves, local rings, Singularities of surfaces or higher-dimensional varieties Resolution of singularities 1860--1999 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The purpose of the course was to provide a simple introduction to the resolution of singularities of curves in analytic geometry and to the basic invariants of these singularities. However, the course is not elementary or self contained in the usual sense that it uses only simple notions and results. On the contrary, taking advantage of the existence nowadays of excellent treatises of complex analytic geometry, the author freely refers to them for some results, for example on normalization. His hope is to entice the reader to study them carefully by showing some applications of some of the material they contain in the fairly intuitive context of singularities of curves. resolution of singularities; curve singularities B. Teissier, Introduction to curve singularities, Singularity theory (Trieste, 1991), World Scientific Publishing (1995), 866-893. Singularities of curves, local rings, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Introduction to curve singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities O. Zariski : Le Problème des Modules pour les Branches Planes . Course given at Centre de Mathematiques de l'Ecole Polytechnique (1973). Global theory and resolution of singularities (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles, Singularities in algebraic geometry, Families, moduli of curves (algebraic), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Power series rings Le problème des modules pour les branches planes. Rédigé par François Kmety et Michel Merle. Avec un appendice de Bernard Teissier | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 study of (algebraic) complete intersections in \(\mathbb{C}^n\) or of germs of analytic complete intersections in \((\mathbb{C}^n,0)\) a very powerful tool is the use of toric varieties, associated to the corresponding Newton polyhedra. The aim of the paper is to obtain results concerning resolutions of singularities and smooth compactifications for complete intersections as above. The results hold in particular when the coefficients of the polynomial equations (resp. analytical equations) are sufficiently general (assuming here that the Newton polyhedra are fixed). A first part of the paper contains a detailed construction of toric varieties (smooth case), involving less algebraic formalism as in the original one [cf. \textit{G. Kempf}, \textit{F. Knudsen}, \textit{D. Mumford} and \textit{B. Saint-Donat}, Toroidal embeddings. I. Berlin etc.: Springer Verlag (1973; Zbl 0271.14017)]. complete intersections; toric varieties; Newton polyhedra; resolutions of singularities A. G. Khovanskii, ''Newton polyhedra (resolution of singularities),'' in:J. Sov. Math.,27, 2811--2830 (1984). Global theory and resolution of singularities (algebro-geometric aspects), Implicit function theorems; global Newton methods on manifolds, Complete intersections, Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects), Complex singularities Newton polyhedra (resolution of singularities) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The aim of this work is to prove the following theorem: Let \(Y\) be a projective variety of dimension 3 over a field \(k\) such that there exists a regular projective variety \(X\) birationally equivalent to \(Y\). Then, there exists a projective morphism \(\pi :X'\to Y\), where \(X'\) is a regular projective variety and \(\pi\) is an isomorphism above \(Y_{reg}\). -- With the help of short references [the author, Duke Math. J. 63, No. 1, 57-64 (1991); the author, \textit{J. Giraud} and \textit{U. Orbanz}, ``Resolution of surface singularities'', Lect. Notes Math. 1101 (1984; Zbl 0553.14003); and \textit{J. Lipman} in Algebraic Geometry, Proc. Sympos. Pure Math. 29, Arcata 1974, 187-230 (1975; Zbl 0306.14007)], this theorem leads to a proof of desingularization in dimension 3 and characteristic \(\geq 7\).
The proof proceeds in three steps. In I.1, we see that we can assume that the birational morphism \(X..\to Y\) is defined everywhere. In III, we build two birational projective morphisms \(p:Y\to Y\) and \(\widetilde \pi :\widetilde X\to\widetilde Y\), with \(\widetilde X\) regular (\(\widetilde Y\) is in general not regular) such that \(p\) is an isomorphism above \(Y_{reg}\) and the indetermination locus of \(\widetilde \pi^{-1}\) is the union of two disjoint closed subsets \(F_ 1\) and \(F_ 2\) with \(p(F_ 1)\subset\text{Sing}(Y)\). In II, we see that we can modify \(\widetilde \pi\) and \(\widetilde Y\) to get \(F_ 2=\emptyset\). Then the modified \(\widetilde \pi\circ p\) and \(\widetilde Y\) give the desingularization of \(Y\). desingularization; threefold birationally equivalent to regular variety Cossart, V.: Modèle projectif régulier et désingularisation, Math. ann. 293, No. 1, 115-122 (1992) Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, \(3\)-folds, Singularities in algebraic geometry Regular projective model and desingularization. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 higher dimensional geometry resolutions are not unique. In dimension two there are still minimal resolutions, but embedded resolutions are already not unique. Threefold theory suggests to look at embedded canonical models, for which the ambient space has also at most canonical singularities. In this paper toric methods are used to construct such models for singularities, defined by functions which are nondegenerate for their Newton diagrams.
The paper first describes the Newton blow-up, the modification defined by the Newton diagram, in terms of an equivariant normalised blow-up, in arbitrary dimensions. The main results concern the surface case; let \(f\to k\) be nondegenerate for its Newton diagram, put \(S=\{f=0\}\), let \(\hat V\to V\) be the Newton blow-up, and let \(\hat S\) be the strict transform of \(S\). The authors prove that \(\hat S\) has only \(A_ k\)- singularities in smooth points of Sing\(\hat V\), and they describe in detail how the surface \(\hat S\) intersects the exceptional set (transversally, which has to be explained in the presence of singular points). This embedded canonical model \(\hat S\subset\hat V\) of \(S\subset V\) is minimal for toric morphisms.
As example the authors work out the case of \(E_ 6\), which has five terminal embedded resolutions, related by flops, but the toric canonical model is unique. \(E_ 6\); models for singularities; embedded canonical model; toric morphisms; flops Gonzalez-Sprinberg (G.) and Lejeune-Jalabert (M.).- Modèles canoniques plongés. I. Kodai Math. J., 14(2) p. 194-209 (1991). Zbl0772.14008 MR1123416 Global theory and resolution of singularities (algebro-geometric aspects), \(3\)-folds, Singularities of surfaces or higher-dimensional varieties, Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry Modèles canoniques plongés. I. (Embedded canonical models. I) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The classification of terminal singularities in dimension 3 is reduced, by a result of M. Reid, to an analysis of quotients of smooth and double points by cyclic group actions.
After the classification of cyclic Gorenstein quotients in dimension \( 4\) [\textit{D. R. Morrison} and \textit{G. Stevens}, Proc. Am. Math. Soc. 90, 15- 20 (1984; Zbl 0536.14003]; \textit{S. Mori}, \textit{D. R. Morrison} and \textit{I. Morrison} tried to describe, in dimension \( 4,\) all terminal cyclic quotient [Math. Comput. 51, No.184, 760-786 (1988)] and an extensive computer study of such singularities lead them to a conjectural classification.
The author proves a part of this conjecture. The method of the proof is elementary and based on astute and complicated calculations. Only a part of the proof is given and the full details may be obtained from the author. No computer has been used. terminal singularities; terminal cyclic quotient G. K. Sankaran, ''Stable quintuples and terminal quotient singularities,''Math. Proc. Cambridge Philos. Soc.,107, No. 1, 91--101 (1990). Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), \(4\)-folds, Modifications; resolution of singularities (complex-analytic aspects) Stable quintuples and terminal quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author proves the following theorem: Fix an infinite base field \(k\) of characteristic \(p >0\), let \(X\) be a subscheme of the regular \(k\)- scheme \(Z\), and let \(x\in X\) be a closed point such that \(X\) has multiplicity \(e(x)<p\) at \(x\). Then there exists, in a neighborhood of \(x\), a regular subscheme \(W\) of \(Z\) which has maximal contact with \(X\) in \(x\) and which has dimension \(\leq\dim X\).
\textit{J. Giraud} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 8, 201--234 (1975; Zbl 0306.14004)] had proved a similar result, showing the existence of a subscheme \(W'\) as above, but which is not in general regular. In the present paper it is shown that Giraud's \(W'\) is contained in a regular \(W\) with the same property.
Using a result of \textit{S. S. Abhyankar} [``Resolution of singularities of embedded algebraic surfaces'', Pure Appl. Math. 24. New York etc.: Academic Press (1966; Zbl 0147.20504)] it follows that every projective 3-dimensional variety (over a field of positive characteristic \(p\)) is birationally equivalent to one which can be desingularized. Samuel stratum; desingularization of threefold; prime characteristic; maximal contact Cossart, V, Contact maximal en caractéristique positive et petite multiplicite, Duke Math. J., 63, 57-64, (1991) Global theory and resolution of singularities (algebro-geometric aspects), \(3\)-folds, Finite ground fields in algebraic geometry, Singularities in algebraic geometry Maximal contact in positive characteristic and small multiplicity. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities As stated in the title we give an affirmative answer to Riemenschneider's conjecture that isolated quotient singularities are stable under deformations. Of course, higher dimensional isolated quotient singularities being rigid (Schlessinger), it is enough to consider the two dimensional case. The proof relies on a characterization of quotient singularities in terms of ''integral parts of \({\mathbb{Q}}\)-divisors'' with support in the exceptional locus of a desingularization. rigid quotient singularities; deformation; exceptional locus of a desingularization Esnault, Two dimensional quotient singularities deform to quotient singularities, Math Ann pp 271-- (1985) Deformations of singularities, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Formal methods and deformations in algebraic geometry Two dimensional quotient singularities deform to quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This note gives a sketch of the proof of the following theorem: \(Let\quad X\) be a 3-dimensional normal complex algebraic variety with at most canonical singularities. Then the ring \(\oplus_{m\geq 0}{\mathcal O}_ X(m.D)\quad is\) finitely generated, where D is a Weil divisor on X.
This result is then applied to prove the existence of minimal models for on-parameter-families of surfaces of non-negative Kodaira-dimension whose degenerate members are reduced and have only simple normal crossings. blowing-ups; threefold; canonical singularities; minimal models for on- parameter-families of surfaces Y. Kawamata, Crepant blowing-ups of \(3\)-dimensional canonical singularities and its application to degenerations of surfaces , Tokyo University, preprint. JSTOR: Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), \(3\)-folds, Divisors, linear systems, invertible sheaves, Singularities in algebraic geometry, Families, moduli, classification: algebraic theory On the crepant blowing-ups of canonical singularities and its application to degenerations of 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 Authors' abstract: In the 70's Igusa developed a uniform theory for local zeta functions and oscillatory integrals attached to polynomials with coefficients in a local field of characteristic zero. In the present article this theory is extended to the case of rational functions, or, more generally, meromorphic functions \(f/g\), with coefficients in a local field of characteristic zero. This generalization is far from being straightforward due to the fact that several new geometric phenomena appear. Also, the oscillatory integrals have two different asymptotic expansions: the ``usual'' one when the norm of the parameter tends to infinity, and another one when the norm of the parameter tends to zero. The first asymptotic expansion is controlled by the poles (with negative real parts) of all the twisted local zeta functions associated to the meromorphic functions \(f/g-c\), for certain special values \(c\). The second expansion is controlled by the poles (with positive real parts) of all the twisted local zeta functions associated to \(f/g\). oscillatory integrals; local zeta functions; asymptotic expansions; motivic zeta functions; exponential sums; meromorphic functions Veys, W.; Zúñiga-Galindo, WA, Zeta functions and oscillatory integrals for meromorphic functions, Adv. Math., 311, 295, (2017) Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Singular and oscillatory integrals (Calderón-Zygmund, etc.), Zeta functions and \(L\)-functions, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Zeta functions and oscillatory integrals for meromorphic functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The aim of this work is to prove the following theorem: Let \(Y\) be a projective variety of dimension 3 over a field \(k\) such that there exists a regular projective variety \(X\) birationally equivalent to \(Y\). Then, there exists a projective morphism \(\pi :X'\to Y\), where \(X'\) is a regular projective variety and \(\pi\) is an isomorphism above \(Y_{reg}\). --- With the help of short references [ the author, Duke Math. J. 63, No. 1, 57-64 (1991); the author, \textit{J.~Giraud} and \textit{U.~Orbanz}, ``Resolution of surface singularities'', Lect. Notes Math. 1101 (1984; Zbl. 553.14003); and \textit{J.~Lipman} in Algebraic Geometry, Proc. Sympos. Pure Math. 29, Arcata 1974, 187-230 (1975; Zbl. 306.14007)], this theorem leads to a proof of desingularization in dimension 3 and characteristic \(\geq 7\).
The proof proceeds in three steps. In I.1, we see that we can assume that the birational morphism \(X..\to Y\) is defined everywhere. In III, we build two birational projective morphisms \(p:Y\to Y\) and \(\widetilde \pi :\widetilde X\to\widetilde Y\), with \(\widetilde X\) regular (\(\widetilde Y\) is in general not regular) such that \(p\) is an isomorphism above \(Y_{reg}\) and the indetermination locus of \(\widetilde \pi^{-1}\) is the union of two disjoint closed subsets \(F_1\) and \(F_2\) with \(p(F_1)\subset{\mathrm Sing}(Y)\). In II, we see that we can modify \(\widetilde \pi\) and \(\widetilde Y\) to get \(F_2=\emptyset\). Then the modified \(\widetilde \pi\circ p\) and \(\widetilde Y\) give the desingularization of \(Y\). See the preview in Zbl. 735.14011 desingularization; threefold birationally equivalent to regular variety Cossart V.: Modèle projectif régulier et désingularisation. Math. Ann. 293(1), 115--122 (1992) Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, \(3\)-folds, Singularities in algebraic geometry Regular projective model and desingularization | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 motivic zeta function \(Z_{\text{mot}}(f;s)\) of \textit{J. Denef} and \textit{F. Loeser} [J. Alg. Geom. 7, 505--537 (1998; Zbl 0943.14010)] is essentially a singularity invariant, associated to a non-constant polynomial \(f \in \mathbb C[x_1,\dots,x_n]\). There is an explicit formula in terms of an embedded resolution \(h\) of \(\{f=0\}\) in affine \(n\)-space; we have in particular that each irreducible component of \(h^{-1}\{f=0\}\) induces a candidate pole of \(Z_{\text{mot}}(f;s)\) and that each pole is obtained in this way. For \(n=2\) there is a geometric criterion to decide whether such a candidate pole is really a pole [\textit{W. Veys}, Manuscr. Math. 87, 435--448 (1995; Zbl 0851.14012)]. Finding nice geometric conditions in higher dimensions, assuring that a candidate pole is really a pole, is a difficult problem. The author developed a conceptual condition, valid in arbitrary dimension [J. Reine Angew. Math. (to appear)].
In this paper he proves a geometric criterion of a different flavour for \(n=3\). It is stated roughly as follows. Fix a non-rational exceptional surface \(E\) of the resolution \(h\) which is mapped to a point by \(h\). If the open part of \(E\), that doesn't belong to any other component of \(h^{-1}\{f=0\}\), is of log general type, then generically the candidate pole associated to \(E\) is a pole. The author explains why it is natural to consider this condition of maximal logarithmic Kodaira dimension, in particular by reformulating the result for \(n=2\).
More precisely the result is in fact proven for the more manageable and concrete \textsl{Hodge zeta function} of \(f\), which is a priori a stronger result than for the motivic one. Hodge and motivic zeta functions; poles Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Modifications; resolution of singularities (complex-analytic aspects) Logarithmic Kodaira dimension and the poles of the Hodge and motivic zeta functions for surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a field. Spivakovsky's theorem [\textit{M. Spivakovsky}, in: Arithmetic and geometry, Vol. II, Prog. Math. 36, 419--432 (1983; Zbl 0531.14009)] on the solution of Hironaka's polyhedral game was extended by \textit{S. Bloch} [J. Algebr. Geom. 3, 537--568 (1994; Zbl 0830.14003)] to show that a morphism \(f\colon Z \rightarrow S\) of finite type \(k\)-schemes can be put in good position with respect to a normal crossing divisor \(\partial S\) on \(S\) by taking the proper transform with respect to an iterated blowing-up of faces of \(\partial S\). We extend these results to schemes of finite type over a regular scheme of dimension one, including the case of mixed characteristic. Global theory and resolution of singularities (algebro-geometric aspects), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Rational and birational maps, Singularities in algebraic geometry Blowing up 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 We present our recent understanding on resolutions of Gorenstein orbifolds, which involves the finite group representation theory. We concern only the quotient singularity of hypersurface type. The abelian group \(A_r(n)\) for \(A\)-type hypersurface quotient singularity of dimension \(n\) is introduced. For \(n=4\), the structure of the Hilbert scheme of group orbits and crepant resolutions of \(A_r(4)\)-singularity are obtained. The flop procedure of 4-folds is explicitly constructed through the process. crepant resolutions; Gorenstein quotient singularities; McKay correspondence Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Singularities of surfaces or higher-dimensional varieties Orbifolds and finite group 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 give an exposition on the one-step desingularization problem by using higher Semple-Nash blowups and \(F\)-blowups. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Higher Semple-Nash blowups and \(F\)-blowups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 few years ago H. Kawanoue announced a program (the \textit{Idealistic Filtration Program}) to resolve singularities of algebraic varieties which might work even if the base field is of positive characteristic. It is expressed in terms of \textit{idealistic filtrations}. If \(U\) is an open set of an affine variety \(W\), with coordinate ring \(R\), an idealistic filtration on \(W\) (or on \(R\)) is an indexed family of ideals \(I_a\) of \(R\), where \(a\) is a non-negative real number, such that \(I_a \subset I_b\) if \(a \geq b\), \(I_a I_b \subset I_{a+b}\) and \(I_0=R\). One defines the support, or singular locus, of an idealistic filtration. Then the goal is to give an algorithm allowing us to resolve the filtration, i.e., to find blowing-ups with smooth centres such that, taking suitable transforms, eventually the support becomes empty. The centres will be given as the set of points where certain \textit{algorithmic resolution functions}, with values in a suitable totally ordered set, reach the maximum value. From this, a similar algorithm to resolve singularities of varieties will follow.
To construct such resolutions functions, given an idealistic filtration \(\mathcal I\) on the variety \(W\) a \textit{strand of invariants} is attached to each point \(P\) of \(W\). Each step of the strand involves a triplet \(\sigma\), \(\tilde {\mu}\) and \(s\) of invariants. In the definition of \(\sigma\) and \(\tilde \mu\) a fundamental role is played by \(L(\mathcal I)\), a certain graded sub-algebra of \(Gr_{M}(A)\), the graded ring of \((A,M)\) (the local ring of \(W\) at a closed point \(P\)). Of particular importance is the notion of \textit{leading generator system}. These are elements in suitable ideals \(I_j\) of \(\mathcal I\) whose initial forms in the graded ring \(Gr_{M}(A)\) are generators of \(L(\mathcal I)\), satisfying certain ``minimality'' conditions. This notion seems essential to avoid the explicit use of hypersurfaces of maximal contact, a usual tool in desingularization techniques which is not always available in positive characteristic. Leading generator systems exist when the filtration \(\mathcal I\) is \(D\)-saturated, that is closed under the application of differential operators.
The resolution functions thus obtained have the fundamental property necessary in problems of this type. Namely, their values strictly decrease after performing a blowing-up with the appropriate algorithmic center. But still, when the characteristic is positive, there is no warranty that the process terminates.
Kawanoue, working in cooperation with K. Matsuki, plans to present the details of the program in a series of four papers. In the first one, \textit{H. Kawanoue} [Publ. Res. Inst. Math. Sci. 43, 819--909 (2007; Zbl 1170.14012)], discussed the main lines of the program, in particular what was mentioned so far. In the second part of the series (the article under review) the authors carefully study properties of the invariants \(\sigma\) and \(\tilde \mu\) (at ``time zero'', i.e., at the beginning of the process). For instance, they check the upper-semicontinuity of the invariants \(\sigma\) and \(\tilde \mu\). They also discuss certain power series expansions, but nor necessarily in terms of a regular system of parameters, but rather relative to a leading generating system; something that will be important later. Many of these results had been announced (without proof) in Part I.
In an Appendix, they include an important improvement of a result in Part I. Over there, they prove what they call \textit{the new non-singularity principle}. It says (among other things) that given an idealistic filtration \(\mathcal I\) which is both \(D\)-saturated and \(R\)-saturated (i.e., closed under the operations of differentiation and extraction of roots) and satisfies, for some closed point, \(\tilde \mu (z) = \infty\), then on a neighbourhood of \(z\) the support of \(\mathcal I\) is non-singular. From this result it follows that the centres involved in the application of the algorithm are indeed regular, which is a non-trivial result. In this Appendix a similar result is proved, but avoiding the hypothesis of \(R\)-saturation. This improvement could be an important step toward proving that their algorithm (or a slight variation thereof) will terminate after a finite number of steps, thus concluding a proof of resolution in positive characteristic. Indeed, the assumption on \(R\)-saturation seems to be an obstacle to prove the termination of the algorithm.
A subsection (numbered 0.3) resumes the current status of the Idealistic Filtration Program. idealistic filtration; leading generator system; invariants; power series expansion; positive characteristic H.KAWANOUEand K.MATSUKI,\textit{Toward resolution of singularities over a field of positive charac-} \textit{teristic (the idealistic filtration program) Part II}: \textit{Basic invariants associated to the idealistic filtration} \textit{and their properties}, Publ. Res. Inst. Math. Sci. 46 (2010), no. 2, 359--422.http://dx.doi.org/ 10.2977/PRIMS/12.MR2722782 Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Local theory in algebraic geometry, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Toward resolution of singularities over a field of positive characteristic (The idealistic filtration program). II: Basic invariants associated to the idealistic filtration and their properties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(M_0\) be a smooth complex threefold with trivial canonical bundle \(\omega_{M_0}\) acted on by a finite group \(G\) of automorphisms acting trivially on \(\omega_{M_0}\). A conjecture of Dixon-Harvey-Vafa-Witten says that there always exists a desingularization \(M_0/G\) with trivial canonical bundle, and predicts its Euler number. This conjecture has a local form from which it follows: If \(G\) is a finite subgroup of \(SL(3,{\mathbb{C}})\), there exists a crepant (i.e. with trivial canonical bundle) desingularization of \({\mathbb{C}}^3/G\), and its Euler number is the number of conjugacy classes of \(G\). The conjecture is now completely solved: There is a list of all finite subgroups of \(SL(3,{\mathbb{C}})\) due to Miller, Blichfeldt and Dickson, and a case-by-case analysis, of which this paper is part, shows that there always exists a crepant resolution, constructed by an explicit sequence of blow-ups. The formula for the Euler number follows from these explicit constructions; it also has an independent general proof by the generalization of the MacKay correspondence to dimension \(3\) by \textit{Y. Ito} and \textit{M. Reid}, in: Higher-dimensional complex varieties, Proc. Int. Conf., Trento 1994, 221-240 (1996; Zbl 0894.14024). The construction of a crepant resolution of \({\mathbb{C}}^3/H_{168}\) is pretty straightforward: One makes \(4\) successive blow-ups of singular curves until all the singular points disappear. The fact that the resolution obtained in this way is crepant follows from a remark of Reid (1979). Calabi-Yau threefolds; McKay correspondence; crepant resolutions; complex threefold; finite group of automorphisms; Euler number G. Markushevich, ''Resolution of \(\mathbb{C}\)3/H 168,''Math. Ann.,308, 279--289 (1997). \(3\)-folds, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Birational automorphisms, Cremona group and generalizations Resolution of \(\mathbb{C}^3/H_{168}\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 deals with a geometric study of finite dimensional modules over a finite dimensional associative algebra. The author proves a cancellation theorem for degenerations of modules and considers its applications to the theory of preprojective modules and modules over tame concealed algebras, the theory of matrix pencils and others. In fact, the main results of the article under review have been improved and generalized in two papers written later but published earlier [\textit{K. Bongartz}, Comment. Math. Helv. 69, No. 4, 575-611 (1994; Zbl 0832.16008) and Ann. Sci. Éc. Norm. Supér., IV. Sér. 28, No. 5, 647-668 (1995; Zbl 0844.16007)]. regular modules; stretched modules; tame concealed algebras; tame quivers; indecomposable quivers; simple quivers; path algebras; finite dimensional modules; finite dimensional associative algebras; cancellation theorems; degenerations of modules; preprojective modules; modules over tame concealed algebras; matrix pencils Bongartz, K., On degenerations and extensions of finite dimensional modules, \textit{Adv. Math.}, 121, 245-287, (1996) Representations of quivers and partially ordered sets, Finite rings and finite-dimensional associative algebras, Representation type (finite, tame, wild, etc.) of associative algebras, Group actions on varieties or schemes (quotients), Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Singularities in algebraic geometry, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers On degenerations and extensions of finite dimensional 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 Trying to resolve singularities of algebraic varieties in positive characteristic, the second author introduced Rees algebras several years ago. A Rees algebra over a smooth variety \(V\) over a field \(k\) is a graded subsheaf \(\mathcal G = \sum _{i=1}^{\infty} I_i W^i\) of \({{\mathcal O}_V} [W]\), locally finitely generated, where \(W\) is an indeterminate. The singular locus \(\mathrm{sg}(\mathcal G)\) is the set of points \(x \in V\) such that \({\nu} _x (I _i)\), the order of the ideal \(I_i{\mathcal O}_{V,x}\) is \(\geq i\). If one is able to solve Rees algebras by means of a finite sequence of suitable monoidal transformations (i.e., reach a situation where the singular locus is empty) it is possible to resolve singularities of algebraic varieties. This is usually done, following Hironaka, with the aid of Hilbert-Samuel functions. Under appropriate assumptions (which are not really restrictive) one may find, given a Rees algebra \({\mathcal G}\) over a smooth \(d\)-dimensional variety \(V\), a \textit{transversal projection} onto a smooth \(d'\)-dimensional variety \(V'\), \(d'<d\), that is a smooth surjective morphism \(\beta : V \to V'\) with some nice properties, and a useful Rees algebra \({\mathcal R}={\mathcal R}_{\mathcal G, \beta}\) over \(V'\), called the \textit{elimination algebra} of \(\mathcal G\), relative to \(\beta\). In characteristic zero one shows that a resolution of \(\mathcal G'\) induces one of \(\mathcal G\), so we may use induction on the dimension to resolve \(\mathcal G\). This method is an alternative to the use of \textit{subvarieties of \(V\) of maximal contact}, a more ``classical'' technique, available in characteristic zero only. If the characteristic is positive, the method of projections still might work, but the situation is more complicated, there are problems that are not fully solved yet. Villamayor and some of his former students (A. Bravo, A. Benito, S. Encinas, etc.) have introduced and studied certain invariants of Rees algebras, trying to overcome those problems.
The most basic one, due to Hironaka, in the context of Rees algebras and using the notation above, is \(\mathrm{ord}(\mathcal G)(x)=\)min\(\{\nu_x(I_n) / n : n \in {\mathbb N}\}\) (for \(x \in \mathrm{sg}(\mathcal G))\). Let \(\tau (x)\) be the Hironaka invariant (for Rees algebras, its definition uses the \textit{tangent cone} of \(\mathcal G\) at \(x\)). If \(\tau(x) \geq e\) for all \(x \in \mathrm{sg}(\mathcal G)\) one gets a transversal projection \(\beta\) as above with \(V'\) of dimension \(d-e\). Then define \(\mathrm{Ord}^{(d-e)}(\mathcal G)(x)=\mathrm{ord}(\mathcal R _{\mathcal G, \beta}(x))\).
In previous work the authors showed useful applications of Ord, but trying to surmount the mentioned difficulties, further refinements seem necessary. In this paper the authors discuss a refinement, the function \(\text{H-ord}^{(d-e)}(\mathcal G):\mathrm{sg}(\mathcal G) \to {\mathbb Q}\). Its definition involves a ``presentation'' of \(\mathcal G\) in terms of the elimination algebra \(\mathcal R _{\mathcal G, \beta}\) and certain auxiliary monic polynomials with coefficients in the completion of the local ring of \(\mathcal O _{V',\beta(x)}\). To show the existence of such presentations the authors prove a variant, or generalization, of Weierstrass' Preparation Theorem. These functions are not necessarily upper semicontinuos (as often invariants are), though they satisfy though the inequality \( {\text{H-ord}} ^{(d-e)}(\mathcal G)(x) \leq {\text{Ord}}^{(d-e)}(\mathcal G)(x) \).
After discussing the necessary background to properly define these functions and proving some of its properties (which is pretty technical work) the authors prove their main result. This is an improvement upon a theorem in [\textit{A. Bravo} and \textit{O. Villamayor U.}, Adv. Math. 224, No. 4, 1349--1418 (2010; Zbl 1193.14019)], that essentially says that, under some extra hypotheses which are not really restrictive, with the technique of transversal projections and elimination algebras, by induction of the dimension, given a Rees algebra \(\mathcal G\) one can find a sequence of permissible transformations
\[
\mathcal G = {\mathcal G}_0 \leftarrow \cdots \leftarrow {\mathcal G}_r
\]
where either \(\mathrm{sg}({\mathcal G}_r)=\emptyset\), or the \(\tau\)-invariant increases, or \({\mathcal R} _{{\mathcal G}_r,\beta_r}\) (the elimination algebra, relative to a suitably induced projection \(\beta_r\)) is \textit{monomial} (a particularly simple type of algebra). In characteristic zero this is enough to obtain, with some extra work, a resolution of \(\mathcal G\). In characteristic \(p >0\) the third (monomial) case is more complicated. In this paper the authors obtain new results. Namely, (assuming the base field is perfect, if the H-ord function of \(\mathcal G_r\) satisfies an equality involving ord of a certain attached monomial algebra, then also in this case we are led to a resolution of \(\mathcal G\) or an increase in the value of \(\tau\). A similar result when \(\tau \geq e=1\) had been obtained in [\textit{A. Benito} and \textit{O. E. Villamayor U.}, Compos. Math. 149, No. 8, 1267--1311 (2013; Zbl 1278.14019)].
With these techniques they prove that it is possible to prove an embedded desingularization theorem for an algebraic surface \(X\) embedded in a smooth variety \(V\), over a perfect field. In previous work the authors had showed a similar result in case \(\dim V =3\). resolution of singularities; Hilbert-Samuel function; Rees algebra; invariant; transversal projection Benito, A., Villamayor U., O.E.: On elimination of variables in the study of singularities in positive characteristic. arxiv.1103.3462 Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Multiplicity theory and related topics, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics On elimination of variables in the study of singularities in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Review of [Zbl 1076.14005; Zbl 1113.14013]. External book reviews, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Book review of: Steven Dale Cutkosky, Resolution of singularities; János Kollár, Lectures on resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this note we attempt to generalize a few statements drawn from the 3-dimensional McKay correspondence to the case of a cyclic group not in \(SL(3,\mathbb{C})\). We construct a smooth, discrepant resolution of the cyclic, terminal quotient singularity of type \({1\over r}(1,1,r-1)\), which turns out to be isomorphic to Nakamura's \(G\)-Hilbert scheme. Moreover we explicitly describe the tautological bundles and use them to construct a dual basis to the integral cohomology of the resolution. McKay correspondence; terminal singularities Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients) Cohomology of the \(G\)-Hilbert scheme for \(\frac 1r(1,1,r-1)\) | 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 The string-theoretic E-functions \(E_{\text{str}}(X;u,v) \) of normal complex varieties \(X\) having at most log-terminal singularities are defined by means of snc-resolutions. We give a direct computation of them in the case in which \(X\) is the underlying space of the three-dimensional \(A\)-\(D\)-\(E\) singularities by making use of a canonical resolution process. Moreover, we compute the string-theoretic Euler number for several compact complex threefolds with prescribed A-D-E singularities. D. I. Dais and M. Roczen, On the string-theoretic Euler numbers of \(3\)-dimensional \(A\)-\(D\)-\(E\) singularities , Adv. Geom. 1 (2001), 373--426. Mixed Hodge theory of singular varieties (complex-analytic aspects), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) On the string-theoretic Euler numbers of 3-dimensional \(A\)-\(D\)-\(E\) 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 See the preview in Zbl 0527.14005. deformations of curve; equisingularity; dimension of tangent space; nodes A. Nobile,Families of curves on surfaces, Math. Z.,187 (1984), pp. 453--470. Singularities in algebraic geometry, Families, moduli of curves (algebraic), Singularities of curves, local rings, Formal methods and deformations in algebraic geometry, Structure of families (Picard-Lefschetz, monodromy, etc.), Deformations of singularities, Global theory and resolution of singularities (algebro-geometric aspects), Deformations of complex singularities; vanishing cycles, Families, moduli, classification: algebraic theory Families of curves on surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be an algebraic scheme over a perfect field \(k\). An important invariant attached to a point \(x \in X\) is its multiplicity \(\mathrm{mult}_X(x)\). This gives rise to a function \(\mathrm{mult}_X: |X| \to \mathbb N\), where \(|X|\) is the underlying topological space of \(X\), which is upper-semicontinuous (a non-trivial fact). The sets \(F_m(X)=\{x: \mathrm{mult}_X(x) =m\}\) stratify \(|X|\) into locally closed sets. This is the \textit{equimultiple stratification} of \(X\). Of particular interest is the closed stratum \(F_n(X)\), where \(n\) is the maximum value of \(\mathrm{mult}_X\). Indeed, if \(f:X_1 \to X\) is the blowing-up of \(X\) with a closed, regular center \(C\subseteq F_n(X)\), then \(F_n(X_1) \subseteq f^{-1}(F_n(X))\). Hence, if we are able to get a sequence of blow-ups of this type such that eventually \(F_n(X_r)=\emptyset\), by induction or iterating we resolve the singularities of \(X\).
In the paper under review the author investigates this stratification, specially the stratum \(F_n(X)\) corresponding to maximal multiplicity. Probably the main contribution is a ``representation via embeddings'' theorem. This means that if \(x \in F_n(X)\) is a closed point, there is an étale neighborhood \((X',x') \to (X,x)\), a closed embedding \(X' \subseteq W\), with \(W\) regular and a Rees algebra \(\mathcal G\) on \(W\) such that \(\mathrm{Sing} (\mathcal G) = F_n(X')\), satisfying other useful properties. Moreover, this situation is preserved if we take sequences of blowing ups \(X =X_0\leftarrow \cdots \leftarrow X_s\) with centers \(C_i \subseteq F_n(X_i)\) (assuming \(F_n(X_i)\) is not empty). In particular, assuming to simplify \(X'=X\), a resolution of the Rees algebra \(\mathcal G\) induces, using the same centers, a sequence of blowing-ups \(X \leftarrow \cdots X_s\) where \(F_i(X_s)=\emptyset\). This is equally true in the general case (where the embedding is étale local only), but this requires some extra work. Resolution for Rees algebras is available in characteristic zero, as proved by the author in previous papers.
To prove the mentioned Representation Theorem one is reduced to an affine situation, \(X' = \mathrm{Spec}(B)\). The main technique is to consider a ``general'' finite surjective projection \(X' \to V\), with \(V=\mathrm{Spec} (S)\) affine, regular (more precisely, a \textit{transversal} projection, something defined in the paper). That is, algebraically we have a finite inclusion of rings \(S \subset B\), satisfying certain conditions. Thus, \(B=S[\theta_1, \ldots, \theta_m]\), and each \(\theta _i\) satisfies a minimal polynomial \(f_i\) with coefficients in \(S\). The construction of \(\mathcal G\) involves classical elimination theory applied to the polynomials \(f_i\).
Although this is applied to algebraic schemes over a perfect field, most of this algebraic work (which takes a good part of the article) is done in a more general setting.
There is an alternative version of the representation theorem, involving (in the notation above) hypersurfaces of \(W\) rather than Rees algebras.
Using these techniques the author gives new proofs of the basic results on multiplicity mentioned before. The author also presents other interesting theorems. For instance, he shows that if \(X\) is a (not necessarily reduced) algebraic scheme, then the equimultiple stratifications of \(X\) and of \(X_{\mathrm{red}}\) coincide (both are partitions of the topological space \(|X|=|X_{\mathrm{red}}|\)). From this point of view, the analog of a regular variety in the case of a non-reduced algebraic scheme \(X\) is to require that there be a unique nonempty stratum in the equimultiple stratification of \(X\). The author shows that, in characteristic zero, in this sense resolution of singularities is also available for general algebraic schemes. multiplicity; stratification; resolution of singularities; Rees algebra; elimination Villamayor U, O.: Equimultiplicity, algebraic elimination and blowing-up. Adv. Math. 262, 313-369 (2014) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Multiplicity theory and related topics, Arithmetic rings and other special commutative rings Equimultiplicity, algebraic elimination, and blowing-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 We shall call a hypersurface isolated singularity defined by a quasihomogeneous polynomial, with ''a quasihomogeneous isolated singularity''. Quasihomogeneous isolated singularities have been studied by several authors [e.g. \textit{P. Orlik} and \textit{P. Wagreich} [Ann. Math., II. Ser. 93, 205-228 (1971; Zbl 0212.537) and \textit{E. Yoshinaga} and \textit{K. Watanabe} [Sci. Rep. Yokohama Natl. Univ., Sect. I 25, 45-53 (1978; Zbl 0464.14009)]. In this paper, we classify the quasihomogeneous isolated singularities with \(p_ g=2\). In section 4, we list the minimal resolutions of our classified singularities. geometric genus; quasihomogeneous isolated singularity; minimal resolutions Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Local complex singularities, Singularities of surfaces or higher-dimensional varieties, Complex singularities, Modifications; resolution of singularities (complex-analytic aspects) Two-dimensional quasihomogeneous isolated singularities with geometric genus equal to two | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In characteristic 0 the theory of maximal contact is essential for desingularization. We recall Narashiman's example to show that there is no hope to get a satisfactory theory in characteristic \(p > 0\) and we illustrate the difficulties that this gap makes in dimension 3 by an original example. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Desingularization: A few bad examples in dim. 3, characteristic \(p>0\). | 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 \(f\) be a polynomial in \(n\) variables over some number field and \(Z\) a subscheme of affine \(n\)-space. The notion of motivic oscillation index of \(f\) at \(Z\) was initiated by \textit{R. Cluckers} [Int. Math. Res. Not. 2008, Article ID rnm118, 20 p. (2008; Zbl 1225.11100)] and Cluckers-Mustaţǎ-Nguyen in [\textit{R. Cluckers} et al., Forum Math. Pi 7, Paper No. e3, 28 p. (2019; Zbl 1454.11146)]. In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta function of \(f\) has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of \(\ell\)-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if \(f\) is a polynomial `of Thom-Sebastiani type' with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of \(A-D-E\) type. In an appendix, we answer affirmatively a recent question of Cluckers-Mustaţǎ-Nguyen in [loc. cit.] on poles of maximal order of twisted Igusa's local zeta functions. exponential sums; Igusa's conjecture; Igusa's local zeta functions; motivic oscillation index; non-rational singularities; analytic isomorphism of singularities Estimates on exponential sums, Zeta functions and \(L\)-functions, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Applications of model theory, Model theory (number-theoretic aspects), Germs of analytic sets, local parametrization On the motivic oscillation index and bound of exponential sums modulo \(p^m\) via analytic isomorphisms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is an expository article on the singularities of nilpotent orbit closures in simple Lie algebras over the complex numbers. It is slanted towards aspects that are relevant for representation theory, including Maffei's theorem relating Slodowy slices to Nakajima quiver varieties in type A. There is one new observation: the results of Juteau and Mautner, combined with Maffei's theorem, give a geometric proof of a result on decomposition numbers of Schur algebras due to Fang, Henke and Koenig [\textit{M. Fang} et al., Forum Math. 20, No. 1, 45--79 (2008; Zbl 1151.20035)]. nilpotent orbit; Slodowy slice; Springer fibre; quiver variety A. Henderson, \textit{Singularities of nilpotent orbit closures}, [arXiv:1408.3888]. Singularities in algebraic geometry, Coadjoint orbits; nilpotent varieties, Global theory and resolution of singularities (algebro-geometric aspects), Representation theory for linear algebraic groups Singularities of nilpotent orbit closures | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 punctual Hilbert scheme has been known since the early days of algebraic geometry in EGA style. Indeed it is a very particular case of the Grothendieck's Hilbert scheme which classifies the subschemes of projective space. The general Hilbert scheme is a key object in many geometric constructions, especially in moduli problems. The punctual Hilbert scheme which classifies the 0-dimensional subschemes of fixed degree, roughly finite sets of fat points, while being pathological in most settings, enjoys many interesting properties especially in dimensions at most three. Most interestingly it has been observed in this last decade that the punctual Hilbert scheme, or one of its relatives, the \(G\)-Hilbert scheme of Itô-Nakamura, is a convenient tool in many hot topics, as singularities of algebraic varieties, e.g McKay correspondence, enumerative geometry versus Gromov-Witten invariants, combinatorics and symmetric polynomials as in Haiman's work, geometric representation theory (the subject of this school) and many others topics.
The goal of these lectures is to give a self-contained and elementary study of the foundational aspects around the punctual Hilbert scheme, and then to focus on a selected choice of applications motivated by the subject of the summer school, the punctual Hilbert scheme of the affine plane, and an equivariant version of the punctual Hilbert scheme in connection with the A-D-E singularities. As a consequence of our choice some important aspects are not treated in these notes, mainly the cohomology theory, or Nakajima's theory. for which beautiful surveys are already available in the current literature [\textit{V. Ginzburg}, Lectures on Nakajima's quiver varieties. Paris: Société Mathématique de France. Séminaires et Congrès 24, pt. 1, 145--219 (2012; Zbl 1305.16009); \textit{M. Lehn}, Lectures on Hilbert schemes. Providence, RI: American Mathematical Society (AMS). CRM Proceedings \& Lecture Notes 38, 1--30 (2004; Zbl 1076.14010); \textit{H. Kakajima}, Lectures on Hilbert schemes of points on surfaces. University Lecture Series. 18. Providence, RI: American Mathematical Society (AMS). xi, 132 p. (1999; Zbl 0949.14001)]. scheme; cluster; group; group action; matrix factorization; quotient scheme; singular point Bertin, J., The punctual Hilbert scheme: an introduction, Proceedings of the Summer School 'Geometric Methods in Representation Theory'. I, 1-102, (2012), Soc. Math. France, Paris Actions of groups on commutative rings; invariant theory, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), McKay correspondence, Cluster algebras The punctual Hilbert scheme: an introduction | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a sequel to Part I of this article, \textit{E. Bierstone} and \textit{P. Milman} [Adv. Math. 231, 3022--3053 (2012; Zbl 1257.14002)], where the authors investigate ``partial resolutions'' of an algebraic variety \(X\) over a characteristic zero field. That is, in the process we avoid modifying points of a certain type (e.g. normal crossings.) More precisely, consider sets \(K\) of equivalence classes of singularities, where we identify two if they become isomorphic after base field extension (if necessary) and completion. Let \(X^K\) denote the set of points of \(X\) which are regular or have a singularity in \(K\). If \(K\) consists of normal crossings points, we write \(X^K=X^{nc}\). In the present paper, assuming \(\dim X = 3\), the authors determine the smallest set \(S\) of classes of singularities, such that there is a morphism \(\sigma:X' \to X\), which is a composition of permissible blowing-ups, such that \({(X')}^S=X'\) and \(\sigma\) induces an isomorphism over \(X^{nc}\). The problem can be reduced to that for embedded surfaces, and they give a list of representatives of the classes in \(S\), defined by particularly simple equations (in four variables), called normal forms. This list \(S\) has seven elements.
It is not true that \(\sigma\) is necessarily an isomorphism over \(X^S\). To obtain such a result, it is necessary to enlarge \(S\). In fact, adding just one more normal form, they get a list \(S'\), so that one obtains a morphism \(\bar \sigma: {\bar X} \to X\) as above, with \({\bar X}^{S'}= {\bar X}\), inducing an isomorphism over \(X^{S'}\).
In Part I the case \(\dim X <3\) was studied, in that situation \(S=S'\).
The main technique used comes from \textit{E. Bierstone}--\textit{P. Milman}'s main resolution paper [Invent. Math. 128, No. 2, 207--302 (1997; Zbl 0896.14006)], specially their function \(inv\). But they need other auxiliary results, possibly of independent interest, which are proved in this paper. E.g., they give a characterization of singularities which are limits of triple normal crossings and cannot be eliminated by means of blowing-ups whose centers do not include normal crossings singularities. They also use a ``cleaning lemma'', proved in part I. birational geometry; resolution of singularities; normal crossings; desingularization invariant; normal form Bierstone, E.; Lairez, P.; Milman, P. D.: Resolution except for minimal singularities II. The case of four variables, Adv. math. 231, 3003-3021 (2012) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Invariants of analytic local rings, Normal forms on manifolds Resolution except for minimal singularities. II. The case of four 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 paper under review provides a way to compute minimal log discrepancies of a pair \((X,Y)\), where \(X\) is \(\mathbb{Q}\)-Gorenstein normal and \(Y\) is a formal combination of proper closed subschemes in \(X\). This extends previous results of \textit{M. Mustaţă} [J. Am. Math. Soc. 15, No. 3, 599--615 (2002; Zbl 0998.14009)] and \textit{T. Yasuda} [Am. J. Math. 125, 1137--1145 (2003; Zbl 1049.14011)]. The first application of this is a proof of the inversion of adjunction conjecture in the case when the ambient variety is smooth. Then the authors obtain a characterization of terminal hypersurface singularities. Finally they prove a semicontinuity for minimal log discrepancies in the case of a smooth ambient variety. jet schemes; minimal log discrepancies; adjunction conjecture; terminal hypersurface singularities Ein, Lawrence; Mustaţă, Mircea; Yasuda, Takehiko, Jet schemes, log discrepancies and inversion of adjunction, Invent. Math., 153, 3, 519-535, (2003) Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Infinitesimal methods in algebraic geometry Jet schemes, log discrepancies and inversion of adjunction | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\mathcal{S}\) be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of \(\mathcal{S}\) by means of the Poincaré polynomials of its strata, obtaining interesting polynomial identities relating Poincaré polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities. Schubert varieties; intersection cohomology Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Topological properties in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Global theory of complex singularities; cohomological properties, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Topological properties of mappings on manifolds Polynomial identities related to special Schubert varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider the following conjecture: on a klt germ \((X,x)\), for every finite set \(I\) there is a positive integer \(\ell \) with the property that for every \({\mathbb R}\)-ideal \(\mathfrak {a}\) on \(X\) with exponents in \(I\), there is a divisor \(E\) over \(X\) that computes the minimal log discrepancy \(\text{mld}_x(X,\mathfrak {a})\) and such that its discrepancy \(k_E\) is bounded above by \(\ell \). We show that this implies Shokurov's ACC conjecture for minimal log discrepancies on a fixed klt germ and give some partial results towards the conjecture. minimal log discrepancy; ascending chain condition Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects) A boundedness conjecture for minimal log discrepancies on a fixed germ | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 flow-box or straightening theorem, in the form relevant here, states that a vector field \(\partial\) on a manifold \(M\), transverse to an analytic subvariety \(X\), is locally given by \(\partial_x\), where \(x\) is a local coordinate vanishing on \(X\). If \(X\) is smooth of codimension one, the transversality is equivalent to saying that \(\varphi_p(t)\not\in X\) for small \(t\) and \(p\) in a compact set \(K\subset X\), where \(\varphi_p(0)=p\) and \(p\in K\setminus\mathrm{Sing}\partial\). However, this is not the same as \(X\setminus\mathrm{Sing}\partial\) being transverse, even for \(X\) smooth, as the authors show. Instead an extra condition is needed: \(\partial^2\mathcal{I}_X\subset \partial\mathcal{I}_X+\mathcal{I}_X\).
The major task of the paper is to generalise all of this to foliations. The link between flow-box and transversality is just the
Frobenius theorem, but to define the appropriate properties and establish the relations among them in the case of singular foliations is more difficult, and requires a detour into sub-Riemannian metrics.
The study is motivated by a question of \textit{J. F. Mattei} [Invent. Math. 103, No. 2, 297--325 (1991; Zbl 0709.32025)], about regularising the action of a Lie group: the need for the extra condition shows that the Lie algebra version is false in general and that Mattei's question is thus global in nature. flow-box theorem; vector field; foliations; resolution of singularities Foliations generated by dynamical systems, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Obstructions to group actions (\(K\)-theoretic aspects), General theory of group and pseudogroup actions, Foliations in differential topology; geometric theory, Equisingularity (topological and analytic), Singularities of holomorphic vector fields and foliations Generalized flow-box property for singular foliations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given an algebraic variety \(V\) over a field \(k\) of characteristic zero, the following are two equivalent formulations of the Zariski-Lipman Conjecture.
\textsl{(1) Let \(p\) be a (closed) point of \(V\) with local ring \(R\). If the module of \(k\)-derivations \(\text{Der}_k(R)\) is a free \(R\)-module, then \(V\) is smooth at \(p\).}
(2) \textsl{If \(V\) is normal and the tangent bundle of the smooth locus of \(V\) is a trivial bundle, then \(V\) is smooth.}
The authors prove instances of (the second formulation of) the conjecture in dimension two, using the theory of non-complete algebraic surfaces by the Japanese school. More precisely, denote by \(V^0\) the smooth locus of an algebraic surface \(V\) and by \(\bar \kappa (V^0)\) its logarithmic Kodaira dimension. Assuming that moreover \(k\) is algebraically closed and that the tangent bundle of \(V^0\) is trivial, the authors show the following. {\parindent=6mm \begin{itemize} \item[(1)] If \(V\) is affine and \(\bar \kappa (V^0) \leq 1\), then \(V\) is smooth. \item [(2)] If \(V\) is projective, then \(\bar \kappa (V^0) \leq 0\) and \(V\) has at most one singularity. \item [(3)] If \(V\) is projective and \(\bar \kappa (V^0) =0\), then \(V\) is smooth.
\end{itemize}} Moreover, if \(V\) is projective and \(\bar \kappa (V^0) =-\infty\), they construct a \(\mathbb P^1\)- fibration \(W \to C\), where \(W\) is a resolution of singularities of \(V\) and \(C\) is a smooth projective curve. In this setting they `almost always\'\ prove that \(V\) is smooth, that is, except when this fibration has a certain very special form. Zariski-Lipman Conjecture; module of derivations; normal algebraic surfaces; non-complete algebraic surfaces Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Regular local rings, Derivations and commutative rings On the Zariski-Lipman conjecture for normal 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 Given a singular projective variety in some projective space, we characterize the smooth curves contracted by the Gauss map in terms of normal bundles. As a consequence, we show that if the variety is not linear, then a contracted line always has local obstruction for the embedded deformation and each component of the Hilbert scheme where the line lies is non-reduced everywhere. Gauss map; normal bundle; Hilbert scheme Projective techniques in algebraic geometry, Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry Curves contracted by the Gauss 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 Let T be a triangulation of a compact topological surface on each edge of which a pair of integers is attached. The author announces that, given such a T satisfying some technical conditions, he is able to construct a 3-dimensional cusp singularity admitting a resolution whose exceptional set is determined by T. The proofs are not given and should appear elsewhere. periodic continued fractions; 3-dimensional cusp singularity; exceptional set H. TSUCHIHASHI, 2-dimensional periodic continued functions and 3-dimentional cus singularities, Proc. Japan Acad., 58(A) (1982), 262-264. Global theory and resolution of singularities (algebro-geometric aspects), Continued fractions, Singularities in algebraic geometry 2-dimensional periodic continued fractions and 3-dimensional cusp singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) an algebraically closed field, \(\mathrm{char}\, k=0\). We study multiplicity-\(r\) structures on varieties for \(r\in \mathbb{N}, r\ge 2\). Let \(Z\) be a reduced irreducible nonsingular \((N-2)\)-dimensional variety such that \(rZ=X\cap F\), where \(X\) is a normal \((N-1)\)-fold of degree \(n, F\) is a smooth \((N-1)\)-fold of degree \(m\) in \(\mathbb{P}^N\), such that \(r\in \mathbb{N}, r\ge 2, Z\cap \text{Sing} (X)\not =\emptyset \). There are effective divisors \(V\) and \(D_1\) on \(Z\) such that \(O_Z(V-(r-1)D_1)\simeq{\omega_Z}^r(-rm-n+(N+1)r)\), where \(\omega_Z\) is the canonical sheaf of \(Z\). Let \(Z \subset \mathbb{P}^N\) be a reduced irreducible subvariety of codimension 2. Let \(Y\) be an irreducible hypersurface in \(\mathbb{P}^N, Z \subset Y\). Let \({\omega^o}_Z\) be the dualizing sheaf of \(Z\). Then, there exists a hypersurface \(X\) in \(\mathbb{P}^N\) such that \(Z=Y\cap X\) is a scheme-theoretical complete intersection if and only if \(\bullet{\omega^o}_Z\simeq \omega_{\mathbb{P^N}}\otimes{\wedge }^2{\mathcal N }_Z|_{\mathbb{P^N}}. \bullet \text{deg}\,Y\) divides \(\text{deg}\,Z. \bullet{\omega^o}_Z\simeq O_Z (\text{deg}\,Y+(\frac{\text{deg}\,Z}{\text{deg}\,Y})-N-1)\). Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, \(3\)-folds, \(4\)-folds, \(n\)-folds (\(n>4\)) Study of multiple structures on projective 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 The dimension of the space of first order infinitesimal deformations of a cusp singularity (X,0) is computed in terms of its resolution graph. One basic feature of the proof is to express this dimension in terms of 1- forms on the singularity (using local duality) and to analyze such 1- forms by expanding them as Fourier series on the universal cover of X- \(\{\) \(0\}\) (which is the neighbourhood of a cusp point in the product of two copies of the upper half plane with respect to a Hilbert modular group). A different proof of the result of this paper was given by \textit{I. Nakamura} [Math. Ann., to appear]. infinitesimal deformations; Zariski-differentials; dimension of the space of first order infinitesimal deformations of a cusp singularity; resolution graph [Be 2] K. Behnke. On the module of Zariski differentials and infinitesimal deformations of dusp singularities. Math. Ann. 271, 133--142, 1985 Deformations of singularities, Local deformation theory, Artin approximation, etc., Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials On the module of Zariski differentials and infinitesimal deformations of cusp singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((X,\mathfrak a)\) be a pair consisting of log canonical variety \(X\) (in particular \(X\) is normal and \(K_X\) is \(\mathbb Q\)-Cartier) and an ideal sheaf \(\mathfrak a\subset \mathcal O _X\). The log canonical threshold \(\text{lct} (\mathfrak a )\) is the largest \(t>0\) such that \((X,\mathfrak a^t)\) is log canonical. Log canonical thresholds are an invariant of fundamental importance in birational geometry. Shokurov's ACC conjecture is one of the main conjectures regarding log canonical thresholds. It states that for any fixed dimension, the set of all possible log canonical thresholds satisfies the ascending chain conditions (i.e. there are no accumulation points from below).
In this paper, building on the results of \textit{T. de Fernex} and \textit{M. Mustaţă} [Ann. Sci. Ec. Norm. Super. (4) 42, No. 3, 491--515 (2009; Zbl 1186.14007)] and \textit{J. Kollár} [Which powers of holomorphic functions are integrable? \url{arXiv:0805.0756}], the authors prove that Shokurov's ACC conjecture holds for all smooth varieties over an algebraically closed field of characteristic \(0\) (and more generally the result holds for l.c.i. varieties). ACC for log canonical thresholds de Fernex, Tommaso; Ein, Lawrence; Mustaţă, Mircea, Shokurov's ACC conjecture for log canonical thresholds on smooth varieties, Duke Math. J., 152, 1, 93-114, (2010) Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry Shokurov's ACC conjecture for log canonical thresholds on smooth 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 A polynomial \(V = V(x) \in k[x_1,\dots,x_m]\) is called a \textit{homogeneous potential} if is it homogeneous of degree \(2\) with respect to some given \(\mathbb Q_+\)-grading. Two such potentials \(V(x),W(y)\) (over the same base field \(k \subset \mathbb C\), but not necessarily the same number of variables \(m\)) are called \textit{orbifold equivalent} if there exists a finite-rank graded matrix factorization of \(W(y) - V(x) \in k[x_1,\dots, x_m,y_1,\dots y_n]\) whose left and right quantum dimensions are nonzero (concrete definitions, as well as a comparison with their more general counterparts in the context of pivotal categories, are given in the paper).
The main result of the paper is a classification of orbifold equivalence classes of potentials over \(k = \mathbb C\) which define ADE singularities. Denoting by \(d\) the central charge, equivalence classes are
\[
\begin{align*}{ &\{ A_{d-1} \}, \text{ for \(d\) odd},\cr &\{ A_{d-1}, D_{d/2+1} \} \text{ for \(d\) even, } d \not\in \{ 12,18,30 \},\cr &\{ A_{11}, D_7, E_6 \}, \{ A_{17}, D_{10}, E_7 \}, \{ A_{29}, D_{16}, E_8 \}.}\end{align*}
\]
The final section of the paper explains the relation of the above with the (conjectural) correspondence between \(\mathcal N = 2\) supersymmetric Landau--Ginzburg models and \(\mathcal N = 2\) superconformal field theories in two dimensions. The results of the paper also shed light on the relation between derived categories of matrix factorizations and Dynkin quiver representations. orbifold equivalence; matrix factorization; quantum dimension; ADE singularities N. Carqueville, A. Ros Camacho, and I. Runkel, Orbifold equivalent potentials. J. Pure Appl. Algebra 220(2016), no. 2, 759--781.MR 3399388 Double categories, \(2\)-categories, bicategories, hypercategories, Singularities in algebraic geometry, Derived categories, triangulated categories, Derived categories and commutative rings, Representations of quivers and partially ordered sets Orbifold equivalent potentials | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite group with a complex symplectic representation \(V\). The symplectic form \(\sigma\) on \(V\) induces a symplectic form \(\overline{\sigma}\) on the regular locus of \(V/G\). A symplectic resolution of \(V/G\) is a proper morphism \(f:Y\to V/G\) such that \(Y\) is smooth and \(f^\ast\overline{\sigma}\) extends to a symplectic form on \(Y\). Let \(T_0\subset \mathrm{SO}(3)\) denote the symmetry group of a regular tetrahedron. The preimage of \(T_0\) under the standard homomorphism \(\mathrm{SU}(2)\to \mathrm{SO}(3)\) is the binary tetrahedral group \(T\). An explicit symplectic resolution for the quotient singularity arising from the four-dimensional symplectic representation of the binary tetrahedral group is given. symplectic variety; symplectic resolution; quotient singularities; binary tetrahedral group M. Lehn, C. Sorger, \textit{A symplectic resolution for the binary tetrahedral group}, in: \textit{Geometric Methods in Representation Theory.} II, Sémin. Congr., Vol. 24, Soc. Math. France, Paris, 2012, pp. 429-435. Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) A symplectic resolution for the binary tetrahedral group | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Ruan's ``Cohomological Hyper-Kähler Resolution Conjecture'' aims to identify the orbifold cohomology of a complex orbifold \(X\) admitting a hyper-Kähler resolution with the ordinary cohomology of the resolution. The orbifold cohomology theory which appears in the conjecture was introduced by Chen and Ruan, and it is closely related to the degree \(0\) Gromov-Witten invariants of the orbifold \(X\).
The current paper deals with the case of global quotients \([Y/G]\), where \(Y\) is a smooth complex manifold with the action of a finite group \(G\). First, the authors define a (larger and non-commutative) ring \(H^{\star}(Y,G)\) with a \(G\) action, whose \(G\) invariant part equals the Chen-Ruan orbifold cohomology. As a vector space \(H^{\star}(Y,G)\) is isomorphic to the sum of cohomologies of the fixed loci \(\bigoplus_{g\in G} H^{\star}(Y^{g})\). The definition of the multiplitive structure is trickier and involves the Euler classes of certain ``obstruction bundles'', carefully defined in the first section of the paper.
Secondly, the authors check a special case of Ruan's conjecture. If \(S\) is a smooth surface with trivial canonical bundle, the Hilbert scheme \(S^{[n]}\) of \(n\) points on \(S\) provides a hyper-Kähler resolution for the symmetric product \(\text{Sym}^{n}S\). The authors compute the orbifold cohomology of the symmetric product \(\text{Sym}^{n}S\) and compare their answer to \textit{M. Lehn} and \textit{C. Sorger}'s calculation of the cohomology ring of the Hilbert scheme [Invent. Math. 152, No. 2, 305--329 (2003; Zbl 1035.14001)]. A closely related result can be found in \textit{B. Uribe}'s paper [Commun. Anal. Geom. 13, No. 1, 113--128 (2005; Zbl 1087.32012)]. In addition, the authors discuss the orbifold cohomology of Beauville's generalized Kummer varieties. orbifold cohomology; Hilbert scheme of points Fantechi, B.; Göttsche, L., Orbifold cohomology for global quotients, Duke math. J., 117, 2, 197-227, (2003) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Generalizations (algebraic spaces, stacks), Global theory and resolution of singularities (algebro-geometric aspects) Orbifold cohomology for global quotients | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0607.00004.]
The author studies the question raised by Zariski in 1971: Does topological equisingularity imply equimultiplicity for hypersurface singularity? [\textit{O. Zariski}, Bull. Am. Math. Soc. 77, 481-491 (1971; Zbl 0236.14002)]. In the present paper, an affirmative answer is supplied for some special class of surface singularities under a little stronger assumption. Namely, let (X,p) be a 2-dimensional hypersurface singularity whose associated Newton polyhedron \(\Lambda_ 0\) is non-degenerate and commode (so that its resolution can be constructed by torus embedding). It is further assumed that \(\Lambda_ 0\) admits a good subdivision in the sense of \textit{K. Altmann} [Invent. Math. 88, 619-634 (1987; see the preceding review)]. Then it is proved that \(\mu\)-constant deformation is equimultiple. The above second assumption assures that equisingular deformation is realized as embedded deformation. deformations with constant Milnor number; equisingularity; equimultiplicity; 2-dimensional hypersurface singularity Karras, U. : Equimultiplicity of deformations of constant Milnor number . In: Proceedings of the Conference on Algebraic Geometry, Berlin 1985. Teubner-Texte zur Mathematik, vol. 92, Leipzig: Teubner 1986. Deformations of singularities, Deformations of complex singularities; vanishing cycles, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Equimultiplicity of deformations of constant Milnor number | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 provide an explicit canonical description of the perverse cohomology sheaves and of the primitive perverse cohomology complexes for the non-small resolution \(\pi:\tilde{\mathcal{S}}\to\mathcal{S}\) of a special Schubert variety \(\mathcal{S}\). For such a resolution, we also discuss a way to obtain an explicit splitting of \(R\pi{{}_*}\mathbb{Q}_{\widetilde{{\mathcal{S}}}}\), in the derived category, by means of Gysin morphisms and cohomology extensions. cohomology extension; Gysin morphism; derived category; intersection cohomology; decomposition theorem; Schubert varieties; resolution of singularities Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Topological properties in algebraic geometry, Global theory of complex singularities; cohomological properties, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Topological properties of mappings on manifolds Explicit decomposition theorem for special Schubert varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A Macaulayfication of a Noetherian scheme \(X\) is a pair of a Cohen-Macaulay scheme \(Y\) and a birational proper morphism from \(Y\) to \(X\). The purpose of the paper is to construct a Macaulayfication of a Noetherian scheme which is separated and of finite type over Spec(\(A\)) where \(A\) is a Noetherian ring possessing a dualizing complex. The Macaulayfication consists of successive blowing-ups and the center of each blowing-up is the ideal generated by a subsystem of a \(p\)-standard system of parameters. The theory of \(p\)-standard systems of parameters and that of unconditioned strong \(d\)-sequences (\(d^+\)-sequence for short) play essential roles in constructing the Macaulayfication.
The author develops the theory of \(p\)-standard systems of parameters in sections 2 and 3. In appendix A, he states some results on \(d^+\)-sequences with proofs from the paper ``The theory of unconditioned strong \(d\)-sequences and modules of finite local cohomology'' by \textit{S. Goto} and \textit{K. Yamagishi}. (In spite of its importance, this paper is not published.) -- The construction of Macaulayfications is done in sections 4 and 5. In section 6, it is shown that a conjecture of Sharp is true for a large class of Noetherian rings including local rings and integral domains. (Sharp's conjecture: A Noetherian ring possessing a dualizing complex is a homomorphic image of a finite dimensional Gorenstein ring.) -- An example of a Noetherian local ring with large non-Cohen-Macaulay locus is given and a Macaulayfication of its spectrum is constructed in appendix B.
[In his recent paper ``On arithmetic Macaulayfication of Noetherian rings'', \textit{T. Kawasaki} shows that a Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all of its formal fibers are Cohen-Macaulay, and he gives an affirmative answer to Sharp's conjecture]. Macaulayfication; dualizing complex; \(p\)-standard system of parameters; \(d\)-sequence; Sharp's conjecture; blowing-up; Cohen-Macaulay scheme; desingularization; Noetherian scheme T. Kawasaki, ''On Macaulayfication of Noetherian schemes,'' Trans. Amer. Math. Soc., 352, No. 6, 2517--2552 (2000). Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) On Macaulayfication of Noetherian schemes. -- Appendix A: \(d^+\)-sequences. Appendix B: An example | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field. For a quiver \(Q=(Q_0 ,Q_1, s, e)\) and a dimension vector \(d\in\mathbb{Z}^{Q_0}\) we let rep\(_Q(d)\) be the vector space of representations corresponding to \(d\). Now \(\text{GL}(d)\) acts on rep\(_Q(d)\) in a natural way, hence given a representation \(V\) or \(Q\) we can consider the \(\text{GL}(d)\)-orbit in rep\(_Q(d),\) which consists of the representations isomorphic to \(V\) -- we shall write this orbit as \(\mathcal{O}_V.\) Let \(M\) and \(N\) be representations in rep\(_Q(d)\) such that \(\mathcal{O}_N\subset\mathcal{\bar{O}}_M,\) where \(\mathcal{\bar{O}}_M\) is the Zariski closure of \(\mathcal{O}_M\) in rep\(_Q(d).\) Write Sing\((M,N)\) for the set of smoothly equivalent classes of pointed varieties. The regular points form a type of singularity which we shall denote Reg. In the case where \(\mathcal{O}_N\) has codimension one in \(\mathcal{\bar{O}}_M\) it is known that Sing\((M,N) =\) Reg; furthermore of \(\mathcal{O}_N\) has codimension two and \(Q\) is a Dynkin quiver then again Sing\((M,N) =\) Reg.
Here the author investigates the codimension two case when \(Q\) is an extended Dynkin quiver. In the case where \(Q\) is the Kronecker quiver one has two types of singularities, namely \(A_r=\) Sing\((\mathcal{A}_{r+1},0)\) and \(C_r=\) Sing\((\mathcal{C}_r,0)\), where \(\mathcal{A}_r=\{ ( uv, u^r, v^r) \in k^3\mid u,v\in k\} \) and \(\mathcal{C}_r=\{(u^r, u^{r-1}v,\dots,v^r) \in k^{r+1}\mid u,v\in k\} .\) Note that \(C_1=\) Reg, \(C_2=A_1\) and the remaining types are all distinct. The main result is that for \(Q\) an extended Dynkin quiver in the codimension two case we have Sing\((M,N) \) is one of the \(A_r\) or \(C_r\)'s. If \(Q\) is a cyclic quiver then Sing\((M,N) \neq C_r\) for \(r\geq3,\) i.e. Sing\((M,N) =A_r\) or Reg. Dynkin quivers; representations of quivers; singularities of representations of quivers Zwara, G.: Codimension two singularities for representations of extended Dynkin quivers. Manuscr. Math. 123(3), 237--249 (2007) Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets Codimension two singularities for representations of extended Dynkin quivers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R\subset S\) be local domains essentially of finite type over a field \(k\) with char \(k=0\) and \(V\) a valuation ring of the quotient field \(K\) of \(S\). Then there exist sequences of monomial transforms \(R\rightarrow R'\), \(S\rightarrow S'\) along \(V\) such that \(R'\), \(S'\) are regular local rings, \(S'\) dominates \(R'\) and there exist regular system of parameters \((y_1,\ldots,y_n)\) in \(S'\), \((x_1,\ldots,x_m)\) in \(R'\), units \(\beta_1,\ldots,\beta_n\in S'\) and a \(m\times n\) matrix \((c_{ij})\) of non-negative integers such that Rank\((c_{ij})=m\) and \(x_i=\Pi_{j=1}^ny_j^{c_{ij}}\beta_i\), \(1\leq i\leq m\).
This is the most general possible relative ``Local Uniformization Theorem for mappings'', the case \(R=k\) being given by Zariski. When \(K\) is a finite extension of the quotient field of \(R\) the result was already stated by the author in [``Local monomialization and factorization of morphisms'', Astérisque 260 (1999; Zbl 0941.14001)]. The above result is used to the construction of a monomialization by quasi-complete varieties, which proves a local version of the toroidalization conjecture of \textit{D. Abramovich, K. Karu, K. Matsuki} and \textit{J. Wlodarczyk} [J. Am. Math. Soc. 15, 531--572 (2002; Zbl 1032.14003)]. resolution of singularities; local uniformization theorem; toroidalization Cutkosky, SD, Local monomialization of transcendental extensions, Ann. Inst. Fourier, 55, 1517-1586, (2005) Global theory and resolution of singularities (algebro-geometric aspects), Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Rational and birational maps Local monomialization of transcendental extensions. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 scheme \(X\) is taid to be \textit{CM-quasi-excellent} if it is locally Noetherian and:
(1) the formal fibers of the local rings of \(X\) are Cohen-Macaulay;
(2) every integral, closed subscheme \(X'\subset X\) has a non-empty, Cohen-Macaulay open subscheme.
A CM-quasi-excellent \(X\) is \textit{CM-excellent} if in addition it is universally catenary. \medskip
For each coherent \({\mathcal O}_X\)-module \({\mathcal M}\), denote by CM\((M)\) the subset of points of \(X\) where the stack of \({\mathcal M}\) is Cohen-Macaulay, and by \(U_{(S_2)}({\mathcal M})\) the subset of points of \(X\) where the stack of \({\mathcal M}\) is \(S_2\) (both can be shown to be open if, for instance, \(X\) is CM-quasi-excellent).
The following main result is proven:
Theorem. For every CM-excellent, Noetherian scheme \(X\) equiped with finitely many cocherent \({\mathcal O}_X\)-modules \({\mathcal M}\) with \(|\text{Supp}({\mathcal M})|=|X|\), there are a composition
\[\widetilde{X}: \mathrm{Bl}_Z(X')\to X' \stackrel{\pi'}{\to} X\]
and for each \(\mathcal M\) a coherent \(\mathcal O_{X'}\)-module \(\mathcal M'\) for which \(|\mathrm{Supp}(\mathcal M')|=|X'|\) such that \(\widetilde{X}\) is Cohen-Macaulay, its coherent modules \(\mathrm{Bl}_Z({\mathcal M}')\) are also all Cohen-Macaulay, and
(1) \(X'\) is CM-excellent and locally equidimensional;
(2) \(\pi'\) is finite, birational, and is an isomorphism over the open \[U:=U_{(S_2)}(X)\cap \left(\bigcap_{\mathcal M}U_{(S_2)}({\mathcal M})\right)\subset X\] that is dense in both \(X\) and \(X'\) and for which \({\mathcal M}'|_U\simeq {\mathcal M}|_{U}\);
(3) \(Z\subset X'\) is a closed subscheme that is disjoint from the dense open \[U':=CM(X')\cap \left( \bigcap_{\mathcal M}CM({\mathcal M'}) \right);\]
(4) \(U'\) is also dense in \(\mathrm{Bl}_Z(X')\), so that, in particular, the map \(\widetilde{X}\to X\) is birational.
If \(X\) itself is CM-excellent and locally equidimensional, then we may choose \[X'=X \ \ \text{ and } \ \ \mathcal M'=\mathcal M.\]
This main result in particular says that for every CM-quasi-excellent, Noetherian scheme \(X\) there are a Cohen-Macaulay scheme \(\widetilde{X}\) and a birational, projective morphism \[\pi: \widetilde{X}\to X\] that is an isomorphism over the Cohen-Macaulay locus \(CM(X)\subset X\).
The following corollaries follow:
Corollary 1. For every integral Dedekind scheme \(S\) with the function field \(K\) and every proper, Cohen-Macaulay \(K\)-scheme \(X\), there is a proper, flat, Cohen-Macaulay \(S\)-scheme \(\mathcal X\) with \(\mathcal X_K\simeq X\). If \(X\) is projective over \(K\), then one may choose \(\mathcal X\) to be projective over \(S\).
Corollary 2. For a Noetherian, Cohen-Macaulay scheme \(X\) and a closed subscheme \(Z\subset X\), there are a Cohen-Macaulay scheme \(\widetilde{X}\) and a projective morphism \(\widetilde{X}\to X\) such that the (scheme-theoretic) preimage of \(Z\) in \(\widetilde{X}\) is divisor and \(\widetilde{X}\to X\) is an isomorphism over the maximal open subscheme \(U\subset X\) on which \(Z\) is already a divisor.
Corollary 3. For every CM-quasi-excellent Noetherian scheme \(S\) and every finite type, separated \(S\)-scheme \(X\) that is Cohen-Macaulay, there is an open \(S\)-immersion \(X\hookrightarrow \overline{X}\) into a proper \(S\)-scheme \(\overline{X}\) that is Cohen-Macaulay such that \(\overline{X}\setminus X\) is a (possibly nonreduced) divisor in \(X\). Cohen-Macaulay; excellence; Macaulayfication; resolution of singularities Global theory and resolution of singularities (algebro-geometric aspects), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Macaulayfication of Noetherian 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 A surface singularity is called quasirational if the exceptional locus ensuing from any birational blowing-up of the singularity, consists of rational curves only. The present article is a study of the relationship between the behaviour of a plane curve \(f(X,Y)=0\) and the nature of the singularity of a surface \(Z^ n-f=0\) at the origin. The main theorem of the article asserts that the curve \(f=0\) has at most a node at the origin if and only if for every positive integer n, the surface \(Z^ n-f=0\) has a quasirational singularity at the origin. The proof is essentially local in nature and involves an analysis of the embedded desingularization process for the plane curves. The key idea is to study the effect of iterated quadratic transformations on the tangent cone of f. In case the singularity of \(f=0\) at the origin is worse than a node, then it is shown in the article that there exists a point P infinitely near to the origin such that f has at least three distinct tangents at P and letting n be the multiplicity of f at P, the surface \(Z^ n-f=0\) has a non- quasirational singularity at the origin. The first part of this article presents the entire proof in an informal geometric style and the second part makes it algebraic. This type of a double exposure is indeed a novel feature of this article. It is very helpful to the reader in general and instructive to the young students of the subject. quasirational surface singularity; node of plane curve; quadratic transformations Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings, Singularities in algebraic geometry On a question of Mumford | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(Q=\left( Q_{0},Q_{1},s,e\right) \) be a finite quiver and fix an algebraically closed field \(k\).\ For any \(\mathbf{d}\in \mathbb{N}^{Q_{0}},\) \(\mathbf{d=}\left( d_{i}\right) \) there is a vector space rep\(_{Q}\left( \mathbf{d}\right) =\prod_{\alpha \in Q_{1}}\mathbb{M}_{d_{e\left( \alpha \right) }\times d_{s\left( \alpha \right) }}\left( k\right) \) where \(\mathbb{ M}_{d^{\prime }\times d^{\prime \prime }}\left( k\right) \) is the ring of \( d^{\prime }\times d^{\prime \prime }\) matrices with values in \(k\). There is a natural action of GL\(\left( \mathbf{d}\right) =\prod_{i\in Q_{0}}\)GL\( _{d_{i}}\left( k\right) \) on rep\(_{Q}\left( \mathbf{d}\right) \) given by \( g\star V=\left( g_{e\left( \alpha \right) }V_{\alpha }g_{s\left( \alpha \right) }^{-1}\right) \alpha \in Q_{1}\) where \(g=\left( g_{i}\right) _{i\in Q_{0}}\) and \(V=\left( V_{\alpha }\right) _{\alpha \in Q_{1}}.\) The orbits of this action correspond with the isomorphism classes of the representations of \(Q\) with dimension \(\mathbf{d}\). Given such a representation \(M\) the corresponding orbit will be denoted \(\mathcal{O}_{M},\) and \(\mathcal{ \bar{O}}_{M}\) denotes the closure of the orbit.
Now let \(Q\) be the Kronecker quiver, and let \(\mathbf{d}=\left( 3,3\right) .\) Let \(\mathcal{V}\) be the set of points \(\left( x,y,z,t\right) \in k^{4}\) such that \(xz=xt=yz=yt=0.\) Thus \(\mathcal{V\;}\)is the intersection of two planes intersecting at the point \(0\). Let \(M=\left( M_{\alpha },M_{\beta }\right) ,\;N=\left( N_{\alpha },N_{\beta }\right) \in \,\)rep\(_{Q}\left( \mathbf{d}\right) \) as follows:
\[
\begin{aligned} M_{\alpha } &=\left[ \begin{matrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} \right] ,\quad M_{\beta }=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} \right] \\ N_{\alpha } &=\left[ \begin{matrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} \right] ,\quad N_{\beta }=\left[ \begin{matrix} 0 & 0 & 0 \\ \lambda _{1} & 0 & 0 \\ 0 & \lambda _{2} & 0 \end{matrix} \right]\end{aligned}
\]
where \(\lambda _{1},\lambda _{2}\) are distinct scalars. The main result of this paper is that \(N\in \mathcal{\bar{O}}_{M}\) and Sing\(\left( \mathcal{ \bar{O}}_{M},N\right) =\,\)Sing\(\left( \mathcal{V},0\right) .\) In other words, the closure of this orbit contains a singularity which is smoothly equivalent to the isolated singularity of two planes crossing at a point. Since \(\mathcal{V\;}\)is not Cohen-Macaulay, it follows that this result gives an example of an orbit closure in a variety of representations of a quiver which is not Cohen-Macaulay. two planes crossing at a point; not Cohen-Macaulay Zwara, G, An orbit closure for a representation of the Kronecker quiver with bad singularities, Colloq. Math., 97, 81-86, (2003) Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets An orbit closure for a representation of the Kronecker quiver with bad singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We present a procedure for computing the log-canonical threshold of an arbitrary ideal generated by binomials and monomials. The computation of the log canonical threshold is reduced to the problem of computing the minimum of a function, which is defined in terms of the generators of the ideal. The minimum of this function is attained at some ray of a fan which only depends on the exponents of the monomials appearing in the generators of the ideal. Multiplier ideals, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies A procedure for computing the log canonical threshold of a binomial ideal | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article under review deals with the following problem: what kind of partial resolution of the singularities of an algebraic variety \(X\) can we get if we do not modify points at which \(X\) has normal crossings?
Of course, the notion of ``normal crossing'' must be clarified. Let us say that \(X\) has \textit{strict normal crossings} (snc) at a point \(x\) if, locally near \(x\), there is an embedding of \(X\) into a regular variety \(U\) so that \(X\) is locally defined by a monomial expression involving a regular system of parameters of \({\mathcal O}_{U,x}\). We say that \textit{X has normal crossing at \(x\)} (nc) if there is an etale neighborhood \((X',x')\) of \((X,x)\) so that \(X'\) has simple normal crossings at \(x'\). We let \(X^{snc}\) and \(X^{nc}\) denote the set of points of \(X\) having snc and nc respectively.
Working throughout over a field of characteristic zero, the authors prove that, given an algebraic variety \(X\), there is a projective morphism \(f:X' \to X\), such has the only singularities of \(X'\) are snc points, and \(f\) induces an isomorphism from \(f^{-1}(X^{snc})\) onto \(X^{snc}\). The morphism \(f\) is a composition of blowing-ups with admissible centers, which can be chosen in a ``canonical'' or ``algorithmic'' way.
A similar statement involving normal crossings rather that simple normal crossings is not true. If we avoid modifying nc points of \(X\) may be we cannot get a partial resolution whose only singularities are nc points. The ``Whitney umbrella'' defined by \(z^2-xy^2=0\) in \({\mathbf C}^3\) is a counter-example.
So, a natural question (posed by Kollar) is: what singularities are unavoidable when we use centers that do not include nc points? The problem is easily reduced to the special case where \(X\) is a hypersurface in a regular scheme \(Z\) of dimension \(n\). The authors solve this problem for \(n \leq 4\). For instance, identifying singularities which become isomorphic on a suitable etale neighborhood, for \(n=3\) the complete list \(\mathcal S\) of particularly simple representatives of the resulting equivalence classes, called normal forms, consists of those defined, in local coordinates \(x, y, z\) by \(xy\), \(xyz\) or \(z^2 +xy^2\) (a pinch point.) They also find, for \(n \leq 4\), the smallest class \(\mathcal S'\) of singularities, containing regular and nc points, such that by suitably modifying a variety \(X\) with centers that avoid points in \(\mathcal S'\) we get a variety \(X'\) whose only singularities are in \(S'\). For \(n \leq 3\), \(\mathcal S = \mathcal S '\), for \(n >3\) the set \(\mathcal S '\) is strictly larger.
The tools used in the proofs come from the authors' fundamental article on resolution [Invent. Math. 128, No. 2, 207--302 (1997; Zbl 0896.14006)]. They involve Hilbert functions, auxiliary marked ideals and, specially, the numerical invariant inv, used to control the resolution process. Another feature is a \textit{Cleaning Lemma}, a simplification technique resembling the one used to resolve monoidal ideals in the basic papers on desingularization. From properties of inv it follows that the resolution sequences obtained are functorial with respect to local isomorphisms and etale maps. The authors do not include proofs for the case \(n=4\), they will appear in a sequel, written in cooperation with P. Lairez. They make some comments on the general case.
The paper contains several interesting examples and an appendix where the authors review their fundamental resolution technique, specially the invariant inv. birational geometry; resolution of singularities; normal crossings; desingularization invariant; normal form Bierstone, E., Milman, P.D.: Resolution except for minimal singularities I (2011). arXiv:1107.5595 Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Invariants of analytic local rings, Normal forms on manifolds Resolution except for minimal singularities. I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a complex polynomial or analytic function \(f\), there is a strong correspondence between poles of the so-called local zeta functions or complex powers \(\int | f|^{ 2{s}}\omega\), where the \(\omega\) are \(C^{ \infty}\) differential forms with compact support, and eigenvalues of the local monodromy of \(f\). In particular Barlet showed that each monodromy eigenvalue of \(f\) is of the form \(\text{exp}(2 \pi \sqrt{-1_s}_{0})\), where \(s_0\) is such a pole. We prove an analogous result for similar \(p\)-adic complex powers, called Igusa (local) zeta functions, but mainly for the related algebro-geometric topological and motivic zeta functions. monodromy; topologicalzeta function; Igusa zeta function Veys, W.: Monodromy eigenvalues and zeta functions with differential forms. Adv. Math. 213(1), 341--357 (2007). MR MR2331246 (2009c:32058) Singularities in algebraic geometry, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Local complex singularities Monodromy eigenvalues and zeta functions with differential forms | 0 |
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