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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 Resolution of singularities is one of the origins of algebraic geometry. There is a long way from Newton's method to determine branches of a plane curve, Puiseux-series', the work of M. Noether, Riemann, the geometers of the Italian school, and many others. In the middle of the 20th century, Zariski and Abhyankar prepared the ground to study the general case of arbitrary dimension which has been settled in characteristic 0 by \textit{H. Hironaka} [Ann. Math. (2) 79, 109--203, 205--326 (1964; Zbl 0122.38603)]. Hironaka's result (which had not been recognized in its full importance over the first years) is meanwhile one of the most famous, frequently used theorems of algebraic geometry, though apparently only few people have gone through all details of its demanding proof. Apart from the still unsolved problem of resolution in positive characteristics -- which motivates a closer look for alternative proofs of the characteristic 0 case -- there are other reasons for further studies: The resolution problem admits modifications, one of them due to \textit{A. J. de Jong} [in: Resolution of singularities. A research textbook in tribute to Oscar Zariski. Prog. Math. 181, 375--380 (2000; Zbl 1022.14005)], which led to a solution of the general problem up to alterations. On the other hand, appearance of computers in the offices of most mathematicians during the last decade of the 20th century has led to increasing interest in algorithmic questions. Refined resolution algorithms have been developed [cf. \textit{O. Villamayor}, in: Real analytic and algebraic geometry. Proc. int. conf. Trento 1992, 277--291 (1995; Zbl 0930.14039)], and there is a continued interest in better understanding the ideas of Hironaka's original proof [cf. \textit{H. Hauser}, Bull. Am. Math. Soc., New Ser. 40, No.3, 323--403 (2003; Zbl 1030.14007)]. Several more recent proofs include results of \textit{E. Bierstone} and \textit{P. D. Milman} [Invent. Math. 128, No.2, 207--302 (1997; Zbl 0896.14006)], \textit{S. Encinas} and \textit{O. Villamayor} [in: Resolution of singularities. A research textbook in tribute to Oscar Zariski. Prog. Math. 181, 147--227 (2000; Zbl 0969.14007) and Rev. Mat. Iberoam. 19, No.2, 339--353 (2003; Zbl 1073.14021)], \textit{T. T. Moh} [Commun. Algebra 20, No.11, 3207--3249 (1992; Zbl 0784.14008)], \textit{O. Villamayor} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No.1, 1--32 (1989; Zbl 0675.14003), Ann. Sci. Éc. Norm. Supér., IV. Sér. 25, No.6, 629--677 (1992; Zbl 0782.14009)], \textit{J. Wlodarczyk} [J. Am. Math. Soc. 18, No.4, 779--822 (2005; Zbl 1084.14018)]. It should be noted, that there exist first approaches to use computer-algebra systems for performing the resolution process of given singularities (cf. the one by \textit{G. Bodnar, J. Schicho} [J. Symb. Comput. 30, No.4, 401--428 (2000; Zbl 1011.14005)] using Maple and another one by \textit{A. Frühbis-Krüger, G. Pfister} [Mitt. Dtsch. Math.-Ver. 13, No.2, 98--105 (2005; Zbl 1084.14036)] for Singular, respectively).
The book under review provides as well an introduction as advanced treatment of the resolution problem. Its modern presentation of meanwhile classical ideas interacts with recent research on the topic (cf. e.g. \textit{J. Kollar} [``Resolution of Singularities - Seattle Lecture'', preprint, \texttt{http://arXiv.org/abs/math/0508332}] and results by the author). After a short introduction, Chapter 2 defines basic notions of smoothness, non-singularity, resolution, normalization and local uniformization, followed by chapter 3, containing a discussion of embedded resolution for curve singularities. Chapter 4 starts constructing the blowing up of an ideal and gives the general notion of resolution. The fifth chapter studies resolution of surface singularities and their embedded resolution (again in characteristic 0). Chapter 6 gives a complete proof for resolution of singularities in arbitrary dimension and characteristic 0, based on the work of Encinas and Villamayor. Chapters 7 and 8 cover additional topics: Local uniformization and resolution of surfaces in positive characteristics (in a modern version of Zariski's original proof) and an introduction to valuation theory in algebraic geometry, together with the problem of local uniformization. An appendix contains technical material on the singular locus and semi-continuity-theorems used in the previous text. This book is pleasant to read and gives with its exercises a well prepared basis for a graduate course. ,\textit{Resolution of Singularities}, Graduate Studies in Mathematics, vol. 63, American Mathematical Society, Providence, RI, 2004.http://dx.doi.org/10.1090/gsm/063.MR2058431 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 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 problem of resolution of singularities -- solved by Hironaka about half a century ago over a field of characteristic 0 -- is still open in full generality. For arithmetical schemes \(\mathcal X\), the general case of \(\dim \mathcal{X} >2 \) remained unsolved until now. The next step is done here by the authors with their
Theorem 1.1. Let \(\mathcal X\) be a reduced and separated Noetherian scheme which is quasi-excellent and of dimension at most 3. There exists a proper birational morphism \(\pi :\mathcal{X}'\to\mathcal{X}\) with the following properties:
\begin{enumerate}
\item[(i)] \(\mathcal{X}'\) is everywhere regular;
\item[(ii)] \(\pi\) induces an isomorphism \(\pi^{-1} (\mathrm{Reg} \ \mathcal{X}) \to\mathrm{Reg}\mathcal{X} \);
\item[(iii)] \(\pi^{-1} (\mathrm{Sing } \ \mathcal{X})\) is a strict normal crossings divisor on \(\mathcal{X}'\).
\end{enumerate}
If furthermore a finite affine covering \(\mathcal{X} = \mathcal{U}_1 \cup \mathcal{U}_2 \cup \dots \cup \mathcal{U}_n \) is specified, one may take \(\pi^{-1}(\mathcal{U}_i) \to \mathcal{U}_i\) projective, \(1\leq i \leq n\).
A proper birational morphism \(\pi\) as above is said to be a resolution if it satisfies properties (i) and (ii) above. If additionally (iii) is satisfied, \(\pi\) is said to be a good resolution.
It is pointed out that the construction of \(\pi\) is not given as a sequence of Hironaka-permissible blowing ups. The authors give evidence for situations, when this can be achieved proving a local version of the theorem which uses only Hironaka-permissible blowing ups. An answer to the question whether this could hold in general is referred to as \textit{widely open}.
The following can be deduced from the theorem.
Corollary 1.2. Let \(A\) be a reduced complete Noetherian local ring of dimension three. Then \(\mathcal{X} := \mathrm{Spec} \ A\) has a good resolution of singularities which is projective.
Corollary 1.3. Let \(\mathcal O\) be an excellent Dedekind domain with quotient field \(F\) and \(\Sigma / F\) be a regular projective surface. There exists a proper and flat \(\mathcal O\)-scheme \(\mathcal X\) with generic fiber \(\mathcal{X}_F = \Sigma\) which is everywhere regular.
In this paper the authors continue their earlier work on resolution of singularities of threefolds. In the first sections they start developping a more general approach to the problem for hypersurface singularities defined by a reduced polynomial \(h=X^p+f_1X^{p-1}+ \dots + f_p \in S[X], f_i\in S\) over an excellent regular local ring \(S\) of any dimension \(\geq 1\), where \(p:= \mathrm{char} (S/\mathbf{m}_S) >0\). It is supposed -- using the notations \(K:= Q(S)\), \(\mathcal{X}:= \mathrm{Spec} (S[X]/(h))\) and \(L\) for the total quotient ring of \(S[X]/(h)\) -- that
\begin{enumerate}
\item[(i)] \(\mathrm{char} (K)=p\) and \(f_i =0\) for \(i<p\), or
\item[(ii)] \(\mathcal{X}\) is \(G\)-invariant, where \(G:= \mathrm{Aut}_K(L) = \mathbb{Z}/(p)\).
\end{enumerate}
Under these assumptions and for \(\mathrm{dim} (S)=3\), the main part of the article gives the following version of a resolution as stated in
Theorem 1.5. Let \(\mu\) be a valuation of \(L\) which is centered in \(\mathbf{m}_S\). There exists a composition of local Hironaka-permissible blowing ups \((\mathcal{X}=: \mathcal{X}_0,x_0 ) \leftarrow (\mathcal{X}_1,x_1 ) \leftarrow \dots \leftarrow (\mathcal{X}_r,x_r ) \), where \(x_i\in \mathcal{X}_i\) is the center of \(\mu\), such that \((\mathcal{X}_r,x_r) \) is regular.
Main combinatorial tool is a variant of Hironaka's characteristic polyhedron attached with the singularity. The inductive procedure is controlled by a numerical function which is different from the ``classical'' pair of multiplicity and slope function used for hypersurface singularities in residue characteristic 0.
Applying an idea which can be traced back to Zariski, Theorem 1.5 implies the above Theorem 1.1. resolution of singularities; arithmetical varieties; Zariski; blowing up; valuations Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Varieties over finite and local fields, Varieties over global fields Resolution of singularities of arithmetical 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 Everywhere in this paper \(K\) is an algebraically closed field of characteristic \(0\), and an embedded algebroid surface \(S\) is defined by the zeros of a polynomial \(F = F(x,y,z)\). The polynomial \(F\) can be considered as given in a Weierstrass form \(F = z^n + \sum_{k=0}^{n-1} a_k(x,y)z^k\), ord\((a_k) \geq n-k\). The equimultiple locus \({\mathcal E}(S)\) of \(S\) is the set of points of \(S\) of multiplicity \(n\), and \({\mathcal E}_o(S)\) is the smooth locus of \({\mathcal E}(S)\). By a classical result of Levi-Zariski, if \(S\) has only normal crossing singularities then blowing up centers lying in \({\mathcal E}_o(S)\) of maximal dimension resolves the singularity at the origin [see \textit{O. Zariski}, Ann. Math. (2) 45, 472--542 (1944; Zbl 0063.08361)]. As shown by \textit{M. Spivakovsky} [Invent. Math. 96, No.1, 181--183 (1989; Zbl 0688.14012)], this result can't be directly extended to higher dimensions, thus further understanding the resolution process in the dimension \(2\) case remains actual. In this paper the behaviour of the equidimensional locus \({\mathcal E}(S)\) under the resolution process, i.e. under a monomial transform along a curve \(P\) from \({\mathcal E}_o(S)\) or a quadratic transform (a blow-up) at the origin \(M\), is studied. In the main theorem of this paper it is proved that after a monomial transform at \(P\), the smooth equidimensional locus \({\mathcal E}_o\) of the new surface is either the preimage of \({\mathcal E}_o(S)\) or the preimage of the complement of \({\mathcal E}_o(S)\) to \(P\). If the resolving transformation is a blow-up at \(M\), then the change of \({\mathcal E}_o(S)\) depends on whether the tangent cone \(K\) to \(S\) at \(M\) is geometrically a plane or not. In the last case the new \({\mathcal E}_o\) is still the preimage of \({\mathcal E}_o(S)\). In the case when \(K\) is a plane the description of the new \({\mathcal E}_o\), made in detail by the authors, is less evident. resolution of surface singularities; blowing-up; equimultiple locus Singularities in algebraic geometry, Complex surface and hypersurface singularities, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Equimultiple locus of embedded algebroid surfaces and blowing-up 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 \(0\in X:=(f=0)\subset {\mathbb C}^n\) be an isolated canonical hypersurface singularity. For a weighting \(\alpha\) denote by \(|\alpha|\) the sum of the weights and by \(\alpha(f)\) the degree of the \(\alpha\)-tangent cone of \(X\). Every crepant valuation is represented by an exceptional divisor on an \(\alpha\)-blowup \(\phi_{\alpha}:X(\alpha)\rightarrow X\), where \(\alpha\) is such that \(|\alpha|=\alpha(f)+1\). Such weighting \(\alpha\) is called crepant. If \(X\) is nondegenerate the number of crepant valuations is computed in terms of crepant weightings and the Newton polyhedron of \(f\). crepant divisors; crepant weightings; Newton polyhedron Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Global theory of complex singularities; cohomological properties, Complex surface and hypersurface singularities On the number of crepant valuations of canonical singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Translation from Mat. Sb., N. Ser. 89(131), 297--312 (1972; Zbl 0226.14003). I. DOLGACEV , The Euler Characteristic of a Family of Algebraic Varieties (Math. U.S.S.R. Sbornik, Vol. 18, 1972 ). MR 48 #6116 | Zbl 0263.14002 Schemes and morphisms, Curves in algebraic geometry, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) The Euler characteristic of a family of algebraic varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities An exposition of the algorithm for canonical resolution of singularities in characteristic zero [cf. \textit{E. Bierstone} and \textit{P. D. Milman}, Invent. Math. 128, No. 2, 207-302 (1997; Zbl 0896.14006)] is given. resolution; desingularization; blowing up Bierstone, E. and Milman, P.: Resolution of Singularities, Several Complex Variables, Berkeley Ca (1995--96), Math. Sci. Res. Institute Publications Vol. 37, Cambridge University Press (1999), pp. 43--78. Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) Resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S=\text{spec} R\) where \(R\) is a Noetherian normal complete local ring containing an algebraically closed field isomorphic to the residue field of \(R\). Such an \(S\) is called a normal surface singularity, and it is called a rational surface singularity if furthermore there is a desingularization \(\pi:X\to S\) such that the stalk of \(R^1\pi_*{\mathcal O}_X\) at the closed point is zero. Let \(\pi:X\to S\) be any desingularization of a rational surface singularity \(S\). In the present paper, normal \(S\)-schemes \(Y\) factoring \(\pi\) are related to complete ideals on \(S\) and the semifactorization theory for the latter is used to get a characterization of \(S\)-isomorphisms between \(S\)-schemes \(Y\). isomorphisms between schemes; normal surface singularity; rational surface singularity; desingularization; complete ideals; semifactorization Cossart, V.; Piltant, O.; Reguera-López, A. J.: On isomorphisms of blowing-ups of complete ideals of a rational surface singularity. Manuscripta math. 98, No. 1, 65-73 (1999) Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Rational and birational maps On isomorphisms of blowing-ups of complete ideals of a rational 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 See the review of the similar (but much longer) paper of the author [Bull. Am. Math. Soc., New Ser. 35, No. 4, 319--331 (1998; Zbl 0928.14012)]. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Alterating singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We give a birational reduction of singularities for one dimensional foliations in ambient spaces of dimension three. To do this, we first prove the existence of a local uniformization in the sense of Zariski. The reduction of singularities is then obtained by a gluing procedure for Local Uniformizations similar to the one used by Zariski. F. Cano C. Roche M. Spivakovsky Reduction of singularities of three-dimensional line foliations Global theory and resolution of singularities (algebro-geometric aspects), Singularities of holomorphic vector fields and foliations, Singularities in algebraic geometry Reduction of singularities of three-dimensional line 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 [For the entire collection see Zbl 0665.00008.]
Let \(X'\) be a covering of \({\mathbb{P}}^ 2\) ramified along a union of lines with \(ramification\quad index\quad n\) and let X be its minimal desingularization, which admits a finite morphism \(X\to F\) where F is a suitable blow-up of \({\mathbb{P}}^ 2\). The Galois group A of X over F acts on X, hence on its cohomology. The main result of this paper is a formula which computes \(e(\alpha)=\sum_{i}\dim H^ i(\alpha) \), where \(\alpha\) is a character of A and \(H^ i(\alpha)\) is the corresponding eigenspace of \(H^ i(X)\). Then the notion of generic character is introduced: roughly speaking this is a character which is non trivial on the inertia subgroups of A at the divisors of F corresponding to the lines of the configuration and the exceptional divisors of the blowing up of \({\mathbb{P}}^ 2\). For such a character \(\alpha\) it is proved that \(H^ 1(\alpha)=H^ 3(\alpha)=0\) and the dimensions of \(H^ 2(\alpha)\) and of its Hodge components \(H^{2,0}(\alpha)\), \(H^{1,1}(\alpha)\), \(H^{0,2}(\alpha)\) are computed. It may be worth remarking that the interest on such coverings of \({\mathbb{P}}^ 2\) was called some years ago by \textit{F. Hirzebruch} [in Arithmetical and geometry Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Proc. Math. 36, 113-140 (1983; Zbl 0527.14033)] who proved that many surfaces \(X'\) of this kind are of general type and have Chern numbers in the non densely populated region \(2c_ 2(X)\leq c_ 1(X)^ 2\leq 3c_ 2(X)\). Hodge numbers; Kummer covering; Miyaoka-Yau-inequality; ramification; minimal desingularization; generic character; Chern numbers Coverings in algebraic geometry, Projective techniques in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Transcendental methods, Hodge theory (algebro-geometric aspects), Singularities in algebraic geometry, Characteristic classes and numbers in differential topology Hodge numbers of a Kummer covering of \({\mathbb{P}}^ 2\) ramified along a line configuration | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Hereafter, we refer to [\textit{R. Hartshorne}, Algebraic geometry. York-Heidelberg-Berlin: Springer-Verlag (1977; Zbl 0367.14001)] for any unexplained terminology.
Let \(\pi: Y\rightarrow X\) be a morphism of reduced schemes of finite type over a field \(\mathbb{K}\) and suppose that \(\mathfrak{a}\) is a sheaf of ideals on \(X\). We say that \(\pi\) is a log resolution of \(\mathfrak{a}\) provided it satisfies the following four conditions. {\parindent=6mm \begin{itemize}\item[1.] \(\pi\) is birational and proper. \item[2.] \(Y\) is smooth over \(\mathbb{K}\). \item[3.] \(\mathfrak{a}\mathcal{O}_Y =\mathcal{O}_Y (-G)\) is an invertible sheaf corresponding to a divisor, namely \(-G\). \item[4.] If \(E\) is the exceptional set of \(\pi\), then \(\operatorname{Supp} (G)\cup E\) has simple normal crossings.
\end{itemize}} Moreover, we say that \(\pi\) is a strong log resolution if it is a log resolution and \(\pi\) is an isomorphism outside of the subscheme \(V(\mathfrak{a})\) defined by \(\mathfrak{a}\). It is well known, by the celebrated results obtained by \textit{H. Hironaka} in [Ann. Math. (2) 79, 109--203, 205--326 (1964; Zbl 0122.38603)], that log resolutions always exist if the characteristic of \(\mathbb{K}\) is zero and strong log resolutions always exist in case \(X\) is smooth and \(\mathbb{K}\) has characteristic zero. However, despite the effort of many researchers the existence of resolution of singularities over a field of prime characteristic is still an open question in general.
Loosely speaking, in characteristic zero Hironaka showed that it is possible to obtain a (not necessarily unique) resolution of singularities through a sequence of normalizations and blowups at smooth centers. It turns out that this strategy does not work in prime characteristic; regardless, it is natural to ask the following:
Question. Is there a notion in prime characteristic which might play the role of blowup in characteristic zero?
Inspired by this question, \textit{T. Yasuda} in [Am. J. Math. 134, No. 2, 349--378 (2012; Zbl 1251.14002)] defined the so-called \(e\)th \(F\)-blowup, with the hope that it might play the same role as the blowup in characteristic zero. Roughly speaking, the idea is, given a (possibly singular) algebraic variety over a field of prime characteristic, to construct another algebraic variety \(Y\) on which the Frobenius endomorphism is flat. The idea is, of course, inspired by the classical result obtained by \textit{E. Kunz} in [Am. J. Math. 91, 772--784 (1969; Zbl 0188.33702)] which characterizes the regularity of a commutative reduced ring \(R\) of prime characteristic in terms of the fact that the Frobenius map on \(R\) is flat.
Yasuda's definition works as follows. Let \(\mathbb{K}\) be an algebraically closed field of prime characteristic, and let \(X\) be an algebraic variety over \(\mathbb{K}\) of dimension \(n\). The \(e\)th \(F\)-blowup \(\operatorname{FB}_e (X)\) of \(X\) is defined to be the closure of the subset
\[
\{ (F^e)^{-1} (x)\mid x\in X\text{ smooth}\}\subseteq\operatorname{Hilb}_{p^{ne}} (X^{(e)}),
\]
where \(X^{(e)}\) is the ringed space \((X,F_*^e \mathcal{O}_X)\) and \(\operatorname{Hilb}_{p^{ne}} (X^{(e)})\) denotes the Hilbert scheme of zero dimensional subschemes of \(X^{(e)}\) of length \(p^{ne}\). We refer to [Zbl 0188.33702] for more details.
From now on, we restrict our attention to the case of surfaces. In this case, given a surface \(S\) it is known that, in any characteristic, there exists a minimal resolution of singularities \(S_0\rightarrow S\). So, it is natural to ask the following:
Question. Let \(\mathbb{K}\) be an algebraically closed field of prime characteristic, let \((S,x)\) be a normal surface singularity and let \(S_0\rightarrow S\) be the minimal resolution. When is \(\operatorname{FB}_e (S)\) equal to the minimal resolution \(S_0\)?
It is true that \(\operatorname{FB}_e (S)=S_0\) for \(e\gg 0\) if either \(S\) is a toric singularity, a tame quotient singularity, or an \(F\)-regular double point.
We wish to introduce an additional notion before starting properly our review. Let \(R\) be an integral domain of prime characteristic \(p\) which is \(F\)-finite. We say that \(R\) is strongly \(F\)-regular if, for any nonzero element \(c\in R\), there exists a power \(q=p^e\) such that the inclusion map \(c^{1/q}R\hookrightarrow R^{1/q}\) splits as an \(R\)-module homomorphism.
In the paper under review, the author shows as main result that if \((S,x)\) is an strongly \(F\)-regular surface singularity, then its \(e\)th \(F\)-blowup \(\operatorname{FB}_e (S)\) coincides with the minimal resolution of \(X\) for \(e\gg 0\). This result generalizes an earlier one obtained by \textit{N. Hara} and \textit{T. Sawada} in [RIMS Kôkyûroku Bessatsu B24, 121--141 (2011; Zbl 1228.13009)] which was proved for \(F\)-rational double points. One of the technical tools developed by the author for that purpose, which is interesting in its own right, is a nice characterization of complete strongly \(F\)-regular rings of dimension \(2\) over \(\mathbb{K}\) (cf. Theorem 2.1). Finally, the author raises several questions related with these topics in order to stimulate further research about \(F\)-blowups and strongly \(F\)-regular rings. Hara, Nobuo: F-blowups of F-regular surface singularities, Proc. amer. Math. soc. 140, 2215-2226 (2012) Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics F-blowups of F-regular surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We define normalized versions of Berkovich spaces over a trivially valued field \( k\), obtained as quotients by the action of \( \mathbb{R}_{>0}\) defined by rescaling semivaluations. We associate such a normalized space to any special formal \( k\)-scheme and prove an analogue of Raynaud's theorem, characterizing categorically the spaces obtained in this way. This construction yields a locally ringed \( G\)-topological space, which we prove to be \( G\)-locally isomorphic to a Berkovich space over the field \( k((t))\) with a \( t\)-adic valuation. These spaces can be interpreted as non-archimedean models for the links of the singularities of \( k\)-varieties, and allow us to study the birational geometry of \( k\)-varieties using techniques of non-archimedean geometry available only when working over a field with non-trivial valuation. In particular, we prove that the structure of the normalized non-archimedean links of surface singularities over an algebraically closed field \( k\) is analogous to the structure of non-archimedean analytic curves over \( k((t))\) and deduce characterizations of the essential and of the log essential valuations, i.e., those valuations whose center on every resolution (respectively log resolution) of the given surface is a divisor. Rigid analytic geometry, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Normalized Berkovich spaces and surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article under review is an introduction to the theory of algorithmic resolution of singularities of algebraic varieties and local principalization of sheaves of ideals. The authors explain the basic notions and discuss some natural properties of these algorithms, such as compatibility with pull-backs via smooth morphisms, equivariance, changes of base field, etc. They emphasize the importance of some fundamental ideas of Hironaka on the subject, such as his ``fundamental invariant'' (a certain fraction, involving the order of an ideal) and a notion of equivalence (requiring equalities of certain closed sets, which are singular loci). The authors present a fairly complete proof a an algorithmic resolution theorem (in characteristic zero), which is a streamlined version (and simplification) of one developed by Villamayor several years ago explained, for instance, in Chapter 6 of [\textit{S. D. Cutkosky}, Resolution of singularities. Graduate Studies in Mathematics 63. Providence, RI: American Mathematical Society (AMS). (2004; Zbl 1076.14005)]. Singularities; resolution of singularities; equivalence; basic object; log-principalization; equivariance Benito, A., Encinas, S., Villamayor, O.: Some natural properties of constructive resolution of singularities. Asian J. Math. 15, 141-192 (2011) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Some natural properties of constructive resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite subgroup of \(SL(3,\mathbb C)\). The Hilbert scheme \(\text{Hilb}^G(\mathbb C^3)\) is by definition the subscheme of \(\text{Hilb}^{\mid G\mid}(\mathbb C^3)\) parametrizing \(G\)-invariant subschemes. It has been proved that \(\text{Hilb}^G(\mathbb C^3)\) is irreducible, smooth and it is a crepant resolution of \(\mathbb C^3/G\). In this paper, the authors study this Hilbert scheme when \(G\) is a non-abelian simple subgroup of \(SL(3,\mathbb C)\). There are two such subgroups, \(G_{60}\) and \(G_{168}\), of order 60 and 168 respectively. \(G_{60}\) is isomorphic to the alternating group of degree 5 and \(G_{168}\) is isomorphic to \(PSL(2,7)\). The authors are particularly interested in giving a precise description of the fibre over the origin of \(\mathbb C^3/G\). It turns out that in the first case this fibre is a connected union of four smooth rational curves and in the second one it is a union of a smooth rational curve and a doubly blown-up projective plane, with infinitely near centres. \(G\)-invariant subschemes; crepant resolution; Hilbert scheme Gomi, Y., Nakamura, I., Shinoda, K.: Hilbert schemes of G-orbits in dimension three. Asian J. Math. 4(1), 51--70 (2000; Kodaira's issue) Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Linear algebraic groups over the reals, the complexes, the quaternions Hilbert schemes of \(G\)-orbits in dimension three | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a normal variety and \(\Delta\) an \(\mathbb{R}\)-divisor on \(X\) such that \(K_X+\Delta\) is \(\mathbb{R}\)-Cartier. Let \(f: Y\to X\) be a resolution such that the support of \(\Delta_Y:=f^*(K_X+\Delta)-K_Y\) is a simple normal crossing. Then in [``Theory of non-lc ideal sheaves: basic properties'', Kyoto J. Math. 50, No. 2, 225--245 (2010; Zbl 1200.14033)], the first named author defined the non-lc ideal sheaf \(\mathcal{J}_{NLC}(X,\Delta)\) as
\[
\mathcal{J}_{NLC}(X,\Delta):=f_*\mathcal{O}_Y(-\lfloor \Delta_Y\rfloor+\Delta_Y^{=1}),
\]
and he proves that \(\mathcal{J}_{NLC}(X,\Delta)\) satisfies the following 4 properties:
{\parindent=10mm \begin{itemize}\item[(A)] The pair \((X,\Delta)\) is log canonical if and only if \(\mathcal{J}_{NLC}(X,\Delta)=\mathcal{O}_X\). \item[(B)] Assume that \(X\) is projective and \(D\) is a Cartier divisor on \(X\) such that \(D-(K_X+\Delta)\) is ample. Then
\[
H^i(X,\mathcal{J}_{NLC}(X,\Delta)\otimes \mathcal{O}_X(D))=0
\]
for each \(i>0\). \item[(C)] Let \(H\) be a general member of a free linear system \(\Lambda\) on \(X\). Then
\[
\mathcal{J}_{NLC}(X,\Delta)=\mathcal{J}_{NLC}(X,\Delta+H).
\]
\item[(D)] Assume that \(\Delta=S+B\) such that \(S\) is a normal prime Weil divisor, \(B\) is an effective \(\mathbb{Q}\)-divisor, and \(S,B\) have no common irreducible components. Then
\[
\mathcal{J}_{NLC}(X,\Delta)|_S=\mathcal{J}_{NLC}(S,B_S)
\]
where \((K_X+\Delta)|_S=K_S+B_S\).
\end{itemize}}
This motivates the authors of the paper under review to pursue a more general theory of ideal sheaves that satisfy properties (A), (B), (C), and (D). To this end, the authors introduce
{\parindent=6mm
\begin{itemize}\item[-] maximal non-lc ideal: \newline \(\mathcal{J}'(X,\Delta):=f_*\mathcal{O}_Y(\lceil K_Y-f^*(K_X+\Delta)+\varepsilon F\rceil)\) with \(0<\varepsilon \ll 1\) and \(F=\mathrm{Supp}(\Delta^{\geq 1}_Y)\).
\item[-] intermediate non-lc ideal: \newline \(\mathcal{J}'_{\ell}(X,\Delta):=f_*\mathcal{O}_Y(-\lfloor \Delta_Y\rfloor +\Delta^{=1}_Y+\sum^{\infty}_{k=2-\ell} {^k\Delta_Y} )\) for \(\ell=0,-1,\dots,-\infty\).
\item[-] non-klt ideal: \newline \(\mathcal{J}_{\ell}(X,\Delta):=f_*\mathcal{O}_Y(-\lfloor \Delta_Y\rfloor +\sum^{\infty}_{k=2-\ell} {^k\Delta_Y} )\) for \(\ell=0,-1,\dots,-\infty\).
\end{itemize}} with the following properties:
{\parindent=6mm
\begin{itemize}\item[1.] \(\mathcal{J}(X,\Delta)=\mathcal{J}'_{-\infty}(X,\Delta)\subset \cdots\subset \mathcal{J}_0(X,\Delta)\subset \mathcal{J}_{NLC}(X,\Delta)\subset \mathcal{J}'(X,\Delta)\), where \(\mathcal{J}(X,\Delta)\) is the multiplier ideal.
\item[2.] When \(\Delta\) is effective, all three aforementioned ideal sheaves satisfy properties (A), (B), and (C).
\item[3.] \(\mathcal{J}'(X,\Delta)\) satisfies (D) when \(X\) is a complete intersection.
\end{itemize}}
In the second part of this paper, a characteristic \(p\) analogue of the maximal non-lc ideal, called non-\(F\)-pure ideal, is introduced. Using \(F\)-adjunction tricks, the authors establish a restriction theorem for non-\(F\)-pure ideals; results on connection between non-\(F\)-pure ideals and maximal non-lc ideals are also discussed. multiplier ideal; non-lc ideal; maximal non-lc ideal; intermediate non-lc ideal; log canonical center; test ideal; non-\(F\)-pure ideal Osamu Fujino, Karl Schwede, and Shunsuke Takagi, Supplements to non-lc ideal sheaves, Higher dimensional algebraic geometry, RIMS Kôkyûroku Bessatsu, B24, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, pp. 1 -- 46. Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Supplements to non-lc ideal sheaves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0653.00006.]
This is a survey describing the results in and the background for the author's papers in J. Reine Angew. Math. 359, 90-105 and 362, 4-24 (1985; Zbl 0603.14006 and 14007)]. The problem is to find criteria for the infinite dimensionality of the so called \(A_ 0(X)\), the kernel of the degree map from the Chow group of zero cycles of a normal projective surface. Bloch's conjecture; Chow group; zero cycles (Equivariant) Chow groups and rings; motives, Algebraic cycles, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes) Zero cycles on a singular surface: An introduction | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is known that every topological nature of a normal complex surface singularity is described in terms of the weighted dual graph of the exceptional set a good resolution.
The main result of this paper is that for a given topological type \(\Gamma\) of a normal surface singularity, though the families of generic hyperplane sections and that of polar curves of the generic plane projections of a singularity with graph \(\Gamma\) depend on the complex analytic type, the topological types of them are finite. In other words, the number of the weighted dual graph for the minimal good resolution which factors through the blowup of the maximal ideal and the Nash transform of a singularity is finite.
One of key points for the proof is that the upper bounds of the multiplicity and the polar multiplicity, which are numerical invariants associated with generic hyperplane sections and the generic plane projections, are given by \(\Gamma\). Many arguments in the proof rely on [\textit{A. Belotto da Silva} et al., Geom. Topol. 26, No. 1, 163--219 (2022; Zbl 1487.32166)] and related results; however, this article includes concise explanations for those ingredients. complex surface singularities; polar curves; hyperplane sections; Nash transform; Lipschitz geometry; multiplicity; Mather discrepancy Singularities in algebraic geometry, Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects) Polar exploration of complex surface germs | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In a previous article [\textit{T. Yasuda} , Compos. Math. 143, 1493--1510 (2007; Zbl 1135.14011)], the author introduced the notion of (simple) \(n\)-th Nash blowup \(Nash_n(X)\) of an algebraic variety \(X\) (defined over a field \(k\), algraically closed field of characteristic zero). This is the closure (in a suitable Hilbert scheme \(H\)) of the points corresponding to (zero-dimensional) subschemes of \(X\) of the form \(x^{(n)}= {\text{Spec}}({\mathcal O}_{X,x}/M_x ^{n+1})\), where \(x\) is a closed regular point of \(X\). In the present paper the author points out (via examples) some shortcomings of that definition, e.g., it does not behave well with respect to products and smooth morphisms \(X' \to X\)) and proposes a variant thereof, which avoids these problems. This object is the flag n-th Nash blowup, \(fNash_n(X)\), which is the closure (in \(H^{n+1}\)) of the sequences \((x^{(0)}, x^{(1)}, \ldots, x^{(n)})\), \(x\) a closed point in the regular set of \(X\), using the previous notation.
There is a natural projection \(p_n:fNash_n(X) \to X\), which is proper and birational. The author also introduce a ``formal'' version of the simple and flag n-blowups , which applies to schemes of the form \({\mathbf X}= {\text{Spec}}(A)\), with \(A\) a complete local noetherian ring with coefficient field \(k\).
Some results proved in this paper are: (a) If \(x \in X\) is a closed point, \((A,M)\) the local ring of \(X\) at \(x\) and \(B\) its completion, \(\mathbf Y = {\text{Spec}}( B)\), then \(fNash_n({\mathbf Y})\) is naturally induced by \(fNash_n(X)\); similarly with \(Nash_n(X)\). (b) The mentioned compatibility of \(fNash_n(X)\) with products and smooth morphisms. (c) Some results on normality, e.g., \(z \in Nash(X)\) is normal if the corresponding subscheme \(Z \subset X\) is not contained in the conductor subscheme; a similar result holds for \(fNash (X)\). In the proof one uses some associated semigroups. (d) Some criteria for the regularity of the \(fNash_n\) of products of formal irreducible curves, involving the semigroups of (c). There are detailed calculations for the formal plane curve defined by \(x^5-y^7\), which provide the examples mentioned before.
The paper concludes with the statement of some problems. One of these asks: is \(Nash_n(X)=fNash_n(X)\) if \(\dim X \geq 2\)? Higher Nash blowing-up; Hilbert scheme of points; singular point; normal point Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) Flag higher Nash 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 [For the entire collection see Zbl 0588.00014.]
Let X be a quasiprojective variety over a field and Y a closed subset containing the singular locus of X and contained in an affine open subset of X. The authors define and study relative Chow groups \(CH^ P(X,Y)\) and relative higher algebraic K-sheaves \({\mathcal K}_ n(X,Y)\). One establishes isomorphisms \(H^ p(X,{\mathcal K}_ p(X,Y))\simeq CH^{p+1}(X,Y)\) and exact sequences
\[
...\to H^{p-1}(X_ Y,{\mathcal K}_ p)\to CH^ p(X,Y)\to H^ p(X,{\mathcal K}_ p)\to H^ p(X_ Y,{\mathcal K}_ p)\to 0
\]
(here \(X_ Y\) is the intersection of all the neighbourhoods of Y which are cut out by divisors). Special emphasis is given to surfaces: one gives examples for which \(CH^ 2(X,Y)\not\cong H^ 2(X,{\mathcal K}_ 2)\) and one shows that \(H^ 2(X,{\mathcal K}_ 2)\simeq SK_ 0(X)\) when X is affine. The paper extends the work of \textit{A. Collino} [Ill. J. Math. 25, 654-666 (1981; Zbl 0496.14005)], which considers varieties with only one singular point. singular varieties; Chow groups; higher algebraic K-sheaves C. Pedrini and C. Weibel, \(K\)-theory and Chow groups on singular varieties , Applications of algebraic \(K\)-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 339-370. Parametrization (Chow and Hilbert schemes), Applications of methods of algebraic \(K\)-theory in algebraic geometry, (Equivariant) Chow groups and rings; motives, Singularities in algebraic geometry K-theory and Chow groups on singular varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0668.00006.]
Let \(W^ m\subset {\mathbb{C}}^{m+1}\) be a local analytic hypersurface given by the equation \(\{f=0\}\). For \(x\in W\), the author considers two limit spaces of tangents \(K(W,x)\subset K(f,x)\subset {\mathbb{P}}^ m\) which are associated in a natural and familiar way to W and f. The variation of these spaces in a family of singularities is studied (mainly in order to detect equisingularity properties). family of singularities; equisingularity Singularities in algebraic geometry, Complex singularities, Global theory and resolution of singularities (algebro-geometric aspects) Spaces of limiting tangent 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 author deals with a complex projective variety \(X\) with quotient singularities. \(X\) is thought of as being the coarse moduli space of a smooth D-M stack \(\mathcal X\), which was first done for the case of moduli spaces of stable curves by \textit{P. Deligne } and \textit{D. Mumford} [Publ. Math., Inst. Hautes Étud. Sci. 36, 75--109 (1969; Zbl 0181.48803)]. Invariants of the singular variety \(X\) can be understood by looking at the geometry of \(\mathcal X\). Consider the inertia stack \({I \mathcal X } \), its Chern class \(c (T_{I \mathcal X } )\), the natural projections \(p: {I \mathcal X } \to \mathcal X \) and \(\pi: \mathcal X \to X \). Take \(c^{SM}(X)\), the Chern-Schwartz-MacPherson class of \(X\), the first main result of the paper is the equality \(c^{SM}(X) = \pi_{\ast} p_ {\ast} c (T_{I \mathcal X } )\). Stringy Chern classes \(c_{str}(X)\) have been defined and studied independently by \textit{P. Aluffi } [in: Trends in Mathematics, 1--13 (2007; Zbl 1120.14004)] and \textit{T. de Fernex}, \textit{E. Lupercio}, \textit{T. Nevins}, and \textit{B. Uribe} [Adv. Math. 208, No. 2, 597--621 (2007; Zbl 1113.14008)]. Let \(q:{II \mathcal X } \to { \mathcal X }\) be the natural projection from the double inertia stack to \( { \mathcal X }\), the second main result proved here says \(c_{str}(X) = \pi_{\ast} q_ {\ast} c (T_{II \mathcal X } )\). Some consequences for the stingy/orbifolds numbers are then drawn, under certain additional hypotheses, among them the request that \(X\) is Gorenstein. Chern classes; algebraic varieties; algebraic stacks Tseng, H. -H.: Chern classes of Deligne-Mumford stacks and their coarse moduli spaces, Amer. J. Math. 133, 29-38 (2011) Stacks and moduli problems, Singularities in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) Chern classes of Deligne-Mumford stacks and their coarse moduli spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the case of positive characteristic, it is known that there exist fibrations whose general fibres have singularities and that such fibrations often have pathological phenomena. Meanwhile, we can easily construct surfaces with such fibration as the quotients of surfaces with smooth fibration by p-closed rational vector fields.
In the present article, we give a description of the singularities of fibres in terms of rational vector fields. - As an application of our theory, we consider surfaces with fibration such that all the fibres are rational curves with one cusp of type \(x^ p+y^ n=0\) \((p=char(k)\), \((p,n)=1)\), which we call generalized Raynaud surface. In particular, we construct a sequence \(\{(x_ n,{\mathcal L}_ n)\}_ n\) consisting of pairs of generalized Raynaud surfaces \(X_ n\) and ample invertible sheaves \({\mathcal L}_ n\) on \(X_ n\) such that \(\lim_{n\to \infty}\dim H^ 1(X_ n,{\mathcal L}_ n^{-1})=\infty\). moving cuspidal singularities; positive characteristic; fibration by p- closed rational vector fields; singularities of fibres; generalized Raynaud surface Takeda, Y.: Fibrations with moving cuspidal singularities. Nagoya Math. J.122, 161-179 (1991) Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Families, fibrations in algebraic geometry, Singularities in algebraic geometry, Singularities of curves, local rings Fibrations with moving cuspidal singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field of characteristic \(p>0\) and \(X_1, X_2\) be two smooth proper connected curves. Let \(\sigma_i: X_i\to X_i\) be an automorphism of order \(p\) and denote by \(\sigma\) the automorphism \(\sigma_1\times \sigma_2: X_1\times X_2\to X_1\times X_2=:Y\). It is proved that the graph of the resolution of any singularity of \(Y/\langle\sigma \rangle\) is a star-shaped graph with three terminal chains when \(X_2\) is an ordinary curve of positive genus. The intersection matrix of the resolution has determinant \(\pm p^2\). The singularity is rational. It is proved that for any \(s>0\) not divisible by \(p\) there are resolution graphs of wild \(\mathbb{Z}/p\mathbb{Z}\) quotient singularities with one node, \(s+2\) terminal chains and intersection matrix having determinant \(\pm p^{s+1}\). product of curves; cyclic quotient singularity; rational singularity; wild; intersection matrix; resolution graph; fundamental cycle Lorenzini, D.: Wild quotients of products of curves (2012, preprint) Singularities in algebraic geometry, Local ground fields in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings, Multiplicity theory and related topics, Singularities of surfaces or higher-dimensional varieties Wild quotients of products of curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(d\) be a positive integer, \(k\) be a field of characteristic zero containing all \(d\)-th roots of unity, and let \(G\) be a finite subgroup of order \(d\) in \(\text{SL}(k,n)\) acting on the affine space \(\mathbb A^n_k\). Consider a resolution \(Y\to X\) of singularities on the quotient \(X=\mathbb A_k^n/G\) and assume that \(Y\) is crepant, i.e. \(K_Y=0\). The McKay correspondence is a connection between irreducible representations of the group \(G\) and cohomology of \(Y\). In one form it says that the Euler number of \(Y\) is equal to the number of conjugacy classes in \(G\) [\textit{M. Reid}, in: Séminaire Bourbaki, Vol. 1999/2000. Astérisque No. 276, 53--72 (2002; Zbl 0996.14006)], and it was proved by \textit{V. Batyrev} [J. Eur. Math. Soc. 1, No. 1, 5--33 (1999; Zbl 0943.14004)]. The present paper is devoted to a proof of the corresponding statement on the motivic level by means of motivic integration.
To be slightly more precise, let \(\mathcal M\) be the Grothendieck group of algebraic varieties over \(k\) with relation \([X]=[Z]+[X-Z]\) for Zariski closed \(Z\) in \(X\) and the product induced by products of varieties. Following the notation of the paper, let \(\mathcal M_{\text{loc}}\) be the localization \(\mathcal M[\mathbb L^{-1}]\), where \(\mathbb L=[\mathbb A^1]\). Let also \(F^m\mathcal M_{\text{loc}}\) be a subgroup generated by \([X]\mathbb L^{-i}\) with \(\dim (X)\leq i-m\). The filtration \(F^m\) gives the completion \(\hat {\mathcal M}\) of \(\mathcal M_{\text{loc}}\). At last, we add the relation \([V/G]=[V]\) for each \(k\)-vector space with a linear action of a finite group \(G\) getting the corresponding quotient ring \(\hat \mathcal M_{\slash }\).
The above rings are closely connected with the category of Chow motives \(\text{CHM}_k\) over \(k\) with coefficients in \(\mathbb Q\). Namely, there exists a function \(\chi _c\) from the set of varieties over \(k\) to \(K_0(\text{CHM}_k)\) satisfying the nice properties listed on page 283 of the paper. The analogous filtration on \(K_0(\text{CHM}_k)\) gives rise to the completion \(\hat K_0(\text{CHM}_k)\). The map \(\chi _c\) induces ring homomorphisms \(\chi _c:\mathcal M\to K_0(\text{CHM}_k)\) and \(\hat \chi _c:\hat \mathcal M\to \hat K_0(\text{CHM}_k)\), which can be factored through \(\mathcal M_{\slash }\) and \(\hat \mathcal M_{\slash }\) respectively.
Given a variety \(X\) over \(k\) let \(\mathcal L(X)\) be the scheme of germs of arcs on \(X\). For any field extension \(K/k\) one has a natural bijection \(\mathcal L(X)(K)\cong \text{Hom}_k(K[[t]],X)\) where \(K[[t]]\) is the ring of formal power series with coefficients in \(K\). If \(B^t\) is a set of \(k[t]\)-semi-algebraic subsets in \(\mathcal L(X)\), then there is a nice measure \(\mu :B^t\to \hat \mathcal M\), called a motivic measure on \(\mathcal L(X)\). Assume that \(X\) is an irreducible normal variety of dimension \(n\), which is Gorenstein with at most canonical singularities at each point. Some appropriate notion of integration with respect to \(\mu \) with values in the ring \(\hat \mathcal M\) gives rise to the notion of motivic Gorenstein measure
\[
\mu^{\text{Gor}}(A)=\int _A{\mathbb L}^{-\text{ord}}_{t\omega _X}{\text{ d}}\mu
\]
of each subset \(A\in B^t\). Here \(\text{ord}_t\omega _X\) is the order of a global section \(\omega _X\) of \(\Omega _X^n\otimes k(X)\) generating \(\Omega _X^n\) at each smooth point of \(X\) (use that \(X\) is Gorenstein with good singularities).
Now we are returning to the quotient \(X=\mathbb A^n_k/G\), where \(G\) is a finite subgroup of \(\text{SL}(k,n)\). Let \(\mathcal L(X)_0\) be the set of arcs whose origins are in the image of the point \(0\) in the quotient \(X\). The main result of the paper expresses the Gorenstein motivic measure \(\mu^{\text{Gor}}(\mathcal L(X)_0)\) in terms of weights \(w(\gamma )\) of conjugacy classes \(\gamma \) in \(G\). Namely, the equality
\[
\mu ^{\text{Gor }}(\mathcal L(X)_0)= \sum _{\gamma \in \text{Conj}(G)}\mathbb L^{-w(\gamma )}
\]
holds in \(\hat \mathcal M_{\slash }\), where \(\text{Conj}(G)\) is the set of conjugacy classes of \(G\). As a corollary, if \(h:Y\to X\) is a crepant resolution, then
\[
[h^{-1}(0)]=\sum _{\gamma \in \text{Conj}(G)}\mathbb L^{n-w(\gamma )}
\]
in the ring \(\hat \mathcal M_{\slash }\). This is already a motivic expression of the McKay correspondence. If \(k=\mathbb C\) and we pass to the Hodge realization, we get the result proved by Batyrev and conjectured by Reid. germs of arcs; motivic measure; irreducible representation; crepant resolution Denef, Jan; Loeser, François, Motivic integration, quotient singularities and the mckay correspondence, Compos. Math., 131, 3, 267-290, (2002) Arcs and motivic integration, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Schemes and morphisms, Generalizations (algebraic spaces, stacks), Group actions on varieties or schemes (quotients) Motivic integration, quotient singularities and the McKay correspondence | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 arbitrary algebraically closed field. For any finite subgroup \(G\) of \(\text{SL}(3,k)\) of order prime to the characteristic of the ground field \(k\) the quotient space \(\mathbb{A}^3_k/G\) is then a normal Gorenstein variety with trivial canonical sheaf. In birational geometry, a natural choice of a crepant smooth resolution of the singular variety \(\mathbb{A}^3_k/G\) would be very convenient, but it seems that the general theories in this framework, such as the theory of minimal models and the theory of torus embeddings, do not canonically provide a resolution of this type.
In the paper under review, the author approaches this problem by studying a particular Hilbert scheme \(\text{Hilb}^G(\mathbb{A}^3_k)\), which finally turns out to be a canonical crepant resolution of the quotient variety \(\mathbb{A}^3_k/G\). The so-called \(G\)-orbit Hilbert scheme \(\text{Hilb}^G:= \text{Hilb}^G(\mathbb{A}^3_k)\) is, by definition, the scheme parametrizing all \(G\)-invariant smoothable \(0\)-dimensional subschemes of \(\mathbb{A}^3_k\) of length \(n:=|G|\). This object, which may be regarded as a certain substitute for the quotient \(\mathbb{A}^3_k/G\) was introduced by the author and \textit{Y. Itô} in [Proc. Japan Acad., Ser. A 72, No. 7, 135--138 (1996; Zbl 0881.14002)] as a suitable tool in the study of resolutions of quotient singularities via the famous McKay correspondence.
In the present paper, the \(G\)-orbit Hilbert scheme \(\text{Hilb}^G\) is described for a finite abelian subgroup \(G\) of \(\text{SL}(3,k)\) resulting in the fact that \(\text{Hilb}^G\) appears then as a smooth torus embedding associated to a crepant fan in \(\mathbb{R}^3\) with apices constructed from the group \(G\). Furthermore, it is shown that the commutativity of \(G\) implies the nonsingularity of that associated fan.
This finally establishes the author's main theorem (Theorem 0.1.) stating the following: For any abelian subgroup \(G\) of \(\text{SL}(3,k)\) of order prime to the characteristic of the ground field \(k\) the \(G\)-orbit Hilbert scheme \(\text{Hilb}^G(\mathbb{A}^3_k)\) is a crepant resolution of the quotient space \(\mathbb{A}^3_k/G\).
The first half of the present article is devoted to describing a \(G\)-orbit Hilbert scheme as a toric variety in arbitrary dimension. This part is based on a fine analysis of the corresponding lattices and \(G\)-graphes, which is highly interesting and important for its own sake. In the second half of the paper, the particular case of dimension three and an abelian subgroup \(G\) of \(\text{SL}(3,k)\) is inspected more closely by means of the special appearing \(G\)-graphs, culminating in the author's main theorem mentioned above. At the end of the article, the author discusses some interesting examples in dimension three and four, thereby illustrating the variety of possibilities that can occur already in those low-dimensional cases.
In a sense, the present work may be regarded as a complement to the related earlier results by \textit{Y. Itô} and \textit{M. Reid} [in: Higher-dimensional complex varieties. Proc. Int. Conf. Trento, Italy, June 15--24, 1994. 221--240 (1996; Zbl 0894.14024)] and by \textit{Y. Itô} and \textit{H. Nakajima} [Topology 39, 1155--1191 (2000; Zbl 0995.14001)]. quotient varieties; quotient singularities; resolution of singularities; toric varieties I. Nakamura, \textit{Hilbert schemes of abelian group orbits}, J. Algebraic Geom. \textbf{10} (2001), no. 4, 757-779. Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Hilbert schemes of abelian group orbits | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The goal of this paper is to present the theoretical basis for an algorithmic implementation of embedded resolution of arithmetical schemes. To this end, the authors use as a starting point \textit{V. Cossart} et al.'s paper [Desingularization: invariants and strategy. Application to dimension 2. With contributions by Bernd Schober. Cham: Springer (2020; Zbl 1477.14003)] on resolution of two-dimensional schemes.
The first step is to construct an upper semi-continuous function that stratifies the singular locus of the variety in a way that the maximum value determines the centers to blow up. In \textit{V. Cossart} et al.'s paper [Desingularization: invariants and strategy. Application to dimension 2. With contributions by Bernd Schober. Cham: Springer (2020; Zbl 1477.14003)] the upper-semi-continuous function is constructed taking into account the Hilbert-Samuel function. However, this function is difficult to implement. Thus the authors use a slightly different strategy following \textit{S. Encinas} and \textit{O. Villamayor}'s approach in [Rev. Mat. Iberoam. 19, No. 2, 339--353 (2003; Zbl 1073.14021)]. There, resolution of algebraic varieties is proved using the order of an ideal as main invariant. However, to use this approach, since the goal is to work with aritmetical schemes, some work is done to show that a suitable stratification of the singular locus can be constructed using the order of an ideal as main invariant.
Next, there is another non-trivial problem to face. How does one \textit{compute} the set of points with a given order of an ideal where there is not a good theory of differential operators at disposal? When working over arbitrary fields, this still can be worked out (e.g. if there are \(p\)-basis). But, If there is no base field, and the base is a Dedeking domain, then a technique is developed to search for the so called \textit{wbad primes}, i.e., primes over wchich singularities can be found (and that cannot be described using differential operators).
The paper is very nicely written and several examples are presented to clarify some difficult points. resolution of singularities; differential operators Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Computational aspects and applications of commutative rings, Software, source code, etc. for problems pertaining to commutative algebra Embedded desingularization for arithmetic surfaces -- toward a parallel implementation | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let X be a normal projective algebraic variety over an algebraically closed field and X' an effective divisor on X such that
(i) X' is isomorphic to a weighted projective space \({\mathbb{P}}\), and
(ii) the conormal sheaf of X' is ample.
The author proves that there is an unique contraction of X' to a point of a normal variety Y such that X is a generalized blowing up of Y.
Furthermore it is proved, under supplementary hypothesis, that all singularities obtained by contracting the same \({\mathbb{P}}\) are analytically isomorphic. blowing-down; effective divisor; contraction; singularities Rational and birational maps, Projective techniques in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Singularities in algebraic geometry Projective contractions along weighted projective spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author considers a finite cyclic group \(G\subset \text{GL}(2,\mathbb C) \) acting freely on \(\mathbb C^2\setminus \{ 0 \} \) and studies the singularity \(X= \mathbb C^2/ G\). The main theorems are:
Let \(I_{G}\) be an ideal of \({ \mathcal O }_{{ \mathbb C^2 }}\) defined by the free \(G\)-orbit. Then the Gröbner fan for the \(G\)-homogeneous ideal \(I_{G}\) determines a toric variety which is isomorphic to the minimal resolution of \(X\).
There is a bijection between irreducible special representations \(\rho_{k } \) of \(G\) and binomial generators in the initial ideal \(in_{w(I)}\).
This can be interpreted as a generalized MacKay correspondence. cyclic quotient singularities; Gröbner fan; MacKay correspondence Ito, Y.: Minimal resolution via Gröbner basis. Algebraic geometry in east Asia (Kyoto, 2001), 165-174 (2002) Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies Minimal resolution via Gröbner basis | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 results of the authors' paper [ibid. 361, No. 3--4, 689--723 (2015; Zbl 1331.14022)], especially Theorems 1.1, 1.4 and Corollary 1.2, are correct as
written. However, the final sentence in the statement of Proposition 1.3 is false when the quiver contains a loop. When this is the case, there exist
points for which the corresponding \(A\)-module \(V_y\) contains a submodule of dimension vector \(S_i\) that is not isomorphic to \(S_i\); note that any such \(V_y\) is not nilpotent.
This situation is very rare, but it does occur. McKay correspondence, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Group schemes, Geometric invariant theory, Derived categories and associative algebras, Representations of quivers and partially ordered sets Correction to: ``Geometric Reid's recipe for dimer models'' | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a report on important recent results on resolution of singularities of algebraic varieties and the theory of semi-stable reduction, found in the paper ``Smoothness, semi-stability and alterations'', by \textit{A. J. de Jong} [Publ. Math., Inst. Hautes Étud. Sci. 83, 51-93 (1996; Zbl 0916.14005)]. The key notion is ``alteration'': An alteration of an integral noetherian scheme \(X\) is a proper, surjective morphism \(\varphi:X'\to X\) (with \(X'\) integral, noetherian) such that, for a suitable open dense set \(U\subseteq X\), the induced morphism \(\varphi_U:\varphi^{-1}(U)\to U\) is finite. For instance, and working (to simplify) with projective varieties over a field \(k\), the author proves: Given an integral variety \(X\) and a closed subvariety \(Z\subset X\), there is an alteration \(\pi:X'\to X\) with \(X'\) regular, such that \(\pi^{-1} (Z)_{\text{red}}\) is a strict normal crossings divisor (i.e., the irreducible components are smooth and meet transversally). Note that there are no restrictions on the base field \(k\).
This is the method of proof: Obtain an alteration \(f:X_1\to X\) such that there is flat morphism \(g:X_1\to T\), with \(T\) regular, \(\dim(T)= \dim(X_1)-1\), such that the general fiber of \(g\) is regular and any fiber is a curve with, at worst, ordinary double points as singularities, moreover the set of points of \(T\) where the fiber of \(g\) is singular is contained in a strict normal crossings divisor. Then, to desingularize such a \(X_1\) by means of monoidal transformations is easy. An alteration as above is obtained by using some classical projective techniques, the theory of moduli for pointed semi-stable curves and an induction hypothesis (applied to \(T\), whose dimension is one less than that of \(X)\).
A. J. de Jong also proved (loc. cit.) a weak version of the general semi-stable reduction theorem where (essentially) one allows to substitute one of the relevant varieties involved by an alteration thereof. These results (of course, precisely stated) are discussed in this report. There is an essentially complete proof of the desingularization theorem and good sketch of the one for the reduction problem. The report is an excellent introduction to these topics.
But there is also a very useful section (the last one) on applications. P. Berthelot discusses three:
(1) O. Gabber's affirmative solution to Serre's problem on multiplicity of intersection for two modules over a local noetherian ring \(A\): ``\(\chi_A(M,N)\geq 0\)'' (no restrictions on the ring);
(2) a proof (due to Berthelot) showing that the Monsky-Washnitzer cohomology groups \(H^n_{MW} (X/K)\) (where \(X\) is a smooth affine scheme over a field \(k\) with \(\text{ch} (k)>0\), \(K\) the fractions of a Cohen ring of \(k)\) are finite dimensional vector spaces over \(K\);
(3) some recent work of Deligne on monodromy actions on étale cohomology groups (specially, an ``independence of \(l\)'' theorem).
All these results use the desingularization theorem of de Jong. semistable curve; moduli of curves; resolution of singularities; alteration; integral variety; monoidal transformations; semi-stable reduction theorem; multiplicity of intersection for two modules; Monsky-Washnitzer cohomology groups; monodromy actions on étale cohomology Berthelot, P., Altérations de variétés algébriques (d'après A.J. de jong), Séminaire Bourbaki, vol. 1995/96, Astérisque, 241, 273-311, (1997), Exp. No. 815, 5 Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Families, moduli of curves (algebraic), \(p\)-adic cohomology, crystalline cohomology, Étale and other Grothendieck topologies and (co)homologies Alterations of algebraic varieties (after A. J. de Jong) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Editorial remark: Due to personal communication of M. Roczen, this article is almost identical to his joint paper with the second author and \textit{B. Dgheim} [in: Topics in algebra. Part 2: Commutative rings and algebraic groups, Pap. 31st Semester Class. Algebraic Struct., Warsaw/Poland 1988, Banach Cent. Publ. 26, Part 2, 27--30 (1990; Zbl 0741.14019)], with the difference that the assumption \(\mathrm{char} (k)\neq 2\) is dropped but necessary since the considered equations do not define isolated singularities for \(\mathrm{char} (k)=2\). Global theory and resolution of singularities (algebro-geometric aspects), \(3\)-folds, Singularities in algebraic geometry, Picard groups, Finite ground fields in algebraic geometry Divisors of the canonical resolutions of some singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is well known that every smooth complex projective curve \(C\) is birational (via a generic linear projection) to a plane curve with no worse than ordinary double points. Similarly, if \(Y\) is a smooth projective variety of dimension \(n\), a suitable linear projection yields a hypersurface \(X\) that is birational to \(Y\). This paper examines the type of singularities lying on \(X\).
The main result of the paper states that if \(Y\) is embedded in \(\mathbb{P}^N\) via the \(d\)-uple embedding, where \(d \geq 3n\) and if \(\pi \colon \mathbb{P}^N \to \mathbb{P}^{n+1}\) denotes a generic linear projection, then \(X=\pi(Y)\) has no worse than semi log canonical singularities for \(n\leq 5\). More precisely, the author proves that \(X\) has Du Bois singularities, and that in this setting Du Bois singularities are semi log canonical. Du Bois singularities are interescting in their own right: by definition, their cohomology is easy to determine and they seem to provide the natural setting for vanishing theorems by Kollár's principle [see \textit{K. Schwede}, Compos. Math. 143, No. 4, 813--828 (2007; Zbl 1125.14002) and \textit{S. Kovács, K. Schwede} and \textit{K. Smith}, ``Cohen-Macaulay semi-log canonical singularities are Du Bois'', \url{arXiv:0801.1541}]. The proof relies on the direct analysis of the local analytic type of singularities that arise in such generic projections in \textit{J. Roberts} [Trans. Am. Math. Soc. 212, 229--268 (1975; Zbl 0314.14003)].
Finally, the author presents a counter example for \(n = 30\); a similar statement cannot therefore be expected to hold for arbitrarily large dimension. linear projection; singularities Doherty, DC, Singularities of generic projection hypersurfaces, Proc. Am. Math. Soc., 136, 2407-2415, (2008) Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Singularities of generic projection 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 Let (R,F,G,H) be a 4-tuple, with R a regular local ring, F,G,H pairwise coprime ideals in R such that GH has a normal crossing at R. The problem is to show that, upon applying a suitable finite succession of blowing ups, FGH will have only normal crossings. Here F is of primarily interest, while G and H play an auxiliary role. To study this problem the author, inspired by Zariski's concept of equisingular point of the hypersurface of local equation F, introduces and studies in great details the concept of a good point of F. Roughly speaking, this is a point of F with the property that to simplify it, it takes the same effort as it does to simplify a point of a plane curve. The prototype of a good point in dimension 2 is the singularity at 0 given by \(Z^ d-X^ aY^ b=0\), with \(a+b\geq d>a'+b'\), where \(a'\) and \(b'\) are the residues of a and b modulo d. Although good points are cruder than equisingular points, the author shows that they are better adapted for the desingularization problem.
The paper has three parts. In the first part (the longest one) the author develops the local theory in dimension 2 aiming to prove that after sufficiently many monoidal transformations applied to a d-fold point of a surface, all the resulting d-fold points are good. The second part is a globalization of the first one, and, as a rather easy application, the author obtains a proof of the desingularization theorem for surfaces (via good points). The last part discusses some open problems.
The author has developed this theory of good points in 1966, and the reason of publishing it after more than twenty years is that in the meantime he succeeded in finding another proof of the desingularization theorem in arbitrary dimension; in this sense the present paper could be useful as a good introduction to higher-dimensional theory of good points. normal crossings; good point; desingularization [Ab1]Abhyankar, S.S., Good points of a hypersurface.Adv. in Math., 68 (1988), 87--256 Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Good points of a hypersurface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathcal{O}^x_{n+1}\) be the \(\mathbb{C}\)-algebra of germs of analytic functions at the origin \(\mathbf{0}\in \mathbb{C}^{n+1}\), defined by the usual equivalence relation arising from identical equality on neighborhoods of \(\mathbf{0}\). Let \(\mu(f) = \dim_{\mathbb{C}} \mathcal{O}^x_{n+1}/J(f)\) be the \textbf{Milnor number}, where \(J(f)\subset \mathcal{O}^x_{n+1}\) is the Jacobian ideal, generated by the partial derivatives of \(f\).
The article studies deformations \(F(x,s)\) of \(f\) preserving this Milnor number \(\mu(f)\). In this respect, \(\mu\)-constant deformations are properly defined. The paper then introduces the basics on the support of a convergent power series, its Newton polyhedron and its Newton boundary, as well as non-degeneracy of a deformation (namely, with respect to its Newton boundary). It then proceeds to define simultaneous embedded resolutions of the deformation of a hyperspace \(V\) by \(F\). An important ancillary proposition characterizes embedded topologically trivial deformations as being \(\mu\)-constant.
A terse, yet precise analysis throws light on the true operative gist of an earlier result by \textit{M. Oka} [Adv. Stud. Pure Math. 8, 405--436 (1987; Zbl 0622.14012)], namely the relation between \(\mu\)-constancy and embedded resolutions. This relation is amplified, generalized and formalized in the article's main theorem, which is is first stated in \S 1 and establishes that a deformation \(W\) is \(\mu\)-constant if and only if it admits a simultaneous embedded resolution. This theorem has a corollary stating that every non-degenerate \(\mu\)-constant deformation is topologically trivial. After a short disquisition on the actual novelty of the corollary and its relation to the Lê-Ramanujan conjecture, the article then spends \S 2 introducing the necessary background and preliminary results and proofs on Newton polyhedra, including Newton numbers \(\nu(P)\) of compact polyhedrons. Section 3 follows suit with the topic Newton non-degenerate \(\mu\)-constant deformations, and contains the proof of the main result (whose statement is repeated in the form of Theorem 3.2). It is also worth noting that \S 2 provides a complete solution to V. I. Arnold's Problem No. 1982-16 relating to the monotonicity of Newton numbers in the presence of convenient Newton polyhedra. The degenerate case is shortly addressed in the final section, wherein a twofold necessary condition on \(\mu\)-constancy is proved. algebraic geometry; singularities of algebraic varieties; deformations of singularities; simultaneous embedded resolutions; Milnor number; Newton number; m-jet spaces. Singularities in algebraic geometry, Deformations of singularities, Local structure of morphisms in algebraic geometry: étale, flat, etc., Global theory and resolution of singularities (algebro-geometric aspects), Embeddings in algebraic geometry, Equisingularity (topological and analytic), Local complex singularities, Invariants of analytic local rings Newton non-degenerate \(\mu\)-constant deformations admit simultaneous embedded 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 For a finite group \(G\subset \text{SO}(3, {\mathbb R})\) let \(\tilde{G}\) denote the inverse image via the double covering \(\text{SU}(2) \to \text{SO}(3,{\mathbb R})\). The moduli space of clusters \(\tilde{G}\text{-Hilb}({\mathbb C}^2)\) is a natural resolution of the quotient singularity \({\mathbb C}^2/\tilde{G}\), and where the exceptional curves correspond to irreducible representations of \(\tilde{G}\). Moreover, any irreducible representation of \(G\) is also an irreducible representation of \(\tilde{G}\). The authors construct a map between the moduli spaces \(\tilde{G}\text{-Hilb}({\mathbb C}^2) \to G\text{-Hilb}(\mathbb{C}^3)\), and they show that there is an induced map of exceptional divisors which contracts components that do not correspond to irreducible representations of \(G\). quotient singularities; McKay correspondence; Hilbert schemes; polyhedral groups Boissière, S.; Sarti, A., Contraction of excess fibres between the McKay correspondences in dimensions two and three, Ann. Inst. Fourier (Grenoble), 57, 1839-1861, (2007) Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Ordinary representations and characters Contraction of excess fibres between the McKay correspondences in dimensions two and three | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We give a formula for the local Denef-Loeser zeta function of a superisolated singularity of hypersurface in terms of the local Denef-Loeser zeta function of the singularities of its tangent cone. We prove the monodromy conjecture for some surface singularities. These results are applied to the study of rational arrangements of plane curves whose Euler-Poincaré characteristic is three. topological zeta function; monodromy conjecture; local Denef-Loeser zeta function; superisolated singularity of hypersurface; rational arrangements of plane curves Artal Bartolo, E.; Cassou-Noguès, Pi.; Luengo, I.; Melle Hernández, A., Monodromy conjecture for some surface singularities, Ann. sci. éc. norm. supér. (4), 35, 4, 605-640, (2002) Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Complex surface and hypersurface singularities, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Monodromy conjecture for some surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities ``In my thesis [``Kleine Auflösungen spezieller dreidimensionaler Varietäten'', Bonn. Math. Schr. 186 (1987)] I have dealt with the existence of algebraic projective small resolutions for three-dimensional complex varieties with only isolated singularities and I have presented examples of nodal double solids and nodal hypersurfaces in \({\mathbb{P}}^ 4({\mathbb{C}})\). In this paper I consider complete intersections of two quadrics with small resolutions in \({\mathbb{P}}^ 5({\mathbb{C}})\). Firstly, those with only ordinary double points will be classified with respect to the homology of their small resolutions by means of the associated curve. Then we deal with higher singularities and consider, at the end, the problem how many nodes a complete intersection of two quadrics in \(P^ n({\mathbb{C}})\) could have for arbitrary n.'' small resolutions; nodal hypersurfaces; complete intersections of two quadrics Complete intersections, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Projective techniques in algebraic geometry Vollständige Durchschnitte zweier Quadriken in \({\mathbb{P}}^ 5({\mathbb{C}})\) und ihre kleinen Auflösungen. (Complete intersections of two quadrics in \({\mathbb{P}}^ 5({\mathbb{C}})\) and their small resolutions) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We prove using jet schemes that the zero loci of the moment maps for the quivers with one vertex and at least two loops have rational singularities. This implies that the spaces of representations of the fundamental group of a compact Riemann surface of genus at least two have rational singularities. This has consequences for the numbers of irreducible representations of the special linear groups over the integers and over the \(p\)-adic integers. Arcs and motivic integration, Singularities in algebraic geometry, Representations of quivers and partially ordered sets Rational singularities, quiver moment maps, and representations of surface groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be a \(\mathbb{C}\)-algebra. A linear map \(\text{tr}\colon A\to\text{Z}(A)\) is said to be a trace map if \(\text{tr}(ab)=\text{tr}(ba)\) and \(\text{tr}(\text{tr}(a)b)=\text{tr}(a)\text{tr}(b)\) for all \(a,b\in A\). An affine \(\mathbb{C}\)-algebra \(A\) is said to be an \(n\)-th Cayley-Hamilton smooth algebra if \(A\) satisfies the \(n\)-th Cayley-Hamilton identity with \(\text{tr}(1)=n\) and the Grothendieck lifting property with respect to trace preserving algebra maps, or equivalently [\textit{C. Procesi}, J. Algebra 107, 63-74 (1987; Zbl 0618.16014)], that the scheme of trace preserving \(n\)-dimensional representations of \(A\) is a smooth affine variety (not necessarily connected).
Assume that \(A\) is a Cayley-Hamilton smooth order, i.e. there is a Zariski open subset of this scheme modulo \(\text{GL}_n\) corresponding to simple \(n\)-dimensional representations. \textit{L. Le Bruyn} has shown [Trans. Am. Math. Soc. 352, No. 10, 4815-4841 (2000; Zbl 0957.16009)] that the center \(\text{Z}(A)=\text{tr}(A)\) of \(A\) is smooth if \(\text{Kdim\,}A\leq 2\). In general, there may be singularities. For example, if \(\text{Kdim\,}A=3\), it turns out that the central singularities are essentially of one type, the conifold singularity \(\mathbb{C}[\![x,y,u,v]\!]/(xy-uv)\). In the above mentioned article, Le Bruyn analyses the étale local structure of \(\text{Z}(A)\) by means of a marked quiver. In the present paper, this method is used to determine the central singularity types up to dimension 6. quiver representations; Cayley-Hamilton smooth orders; central singularities of smooth orders DOI: 10.1142/S0219498803000623 Representations of orders, lattices, algebras over commutative rings, Singularities in algebraic geometry, Representations of quivers and partially ordered sets Smooth order singularities. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f\in \mathbb C[x_1, \dots, x_n]\) and \(h: X\to \mathbb C^n\) be an embedded resolution of \(f^{-1}(0)\). Let \(\{E_i\}_{i\in S}\) be the set of irreducible components of \(h^{-1}(f^{-1}(0))\) and \(N_i\) (resp. \(v_i-1\)) be the multiplicities of \(E_i\) in the divisor on \(X\) of \(f\circ h\) (respectively of \(h^\ast (dx_1\wedge \dots \wedge dx_n)\)). For \(I\subset S\) let \(E_I=\underset{i\in I}{\bigcap} E_i\) and \(E_I^\circ=E_I\smallsetminus(\underset{i\notin I} \bigcap E_j)\). The local topological zeta function associated to \(f\) is defined by
\[
Z_{\text{top}, f}(s)=\sum\limits_{I\subset S}\chi(E_I^\circ \bigcap h^{-1}(0))\prod\limits_{i\in I}\frac{1}{N_is+v_i}.
\]
Here \(\chi\) denotes the topological Euler-Poincaré characteristic.
Let \(\mathcal P_n=\{s\;| \;\exists f\in\mathbb C[x_1,\dots, x_n], Z_{\text{top}, f} \text{ has a pole in }s \}\). It is proved that for \(n\geq 2\;[-(n-1)/2, 0)\cap \mathbb Q\subseteq \mathcal P_n\). embedded resolution; hypersurface singularity Lemahieu, A., Segers, D., Veys, W.: On the poles of topological zeta functions. Proc. Amer. Math. Soc. 134, 3429--3436 (2006) Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Global theory and resolution of singularities (algebro-geometric aspects), Complex surface and hypersurface singularities On the poles of topological 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 Let \(X\) be a smooth projective curve of genus \(g\geq3\) defined over the complex numbers and let \({\mathcal N}\) denote the moduli space of stable bundles of rank \(2\) and determinant \({\mathcal O}_X(-x)\) for some fixed point \(x\in X\). The author proves that the moduli space \(\overline{\mathbf M}_{0,0}({\mathcal N},2)\) of stable maps of degree \(2\) from \({\mathbb P}^1\) to \({\mathcal N}\) has two irreducible components intersecting transversely. The first component, which he calls the \textit{Hecke component}, can be identified with Kirwan's partial desingularisation \(\widetilde{\mathcal M}_X\) of the moduli space \({\mathcal M}_X\) of semistable bundles of rank \(2\) with determinant isomorphic to \({\mathcal O}_X(y-x)\) for some \(y\in X\). The generic point of \(\widetilde{\mathcal M}_X\) corresponds to a Hecke curve; these were introduced by \textit{M.~S.~Narasimhan} and \textit{S.~Ramanan} [in: C. P. Ramanujam. -- A tribute. Collect. Publ. of C.P. Ramanujam and Pap. in his Mem., Tata Inst. fundam. Res., Stud. Math. 8, 291--345 (1978; Zbl 0427.14002)] in connection with a desingularisation of this moduli space. A Hecke curve is obtained by fixing a bundle \(E\) in the stable part \({\mathcal M}_X^s\) of \({\mathcal M}_X\). Then, for any \(\nu\in{\mathbb P}E_y^\vee\cong{\mathbb P}^1\), let \(E^\nu\) denote the kernel of the composition \(E\to E_y\smash{\mathop{\rightarrow}\limits^{\tilde{\nu}}}\mathbb{C}\), where \(\tilde\nu\) is any lift of \(\nu\) to \(E^\vee_y\). As \(\nu\) varies, the \(E^\nu\) form a family of stable bundles of determinant \({\mathcal O}_X(-x)\) parametrised by \({\mathbb P}^1\) and hence a morphism \({\mathbb P}^1\to {\mathcal N}\); this morphism is an embedding and has degree 2.
The second component is called the \textit{extension component} and a generic point of this corresponds to a curve which is obtained as follows. Let \(\xi\in\text{Pic}^0X\); then every non-trivial extension \(0\to\xi^{-1}(-x)\to E\to\xi\to0\) defines a bundle \(E\in{\mathcal N}\). Thus we have a morphism \({\mathbb P}\text{Ext}^1(\xi,\xi^{-1}(-x))\to{\mathcal N}\) which is an embedding of degree \(1\). For any conic in \({\mathbb P}\text{Ext}^1(\xi,\xi^{-1}(-x))\), we thus obtain a morphism \({\mathbb P}^1\to{\mathcal N}\) of degree \(2\). It turns out that the extension component \(\widetilde{Q}_J\) can also be identified with a partial desingularisation of a GIT quotient. In fact \(\widetilde{Q}_J\) is itself a moduli space, namely \(\overline{M}_{0,0}({\mathbb P}{\mathcal W}/J,2)\), where \(J=\text{Pic}^0X\) and \({\mathcal W}=R^1\pi_{J*}{\mathcal L}^{-2}(-x)\), with \({\mathcal L}\) being a Poincaré bundle on \(J\times X\). The intersection \(\widetilde{\mathcal M}_X\cap\widetilde{Q}_J=\widetilde{Q}_{\widetilde{X}}\) is yet again a moduli space, this time \(\widetilde{Q}_{\widetilde{X}}=\overline{M}_{0,0}({\mathbb P}{\mathcal W}_0/\widetilde{X},2)\) where \(\widetilde{X}=\{\xi\in J| \xi^2\cong{\mathcal O}_X(y-x)\text{ for some }y\in X\}\) and \({\mathcal W}_0\) is the restriction of \({\mathcal W}\) to \(\widetilde{X}\).
After an introduction which describes in a very clear way the objectives of the paper, the author gives a description of Hecke curves and extension curves and generalises these to higher degree. This is followed by a classification of rational curves in \({\mathcal N}\) (of any degree) which depends on a result of \textit{J. E. Brosius} [Math. Ann. 265, 155--168 (1983; Zbl 0503.55012); Theorem 1]. The classification is spelt out in detail for degree \(\leq4\). Section 4 discusses stable maps to projective space and gives the identification of \(\widetilde{Q}_J\) with \(\overline{M}_{0,0}({\mathbb P}{\mathcal W}/J,2)\). The author then turns to the Hecke curves and partial desingularisations in order to identify \(\widetilde{\mathcal M}_X\) with the Hecke component of \(\overline{M}_{0,0}({\mathcal N},2)\); this involves a modification of a construction of \textit{I.~Choe, J.~Choy} and the author [Topology 44, No. 3, 585--608 (2005; Zbl 1081.14045); \S\S5, 6]. The proof of the theorem is completed in section 6. In the final section, the author shows how the Hilbert scheme \({\mathbf H}\) and the Chow scheme \({\mathbf C}\) of conics in \({\mathcal N}\) are related to \(\overline{M}_{0,0}({\mathcal N},2)\). In particular \({\mathbf H}\) has two components, both smooth, one of which (the Hecke component) is the desingularisation of \({\mathcal M}_X\) constructed by Narasimhan and Ramanan [loc. cit.].
Some related results have been obtained by \textit{A.-M.~Castravet} [Int. J. Math. 15, No. 1, 13--45 (2004; Zbl 1092.14041)]. The two papers are independent and the only major overlap is the use of the results of Brosius [loc. cit.]. Hecke correspondence; stable maps; desingularisation; moduli spaces; Hilbert scheme; Chow scheme Kiem, Y-H, Hecke correspondence, stable maps, and the Kirwan desingularization, Duke Math. J., 136, 585-618, (2007) Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Hecke correspondence, stable maps, and the Kirwan 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 authors give a proof of a theorem that, according to them, Jean Giraud ``forgot'' to state.
The situation is the following one: let \(X\) be an excellent scheme and let \(H_X\) denote the modified Hilbert-Samuel function of \(X\). It is known that this function cannot increase under permissible blowing ups and that it is upper semi-continuous. The result proved in this paper is a strengthened version of this statement.
More precisely, let us assume that \(X\) is embedded in some regular scheme. Then for every \(x\in X\) let us denote by \(\tau_{st}(x)\) the codimension of the ridge of the tangent cone of \(X\) at \(x\) (the ridge of the tangent cone is the largest additive group that preserves the tangent cone by translation). Then the function
\[
\iota : X\longrightarrow \mathbb N^{\mathbb N}\times -\mathbb N\qquad
\]
\[
\qquad x\longmapsto (H_X(x),-\tau_{st}(x))
\]
does not increase under permissible blowing ups for the lexicographic ordering and it is upper semi-continuous.
When \(X\) is not assumed to be embedded in some regular scheme, the authors proves an analogue of this statement by replacing \(\tau_{st}\) by a modified version of this codimension. resolution of singularities; Hilbert-Samuel function Cossart, V; Piltant, O; Schober, B, Faîte du cône tangent à une singularité: un théorème oublié, C. R. Acad. Sci. Paris, Ser. I, 355, 455-459, (2017) Singularities in algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc., Global theory and resolution of singularities (algebro-geometric aspects) Ridge of the tangent cone of a singularity: a forgotten theorem | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Over a field of characteristic zero the geometry of orbit closures for equioriented \(A_n\) quiver was first studied by \textit{S. Abeasis} et al. [Math. Ann. 256, 401-418 (1981; Zbl 0477.14027)] where it was established that the orbit closures are normal, Cohen-Macaulay, and have rational singularities. This result was generalized to the case of a quiver \(A_n\) with an arbitrary orientation by \textit{G. Bobiński} and \textit{G. Zwara} [Manuscr. Math. 105, No. 1, 103-109 (2001; Zbl 1031.16012)].
In the paper under review orbit closures for the non-equioriented \(A_3\) quiver are investigated. Namely, a minimal free resolution of the defining ideal of an orbit closure is explicitly constructed, a description of a minimal set of generators of the defining ideal is obtained, a classification of orbits closures which are Gorenstein is established. Gorenstein orbit closures; Lascoux resolution; Cohen-Macaulay varieties; Dynkin quivers; geometry of orbit closures; Bott vanishing theorem Representations of quivers and partially ordered sets, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Determinantal varieties, Singularities in algebraic geometry, Actions of groups on commutative rings; invariant theory, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Resolutions of defining ideals of orbit closures for quivers of type \(A_3\). | 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 [For the entire collection see Zbl 0653.00009.]
Let X be the canonical resolution of one of the 3-dimensional simple singularities and E its exceptional locus. The author shows how to find the Picard group of the 3-dimensional formal scheme \(X_ E^{\wedge}\) in terms of the resolution diagram. More precisely, let \(E_ 1,...,E_ m\) be the irreducible components of E, \(\mu \in {\mathbb{Z}}^ m\) such that \(\mu\) \(E\hookrightarrow X\) is a negative embedding, then \(Pic(\mu E)=Pic(E)\). The paper contains also the computation of Pic(E) in the case of the singularity \(A_ n\). resolution; 3-dimensional simple singularities; Picard group Picard groups, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) The Picard group of the canonical resolution of a 3-dimensional simple singularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article is the first part of the interesting and rich survey of numerous remarkable results obtained recently concerning singularities in nonlinear infinite-dimensional problems of analysis, the theory of differential and integral equations, etc. This first part presents numerous results about mappings of fold type or, in other words, mappings \(F\), which in a suitable (nonlinear) system of ``coordinates'' \(\mathbb{R} \times E\) can be represented in the form \(F(t,v) =(t^2,v)\) or \(F(t,v) =(-t^2,v)\). The account of concrete results is based on the abstract global characterization of the fold-like mappings which was obtained by the authors in 1992. As a result, this survey presents numerous theorems about fold-like mappings from the unique point of view.
The contents of this part are: (1) Introduction; (2) Fréchet derivatives; (3) Fredholm maps; (4) Local structure of folds (Local characterization and Ambrosetti-Prodi local folds); (5) Abstract global characterization of the fold map (global structures, tools and examples); (6) Ambrosetti-Prodi and Berger-Podolak-Church fold maps; (7) McKean-Scovel fold maps (Riccati operator and a one-dimensional elliptic operator with \(u^2\) nonlinearity); (8) Giannoni-Micheletti fold maps; (9) Mandhyan fold map; (10) Oriented global fold maps; (11) A second Mandhyan fold map; (12) Jumping singularities.
This survey is both useful for specialists in the field as well as for all who want to study singularities of infinite-dimensional mappings.
[For Part II, see the following review, Zbl 0879.58008]. survey; singularities; fold-like mappings; infinite-dimensional mappings P.T. Church, J.G. Timourian, Global structure for nonlinear operators in differential and integral equations, I. Folds, Topological Nonlinear Analysis, II, 109 -- 160, Prog. Nonlinear Differ. Eq. Appl., vol. 27, Birkhaüser Boston, Boston, MA, 1997. Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Equations involving nonlinear operators (general), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Global structure for nonlinear operators in differential and integral equations. I: Folds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 4-dimensional \(A_n\)-singularities over a field of positive characteristic. We construct the canonical resolutions \(X\) of these singularities by giving a full description of the exceptional loci \(E=E_1+\cdots+E_m\) (\(E_i\) the smooth irreducible components) by the isomorphic types of \(E_i\) and determine the intersections \(E_i\cdot E_j\) in the Chow ring \(A_*(X)\), i.e., we find the graphs of \(E_i\). 4-dimensional \(A_n\)-singularities; positive characteristic; canonical resolutions; exceptional loci Global theory and resolution of singularities (algebro-geometric aspects), \(4\)-folds, Singularities in algebraic geometry Description of the canonical resolutions of the 4-dimensional \(A_n\)-singularities. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is reviewed together with its second part [Izv. Math. 68, 355--364 (2004; Zbl 1076.14050)]. log minimal model program Kudryavtsev, S.: Classification of three-dimensional exceptional log-canonical hypersurface singularities. I. Izv., Math. 66, 949--1034 (2002) \(3\)-folds, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Hypersurfaces and algebraic geometry Classification of three-dimensional exceptional log-canonical hypersurface singularities. I. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(V\) be a finite dimensional complex vector space and \(G\subset \text{SL}(V)\) a finite subgroup. Let \(X:=V/G\). Then two natural questions arise: 1) when does \(X\) admit a crepant resolution of singularities \(f\colon Y\to X\), and 2) if such a resolution exists, what can be said about the homology \(H_*(Y,\mathbb Q)\)? In dimension \(2\) J. McKay proved that there always exists a crepant resolution \(f\colon Y\to X\) such that the fiber \(f^{-1}(0)\) over the singularity of \(X\) is a rational curve whose components are numbered by the conjugacy classes of \(G\); moreover, the homology classes of these components freely generate \(H_2(X,\mathbb Q)\) and \(H_i(X,\mathbb Q)=0\) for \(i>0\) and \(i\neq 2\). This is the so-called McKay correspondence [cf. \textit{J. McKay}, Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 183--186 (1980; Zbl 0451.05026)]. If \(\text{dim}(V)=3\) the first question was solved affirmatively by several people independently, while the second question was solved by \textit{Y. Ito} and \textit{M. Reid} who proved that the same result holds true as in dimension \(2\) [in: Higher dimensional complex varieties. Proceedings of the International Conference, Trento, Italy, June 15--24, 1994, 221--240 (1996; Zbl 0894.14024)]. In the paper under review the author imposes an additional assumption of the pair \((V,G)\), namely he assumes that \(V\) has a nondegenerate symplectic form and the inclusions \(G\subset \text{Sp}(V)\subset \text{SL}(V)\) preserve not only the volume form in \(V\) but also the symplectic form. Under these hypotheses he proves a higher-dimensional analogue of the McKay correspondence. Crepant resolutions; finite subgroups of \(\text{SL}(V)\) Kaledin, D.: McKay correspondence for symplectic quotient singularities. Invent. Math. 148(1), 151--175 (2002) Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Singularities in algebraic geometry McKay correspondence for symplectic 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 Let \(X\) be a complex, irreducible, quasi-projective variety, and \(\pi :{\widetilde{X}}\rightarrow X\) a resolution of singularities of \(X\). Assume that the singular locus \({\text{Sing}}(X)\) of \(X\) is smooth, that the induced map \(\pi ^{-1}({\text{Sing}}(X))\rightarrow {\text{Sing}}(X)\) is a smooth fibration admitting a cohomology extension of the fiber, and that \(\pi ^{-1}({\text{Sing}}(X))\) has a negative normal bundle in \({\widetilde{X}}\). We present a very short and explicit proof of the Decomposition Theorem for \(\pi \), providing a way to compute the intersection cohomology of \(X\) by means of the cohomology of \({\widetilde{X}}\) and of \(\pi ^{-1}({\text{Sing}}(X))\). Our result applies to special Schubert varieties with two strata, even if \(\pi \) is non-small. And to certain hypersurfaces of \({\mathbb {P}}^5\) with one-dimensional singular locus. projective variety; smooth fibration; resolution of singularities; derived category; intersection cohomology; decomposition theorem; Poincaré polynomial; Betti numbers; Schubert varieties Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Global theory of complex singularities; cohomological properties, Modifications; resolution of singularities (complex-analytic aspects), Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Topological properties of mappings on manifolds On a resolution of singularities with two strata | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 morphisms given by composition of a sequence of point blow ups of smooth \(d\)-dimensional varieties in terms of combinatorial information coming from the \(d\)-ary intersection form on divisors with exceptional support. The other combinatorial objects under consideration are the so-called weighted dual polyhedron, the weighted dual graph, and the weighted dual tree associated to such morphism. The weighted dual polyhedron and graph are appropriate generalizations of the weighted dual graph for \(d = 2\), the weights being only intersection numbers. The weighted tree is a way to represent combinatorially the sequence of the corresponding blowing ups. The main result of the paper claims the equivalence of the above objects; it follows from this result that the intersection form, the weighted dual polyhedron or the weighted dual graph determine the decomposition of the morphism as the sequence of blowing ups. point blow ups; weighted dual polyhedron; weighted dual tree; intersection numbers Campillo, A., Reguera, A.: Combinatorial aspects of sequences of point blowing ups. Manuscripta Mathematica 84(1), 29-46 (1994) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Combinatorial aspects of sequences of point blowing ups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The articles of this volume will be reviewed individually. For the preceding symposium see [Zbl 1255.14001]. Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Fibrations, degenerations in algebraic geometry, Special algebraic curves and curves of low genus, Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities, Deformations of complex singularities; vanishing cycles, Milnor fibration; relations with knot theory, Singularities of differentiable mappings in differential topology, Proceedings of conferences of miscellaneous specific interest Singularities in geometry and topology 2011. Proceedings of the 6th Franco-Japanese symposium on singularities, Fukuoka, Japan, September 5--10, 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 numerical data of an embedded resolution determine the candidate poles of Igusa's \(p\)-adic zeta function. We determine in complete generality which real candidate poles are actual poles in the curve case. Igusa's \(p\)-adic zeta function Zeta functions and \(L\)-functions, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Determination of the real poles of the Igusa zeta function for curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper deals with the following problem: Given a finitely generated associative algebra over an algebraically closed field, the algebraic variety of \(d\)-dimensional modules is endowed with a natural \(\text{Gl}_d\)-action. Various problems associated with this action are studied in the paper. Some strong results are obtained for algebras of finite representation type.
The author considers pointed varieties of the form \((\overline {O(m)},n)\) where \(\overline {O(m)}\) is the closure of the orbit of a module \(M\) and \(n\) is a minimal degeneration of \(M\) (see the paper for definitions). He calls the pointed varieties of the form \((\overline {O(m)}, n)\) minimal singularities.
The main theorem asserts that all minimal singularities occurring in representations of Dynkin quivers are very smoothly equivalent to \((D(p,q),0)\) where \(D(p,q)\) is the set of \(p \times q\) matrices with rank \(\leq 1\). The theorem at the end of section 1 relates degenerations of two distinct finite dimensional modules. It is a fundamental result used in the rest of the paper.
With this result the author derives the famous result of \textit{H. Kraft} and \textit{C. Procesi} on minimal singularities of conjugacy classes of nilpotent matrices [which appeared in Invent. Math. 53, 227-247 (1979; Zbl 0434.14026)] and states that in this setting any minimal singularity is equivalent to the subregular singularity inside the set of nilpotent matrices of some smaller size or to the singularity at 0 inside the set of all nilpotent matrices of rank at most one.
Section 4 deals with tilting modules, in particular corollary 1 shows a very close relation between the \(\text{Gl}_d\)-stable subsets of \(Y(\underline {d})\) and \(\text{Gl}_e\)-stable subsets of \(Y(\underline {e})\). Here a tilting module \(Y(\underline {d})\) consists of the category of torsion \(A\)-modules of vector dimension \(\underline {d}\). \(Y\) is the subcategory of \(B\)-mod corresponding to \(\tau\) (the torsion free part), and \(Y(\underline {e})\) is the full subcategory of \(Y\) whose objects have vector dimension \(\underline {e}\).
If \([M] = \underline {d}\) then \(\underline {e} = [\text{Hom} (T,M)] - [\text{Ext}^1 (T,M)]\). Theorem 3 and its corollary are very beautiful applications to tilting theory. Section 5 studies possible reduction of the underlying Gabriel quiver. Under reduction of the Gabriel quiver with some technical hypotheses the author gets associated pointed varieties which are very smoothly equivalent. Section 6 studies the minimal degeneration in the cases where the partial orders \(\leq\) and \(\leq_{\text{Ext}}\) are equivalent. This equivalence of the partial orders \(\leq_{\text{Ext}}\) and \(\leq\) is valid for preprojective modules; the author uses this fact and gets that any minimal degeneration of representations of a Dynkin quiver is of codimension one and then gets the main result which is Theorem 6. The proof of this result is very technical and complex.
The entire paper uses techniques of algebraic geometry applied to the representation theory of algebras. This paper is certainly a very nice one, although it requires from the reader a good knowledge of representation theory and algebraic geometry. finitely generated algebras; algebraic variety; \(\text{Gl}_ d\)-actions; algebras of finite representation type; pointed varieties; minimal degenerations; minimal singularities; representations of Dynkin quivers; nilpotent matrices; tilting modules; tilting theories; Gabriel quivers; preprojective modules Bongartz, K.: Minimal singularities for representations of Dynkin quivers. Comment. Math. Helv. 69(4), 575--611 (1994) Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Group actions on varieties or schemes (quotients), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Singularities in algebraic geometry Minimal singularities for representations of 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 \(f:({\mathbb C}^{d+1},0)\to ({\mathbb C},0)\) be a hypersurface singularity. It is quasi-ordinary if in some local coordinates the projection \((\{f=0\},0)\to ({\mathbb C}^d,0)\) , \((x_1,\dots, x_d,x_{d+1})\to (x_1,\dots, x_d)\) is finite and its (reduced) discriminant is included in \((\{x_1\cdots x_d=0\},0)\). If \(f\) is quasi-ordinary then after reordering the coordinates the zeta function of \(f\) is the zeta function of the plane curve singularity \(f(x_1,0,\dots,0,x_{d+1})\), roughly speaking the quasi-ordinary singularities are generalizations of the plane curve singularities. Milnor fiber; zeta-function; Newton polyhedron; total transform González Pérez, P.D., McEwan, L.J., Némethi, A.: The zeta-function of a quasi-ordinary singularity. II, Topics in algebraic and noncommutative geometry (Luminy/Annapolis, MD, 2001). In: Contemp. Math., vol. 324, pp. 109--122. Amer. Math. Soc. Providence (2003) Singularities in algebraic geometry, Global theory of complex singularities; cohomological properties, Global theory and resolution of singularities (algebro-geometric aspects), Milnor fibration; relations with knot theory, Modifications; resolution of singularities (complex-analytic aspects) The zeta-function of a quasi-ordinary singularity. II. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We prove that the higher Nash blowup of a normal toric variety defined over a field of positive characteristic is an isomorphism if and only if it is non-singular. We also extend a result by \textit{R. Toh-Yama} [Commun. Algebra 47, No. 11, 4541--4564 (2019; Zbl 1467.14122)] which shows that higher Nash blowups do not give a one-step resolution of certain toric surface. These results were previously known only in characteristic zero. \(A_3\)-singularity; higher Nash blowups; normal toric varieties Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Higher Nash blowups of normal toric varieties in prime characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X,x\) be a germ of an analytic variety (over the complex numbers) which is a complete intersection isolated singularity. The author associates to \(X,x\) a sequence \(\mu^*=(\mu_ 0,\mu_ 1,\ldots)\) of numerical invariants by taking \(\mu_ i\) to be the minimal value of the Milnor numbers \(\mu(X_ i,x_ i)\) for all deformations \((X_ i,x_ i)\to(S_ i,s_ i)\) of \(X,x\) with \(\dim S_ i=i\). One has \(\mu_ 0=\mu(X,x)\) and \(\mu_ i=0\) if \(i\) is bigger than the embedding codimension of \(X,x\). On the other hand the author defines the topological type of \(X,x\) as the class of homeomorphism of any sequence of germs \((X,x)=(X_ 0,x)\subset(X_ 1,x)\subset\cdots\subset(X_ k,x)\) where \(k\) is the embedding codimension of \(X,x\), for each \(i\), \(\mu(X_ i,x)=\mu_ i(X,x)\) and then the homeomorphism class does not depend on the \(X_ i\).
The main result in the paper says that a \(\mu^*\)-constant family of isolated complete intersection singularities of dimension different from two is topologically equisingular. A sufficient condition for the members of the family to have isomorphic monodromy fibrations is also given. equisingularity; topological type of singularity; complete intersection isolated singularity; Milnor numbers; isomorphic monodromy fibrations Parameswaran, AJ, Topological equisingularity for isolated complete intersection singularities, Compos. Math., 80, 323-336, (1991) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Complete intersections, Germs of analytic sets, local parametrization Topological equisingularity for isolated complete intersection singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((R,\mathfrak{m})\) be a regular local ring, \(J\subset R\) a non-zero ideal and \(u=(u_1,\ldots,u_d)\) a system of regular elements that can be extended to a regular system of parameters of \(R\). To this data Hironaka associates the characterisctic polyhedron \(\Delta(J;u)\) (see [\textit{H. Hironaka}, J. Math. Kyoto Univ. 7, 251--293 (1967; Zbl 0159.50502)]). It is proved that the polyhedron depends only on \(R/J\) and \(u\) mod \(J\). This implies that numerical data obtained from the polyhedron are invariants of the singularity \(R/J\). invariants for singularities; Hironaka's characteristic polyhedra; resolution of singularities Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Polytopes and polyhedra, Modifications; resolution of singularities (complex-analytic aspects) Invariance of Hironaka's characteristic polyhedron | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 thesis is organized as follows. In Section 1 we give a quick review of the definition of Kleinian singularities. In Section 2 we introduce the deformation and resolution theory of singularities. In Section 3 we collect the basic notations and definitions of the representation theory of quivers. In Section 4 we present the main results from the deformation and resolution theory of the Kleinian singularities which are related to quiver varieties. These results are given by Kronheimer, Cassens and Slodowy. In Section 5 we speak about the nilpotent and stable representations of quivers and related results. These results are given by Lusztig and Hille. We explain how these results apply to a description of the exceptional set of the minimal resolution of a Kleinian singularity. In Section 6 we describe explicitly the intersection diagram \(\Gamma(\widetilde\mathbb{A}_{n-1})\) and consider action of the Weyl group on the space of weights \(\mathbb{H}(\delta)\). In Section 7 we describe the intersection diagrams \(\Gamma(\widetilde\mathbb{D}_4)\) and \(\Gamma(\widetilde\mathbb{D}_5)\). Kleinian singularities; deformations; resolutions; representations of quivers; quiver varieties; intersection diagrams; weights Representations of quivers and partially ordered sets, Deformations of singularities, Singularities in algebraic geometry, Deformations of associative rings McKay quivers and the deformation and resolution theory of Kleinian singularities. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A variety \(W\) is said to have symplectic singularities if there exists a holomorphic symplectic 2-form \(\omega \) on the regular part of \(W\) such that for any resolution of singularities \(\pi :X\to W\) the 2-form \(\pi ^*\omega \) can be extended to a holomorphic 2-form on \(X\). If this extension is symplectic then one says that \(W\) admits a symplectic resolution. The authors prove that any crepant (i.e. symplectic) resolution of \(W=W_1\times \ldots \times W_k\) which is locally \({\mathbb Q}\)-factorial, where the \(W_i\) are normal locally \({\mathbb Q}\)-factorial singular varieties which admit a crepant resolution \(\pi _i:X_i\to W_i\), is isomorphic to the product \(\pi =\pi _1\times \ldots \times \pi _k:X=X_1\times \ldots \times X_k\to W_1\times \ldots \times W_k\). The theorem on symplectic resolution in the case of nilpotent orbits in a complex semisimple Lie algebra is also proved. symplectic manifold; crepant resolution; symplectic singularities Bourbaki, N.: Éléments de mathématique. Hermann, Paris, 1975. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles, No. 1364 Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Uniqueness of crepant resolutions and symplectic singularities. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a very nice foundational article about singularities of normal varieties. The authors define generalizations of the notions of multiplier ideals, adjoint ideals, log canonical, log terminal, canonical, and terminal singularities without any additional assumptions of the ambient normal variety \(X\). Prior to their work, a common additional assumption was that the canonical divisor \(K_X\) is \(\mathbb{Q}\)-Cartier, or some \(\mathbb{Q}\)-Cartier perturbation of it, \(K_X+\Delta\), was used. These assumptions are needed to define the relative canonical divisor of a resolution of singularities of \(X\). The issues are: what to do in the case when \(K_X\) is not \(\mathbb{Q}\)-Cartier, or if there is some geometrically-meaningful way of choosing \(\Delta\). These issues are clarified in this paper.
As consequence, the authors obtain generalizations of well-known results to this setting, such as: ``log terminal \(\Rightarrow \) rational'', ``log canonical \(\Rightarrow\) Du Bois'', subadjunction, deformation invariance of canonical singularities, plurigenera, and numerical Kodaira dimension.
The method of paper is the following. Let \(f:Y\rightarrow X\) be a resolution of singularities. For prime divisors \(E\) on \(Y\), let \(\mathrm{val}_E\) denote the valuation corresponding to \(E\) on the rational functions on \(X\). For a coherent fractional ideal sheaf \(\mathcal{I}\subset \mathcal{K}_X\), define
\[
\mathrm{val}_E(\mathcal{I}):=\min\{\mathrm{val}_E\phi\;|\;\phi \in\mathcal{I}(U),\;U\cap f(E)\neq\emptyset\;\}.
\]
For a divisor \(D\) on \(X\), define
\[
f^*D:=\sum_E \left(\lim_{k\rightarrow\infty}\frac{\mathrm{val}_E(\mathcal{O}_X(-k!D))}{k!}\right)\cdot E.
\]
Then the issue about \(K_X\) not being \(\mathbb{Q}\)-Cartier is resolved by using \(f^*(-K_X)\) and \(-f^*(K_X)\) to define two ``relative canonical divisors'' \(K_{Y/X}\) and \(K^-_{Y/X}\), respectively. About the second issue, the choice of \(\Delta\), it is shown that the newly-defined multiplier ideal of a pair \((X,Z)\) is the unique maximal element of the set of multiplier ideals defined as usual using \(\mathbb{Q}\)-Cartier divisors \(K_X+\Delta\).
There are various technical difficulties encountered (e.g. the use of limiting relative canonical divisors), but dealt with by the authors in a very readable manner. The naturality of their construction, and the generalizations of the results mentioned above, follows from the two characterization theorems of the newly-defined notions of multiplier ideals and canonicity in terms of older, more familiar, terminology.
The authors point out some open questions: the relation between their multiplier ideals and the generalized test ideals (they agree in the toric case); the discreteness of the jumping numbers; does canonical imply rational, or log canonical, in this more general setting as well? divisorial valuation; relative canonical divisor; singularities of pairs; multiplier ideals Liu, Y.: Upper bounds for the volumes of singular Kahler-Einstein fano varieties. Compos. Math. arXiv:1605.01034 (\textbf{to appear}) Singularities in algebraic geometry, Multiplier ideals, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Singularities on normal varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article is a translation by T. Krasiński a contribution (in Polish) to the proceedings of the Xth workshop on Theory of Extremal Problems (1989). From the introduction: ``The article does not pretend to any originality. In the literature there exists a number of descriptions of desingularizations in the case of curves. Deciding for this description the author think it is worth looking in details into this fascinating topic in an easily accessible case, namely -- in the effects of multi blowings-up for curves in manifolds and for coherent sheaves on 2-dimensional manifolds.'' blowing up; resolution of singularities; coherent analytic sheaf Modifications; resolution of singularities (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Plane and space curves Geometric desingularization of curves in manifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field of characteristic \(0\) and \(X\subseteq \mathbb{P}^N\) an \(n\)--dimensional normal variety with canonical (resp. termial) singularities. Let \(Y\) be a generic nonsingular subvariety in \( \mathbb{P}^N\) intersecting the singular locus of \(X\). It is proved that \(X\cap Y\) is also normal with canonical (resp. terminal, log terminal) singularities. Let \(Z\) be an irreducible nonsingular \((n-1)\)--dimensional variety such that \(2Z=X\cap F\), where \(F\) is a hypersurface in \(\mathbb{P}^N\). The singularities of \(X\) which are on \(Z\) are studied. canonical singularities; terminal singularities; normal variety Gonzalez-Dorrego, M.R.: Smooth double subvarieties on singular varieties. RACSAM 108, 183-192 (2014) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Complex surface and hypersurface singularities Smooth double subvarieties on singular varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let (V,p) be a 2-dimensional normal singularity and \(\pi: M\to V\) be the minimal resolution of (V,p). When the exceptional set \(A:=\pi^{-1}(p)\) is weakly elliptic and its connected proper subvariety A' is rational, Laufer named (V,p) a ''minimally elliptic'' singularity. He showed that (V,p) is minimally elliptic if and only if the geometric genus \(p_ g:=\dim_ C(R^ 1\pi_*O_ M)_ p=1\) and its local ring \(\nu O_ p\) is a Gorenstein ring [\textit{H. B. Laufer}, Am. J. Math. 99, 1257-1295 (1977; Zbl 0384.32003)]. Also in this paper, all minimally elliptic hypersurface singularities were completely classified using weighted dual graphs and their typical defining equations were found. In this work, we shall classify all weighted dual graphs of minimally elliptic complete intersection singularities (except for hypersurface cases). Since any complete intersection singularity is Gorenstein, we shall find all 2- dimensional complete intersection singularities with \(p_ g=1\). geometric genus; weighted dual graphs of minimally elliptic complete intersection singularities Singularities in algebraic geometry, Complete intersections, Global theory and resolution of singularities (algebro-geometric aspects) Classification of weighted dual graphs of minimally elliptic complete intersection singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper surveys the representation theory of rational Cherednik algebras. We also discuss the representations of the spherical subalgebras. We describe in particular the results on category \(\mathcal O\). For type \(A\), we explain relations with the Hilbert scheme of points on \(\mathbb C^2\). We insist on the analogy with the representation theory of complex semisimple Lie algebras. rational Cherednik algebras; category \(\mathcal O\); Hilbert schemes of points; complex semisimple Lie algebras Raphaël Rouquier, Representations of rational Cherednik algebras, Infinite-dimensional aspects of representation theory and applications, Contemp. Math., vol. 392, Amer. Math. Soc., Providence, RI, 2005, pp. 103 -- 131. Hecke algebras and their representations, Simple, semisimple, reductive (super)algebras, Parametrization (Chow and Hilbert schemes), Representations of quivers and partially ordered sets, Noncommutative algebraic geometry Representations of rational Cherednik algebras. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The aim of this paper is to give some information about the resolution graphs of wild cyclic quotient surface singularities. Let \((B,n,l)\) be a 2-dimensional regular local ring on which a cyclic group \(H\) acts freely off \(\{n\}\). Assume that the characteristic of \(l\) is \(p>0\) and let \((A,m,k)\) be the invariant subring of \(B\) by \(H\). The cyclic surface singularity \(Z= \text{Spec}(A)\) is said to be wild if the order of \(H\) is divisible by \(p\). Assume that there exists a resolution of the singularity \(f: X\to Z\) which is minimal with the irreducible components \(C_i\) of the exceptional divisor \(E= f^{-1}(m)\) being smooth with normal crossing. Denote the intersection matrix \((C_i\cdot C_j)\) and the resolusion graph of \(f\) by \(N\) and \(G\), respectively. First, the author gives under suitable assumtions three results on wild cyclic singularities:
(1) the irreducible components of \(E\) are rational and the resolution graph \(G\) is a tree,
(2) the multiplicity \(e(A)\) of \(A\) is less or equal to \(|H|\),
(3) the order of any element of the group \(\Phi_N= \mathbb{Z}^n/N(\mathbb{Z}^n)\) divide \(|H|\).
Then, towards the explicit determination of invariants of wild cyclic surface singularities, the author studies detailed combinatorial properties of intersection matrix \(N\). cyclic quotient surface singularity; resolution graph; intersection matrix Lorenzini, D, Wild quotient singularities of surfaces, Math. Zeit., 275, 211-232, (2013) Singularities in algebraic geometry, Local ground fields in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings, Multiplicity theory and related topics, Singularities of surfaces or higher-dimensional varieties Wild quotient singularities 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 The classical Plücker formula for a plane curve was generalized by Teissier to the case of a hypersurface with isolated singularities and further by Kleiman to the case of an arbitrary \(n\)-dimensional projective variety \(V\) with isolated singularities. The formula relates the zero-th rank of \(V\) (the degree of the dual variety) to the Segre numbers of the conormal module and certain Buchsbaum-Rim multiplicities associated to the singular points of \(V\). We describe, for a projective variety \(V\) with arbitrary singularities, a natural class in the Chow group of the singular locus whose top dimensional part is given by Buchsbaum-Rim multiplicities, and we obtain generalizations of both formulas. The formulas are equations in the Chow group of \(V\). They imply numerical formulas for all the ranks of \(V\). Plücker formula; Segre numbers; Buchsbaum-Rim multiplicities; Chern class of a desingularization; cuspidal divisor; Chow group of the singular locus Anders Thorup, Generalized Plücker formulas, Recent progress in intersection theory (Bologna, 1997) Trends Math., Birkhäuser Boston, Boston, MA, 2000, pp. 299 -- 327. Singularities in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Enumerative problems (combinatorial problems) in algebraic geometry Generalized Plücker formulas | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper is an exposition of some possible generalization of the Hirzebruch-Riemann-Roch theorem: \(\chi({\mathcal E})=\deg (ch({\mathcal E})\cdot td(T_ X))_ n\) where \({\mathcal E}\) is a locally free sheaf of a smooth projective variety of dimension n over an algebraically closed field. Grothendieck's formulation of the R-R-theorem casts lights upon possible extensions to a more general setting, i.e. to larger categories of schemes and relative morphisms. The first problem is the definition of a functor A from this category to rings (which will play the role of the classical Chow ring) and a natural transformation from the Grothendieck K-functor to A (which will replace the Chern character). For instance, in the case of the category of quasi-projective (possible singular) varieties \textit{Baum}, \textit{Fulton} and \textit{McPherson} have defined Chow groups and intersection theory which give a R-R-theorem for proper morphism. This has been extended to arbitrary algebraic schemes by \textit{Fulton} and \textit{Gillet}. Another possible approach is to introduce a suitable filtration on \(K_ 0(X)\) and take \(A(X)\) to be the graded ring associated to this filtration (after tensoring with \({\mathbb{Q}})\). The author reviews the \(\lambda\)-ring structure on \(K_ 0(X)\) and the associated \(\chi\)-filtration and puts it in relation with the ``topological'' filtration and the Fulton-Lang filtration presenting some new results, too. Using the Quillen filtration on \(K_ 0(X)\) defined by cohomology groups, the author presents Bloch's formula for quasi projective non-singular varieties proved by Quillen and for varieties with isolated singularities proved by the author and Weibel. The paper contains some examples and open questions. singular varieties; Hirzebruch-Riemann-Roch theorem; Chow ring; K- functor; Chern character [P2]C. Pedrini: ``Riemann-Roch and Chow theories for singular varieties{'' Preprint (1988).} Parametrization (Chow and Hilbert schemes), Riemann-Roch theorems, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Singularities in algebraic geometry Riemann-Roch and Chow theories for singular varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let X be a nonsingular quasiprojective algebraic variety over an algebraically closed field k, let \(CH^ 2(X)\) be the group of codimension 2 cycles modulo rational equivalence, and let \({}_ nCH^ 2(X)\) be the kernel of multiplication by n in \(CH^ 2(X)\). From a result of \textit{A. S. Merkur'ev} and \textit{A. A. Suslin} on \(K_ 2\) [cf. Math. USSR, Izv. 21, 307-340 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.5, 1011-1046 (1982; Zbl 0525.18008)] it follows that if n is prime to char k, then \({}_ nCH^ 2(X)\) is finite. The author extends this result to (a suitably defined) Chow group of algebraic varieties with isolated singularities.
As an application, the author shows that if Y is a complete surface with isolated singularities, there is an isomorphism \(CH^ 2(Y)(\ell){\tilde \to}Alb(Y')(\ell)\), where Y' is the desingularization of Y and \(CH^ 2(Y)(\ell)\) is the subgroup of elements whose order is a power of \(\ell\). codimension 2 cycles modulo rational equivalence; Chow group of algebraic varieties with isolated singularities DOI: 10.1016/0022-4049(84)90033-1 Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry, (Equivariant) Chow groups and rings; motives, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) Torsion in the Chow group of codimension two: The case of varieties with isolated singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Die vorliegende Arbeit beschäftigt sich mit Automorphismengruppen von Singularitäten. Speziell werden Levi-Untergruppen, d.h. maximale reduktive Untergruppen der Automorphismengruppen betrachet. Insbesondere werden Levi-Untergruppen von Quotientensingularitäten berechnet. Die Levi-Untergruppen werden in Abhängigkeit der definierenden Gruppe \((H\leq GL_n)\) angegeben, sie ist nämlich gerade isomorph zum Normalisator von \(H\) in \(GL_n\). Außerdem wird für die Quotientensingularitäten ein Zusammenhang hergestellt zwischen den Levi-Untergruppen und den graduierungstreuen Automorphismen der entsprechenden Invariantenalgebra: Die graduierungstreuen Automorphismen sind isomorph zur Levi-Untergruppe der Singularität. Für alle 2-dimensionalen Quotientensingularitäten werden die Levi-Untergruppen konkret berechnet. Vergleicht man nun die in dieser Arbeit ermittelten Levi-Untergruppen der 2-dimensionalen Quotientensingularitäten und die jeweilige Automorphismengruppe des entsprechenden dualen Auflösungsgraphen der minimalen guten Auflösung, so stellt man fest, daß die Automorphismengruppe des Graphen isomorph zu \(G/G^0\) ist, wobei \(G^0\) die Zusammenhangskomponente der Eins bedeutet. Dieser Sachverhalt wird im Rahmen der Arbeit für 2-dimensionale, normale Singularitäten \((X,0)\) weiter untersucht.
Alle Automorphismen einer Singularität induzieren in natürlicher Weise Automorphismen des entsprechenden Auflösungsraumes und damit insbesondere Automorphismen des Graphen. Es wird gezeigt, daß \(G^0\) trivial auf dem Graphen operiert, d.h. es gibt insbesondere eine natürliche Abbildung von \(G/G^0\) in die Automorphismengruppe des Graphen. automorphism group of singularities; Levi subgroups; Brieskorn singularities; isolated singularities Aumann-Körber, E.: Reduktive automorphismengruppen von singularitäten. Dissertation (1995) Singularities in algebraic geometry, Representations of groups as automorphism groups of algebraic systems, Birational automorphisms, Cremona group and generalizations, Homogeneous spaces and generalizations, Modifications; resolution of singularities (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Reductive groups of automorphisms 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 articles of this volume will be reviewed individually. Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Singularities in algebraic geometry, Deformations of singularities, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Computational aspects of algebraic surfaces, Collections of articles of miscellaneous specific interest Deformations of surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main goal of the paper is to classify all finite subgroups \(G\) of \(\mathrm{Sp}(V)\) for which \(V/G\) admits a symplectic resolution. Let \(V\) be a symplectic vector space and \(G \subset \mathrm{Sp}(V)\) a finite group. The authors study singularities of the quotient \(V/G\). If there exists a projective resolution of singularities \(X \to V/G\) such that \(X\) is a symplectic manifold, then they say that \(V/G\) admits a projective symplectic resolution. Consider \(V\) as a symplectically irreducible representation of \(G\). The authors classify all such pairs \((V,G)\), which admit a projective symplectic resolution, in the case \(\dim V \neq 4\), except for four singularities, occurring in dimensions at most 10, for which the question remains open.
Section 2 contains the definition of symplectic variety and symplectic resolutions, and some criteria are given for the (non)existence of projective symplectic resolutions. In Section 3, they recall the Kleinian group, in Section 4, Cohen's classification of symplectic reflection groups. Section 5 deals with two general criteria for the non-existence of projective symplectic reflections for the group \(G(K,H,\alpha)\) with \(H \neq {1}\). In the next section, they prove that the symplectically imprimitive, irreducible symplectic reflection groups of the above type satisfy at least one of the criteria. Section 7 gives the following result: If \(n>2\), then the symplectic quotient \(C^{2n}/G_n(K,H)\) admits a projective symplectic resolution if and only if \(K=H\). The paper ends with some open questions. symplectic resolution; symplectic smoothing; symplectic reflection algebra; Poisson variety; quotient singularity; McKay correspondence Bellamy, G.; Schedler, T., On the (non)existence of symplectic resolutions of linear quotients, Math. Res. Lett., 23, 1537-1564, (2016) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, McKay correspondence, Deformations of associative rings, Poisson algebras On the (non)existence of symplectic resolutions of linear 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 Given an ideal \(\mathfrak{a} \subseteq R\) in a (log) \(\mathbb{Q}\)-Gorenstein \(F\)-finite ring of characteristic \(p > 0\), we study and provide a new perspective on the test ideal \(\tau(R, \mathfrak{a}^t)\) for a real number \(t > 0\). Generalizing a number of known results from the principal case, we show how to effectively compute the test ideal and also describe \(\tau(R, \mathfrak{a}^t)\) using (regular) alterations with a formula analogous to that of multiplier ideals in characteristic zero. We further prove that the \(F\)-jumping numbers of \(\tau(R, \mathfrak{a}^t)\) as \(t\) varies are rational and have no limit points, including the important case where \(R\) is a formal power series ring. Additionally, we obtain a global division theorem for test ideals related to results of \textit{L. Ein} and \textit{R. Lazarsfeld} [Invent. Math. 137, No. 2, 427--448 (1999; Zbl 0944.14003)] from characteristic zero, and also recover a new proof of Skoda's theorem for test ideals which directly mimics the proof for multiplier ideals. test ideals; jumping numbers; vanishing theorem; multiplier ideal; Skoda complex; global division theorem ScTu2 K.~Schwede and K.~Tucker, Test ideals of non-principal ideals: Computations, Jumping Numbers, Alterations and Division Theorems, J. Math. Pures Appl. (9) \textbf 102 (2014), no. 5, 891--929. Multiplier ideals, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Test ideals of non-principal ideals: computations, jumping numbers, alterations and division theorems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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. Watanabe} [Math. Ann. 250, 65-94 (1980; Zbl 0414.32005)] introduced the pluri-genera \(\{ \delta_m (X,x)\}_{m \in \mathbb{N}}\) of a normal isolated singularity \((X,x)\). They can be computed from a good resolution \(f:M\to X\) (i.e. a resolution whose exceptional locus \(A\) is a divisor with normal crossings) as follows:
\[
\delta_m (X,x)= \dim_{\mathbb{C}} H^0 ({\mathcal O}_U(mK)) / H^0 ({\mathcal O}_M (mK+(m-1)A))
\]
where \(K\) is a canonical divisor on \(M\) and \(U=X \setminus \{x\} \cong M \setminus A\). In particular, \(\delta_1 (X,x) = p_g (X,x)\).
The author studies the pluri-genera of normal surface singularities over \(\mathbb{C}\). For rational surface singularities whose dual graph for the minimal resolution is a star-shaped graph, he describes the pluri-genera in terms of some divisors on the central exceptional curve. Using this, he proves that, if for a normal surface singularity \((X,x)\) over \(\mathbb{C}\), \(\delta_m(X,x)=0\) for \(m=4,6\) then \((X,x)\) is a quotient singularity. Since K. Watanabe proved that \((X,x)\) is a quotient singularity if and only if \(\delta_m(X,x)=0\) for all \(m \in \mathbb{N}\), the preceding result characterizes the quotient singularities. Purely elliptic singularities (i.e. those for which \(\delta_m(X,x)=1\) for all \(m \in \mathbb{N}\)) are also characterized in the paper. It is proved that a normal surface singularity \((X,x)\) over \(\mathbb{C}\) is purely elliptic if and only if \(\delta_m(X,x)=1\) for \(m=1,4,6\). Applying the result of S. Ishii which assures that \((X,x)\) is purely elliptic if and only if it is a cusp or a simple elliptic singularity, the result in the paper gives a criterion to characterize these singularities. S. Ishii had also proved that \((X,x)\) is log-canonical if and only if \(\delta_m(X,x) \leq 1\) for all \(m \in \mathbb{N}\). In the paper, the author proves that if \(\delta_{14}(X,x)=0\) then \((X,x)\) is log-canonical. pluri-genera; normal surface singularities; good resolution; purely elliptic singularities; log-canonical singularities Okuma T. The plurigunera of surface singularities. Tohoku Math J, 1998, 50: 119--132 Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Complex surface and hypersurface singularities The pluri-genera of surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0549.00004.]
This article surveys the status, as of 1982, of the desingularization problem in the positive characteristic case. In explains the backgrounds of the results on Hironaka group schemes and the author's easy (and hard) polyhedral games [cf. the author, Ann. Math., II. Ser. 92, 327-334 (1970; Zbl 0228.14007), \textit{T. Oda}, Number theory, Algebr. Geom. Commut. Algebra, in Honor of Y. Akizuki, 181-219 (1973; Zbl 0287.14014) and Publ. Res. Inst. Math. Sci. 19, 1163-1179 (1983; Zbl 0569.14021) and \textit{M. Spivakovsky}, Proc. Am. Math. Soc. (to appear)]. desingularization problem; positive characteristic; Hironaka group schemes; polyhedral games Global theory and resolution of singularities (algebro-geometric aspects), History of algebraic geometry, History of mathematics in the 20th century, Singularities in algebraic geometry, Other algebraic groups (geometric aspects) The present state and prospect of the problem of 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 Let \(Q\) be a quiver consisting of a single vertex along with two loops \(\alpha\), \(\beta.\) For \(k\) an algebraically closed field, let \(kQ\) be the quiver algebra for \(Q\). For a given dimension \(d\) one has the quiver variety rep\((Q,d) =k^{d^{2}}\times k^{d^{2}}\) (where each term on the right corresponds to a matrix representation of a linear operator on \(\alpha\) and \(\beta\) respectively) upon which \(\text{GL}(d)\) acts: the orbits of this action correspond to isomorphism classes of representations. For nonzero \(q\in k\) let \(I_{q}\) be the ideal generated by \(\{\alpha^{2},\beta^{2},\beta\alpha+q\alpha\beta\}\): one then can define the closed subset rep\((Q,I_{q},d)\) of quiver representations which are trivial on \(I_{q}.\)
The work being reviewed is a study of the irreducible components of rep\((Q,I_{q},d).\) Initially, these components are described in terms of orbit closures of modules related to the Kronecker quiver (two vertices, \(1,2\) and two arrows \(\alpha,\beta:1\rightarrow2\)), but this formulation is shown to be equivalent to a more direct description involving representations satisfying numerical criteria depending on the parity of \(d\). Along with this description, results are obtained concerning intersections of irreducible representations; in particular it is shown that a nontrivial intersection of irreducible components of rep\((Q,I_{q},d)\) is irreducible.
The class of examples presented here are noteworthy for two reasons. First, it provides an description of irreducible components in terms of equations for modules over an algebra which is neither representation finite nor hereditary. Second, applying the results to the case \(q=-1\) and \(d=4\) provides for a description of a famous example by Carlson -- see [\textit{C. Riedtmann}, Ann.\ Sci. Èc. Norm. Supér. (4) 19, No. 2, 275--301 (1986; Zbl 0603.16025)]. quiver representations; irreducible representations Riedtmann, C; Rutscho, M; Smalø, SO, Irreducible components of module varieties: an example, J. Algebra, 331, 130-144, (2011) Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets Irreducible components of module varieties: 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 Every flat family of Du Val singularities admits a simultaneous minimal resolution after a finite base change. We investigate a flat family of isolated Gorenstein toric singularities and prove that there exists a simultaneous partial resolution. Du Val singularities; simultaneous minimal resolution; Gorenstein toric singularities Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Simultaneous minimal models of homogeneous toric 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 Let X be the spectrum of a quasi-unmixed local Nagata ring over a field of characteristic zero, and let Y be a reduced closed subscheme of X. The main result of the paper is the following: if x is the closed point of X and if X is equimultiple along Y, then the embedding dimension of Y at x is less than or equal to dim(X). Then the author applies this result to get a stability theorem for standard bases and a structure theorem for tangent cones. multiplicity; spectrum of a quasi-unmixed local Nagata ring; embedding dimension; standard bases; tangent cones Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Multiplicity theory and related topics, Relevant commutative algebra On equimultiple subschemes of a local scheme over a field of 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 The author introduces intersection multiplicities between singular differential 1-forms in the plane and valuations centered at the plane. He proves a ``Noether formula'' showing the behaviour of the intersection multiplicity under quadratic transformations. Several methods to compute this number are derived. In the last part of the paper, intersection numbers are used to give a description of the desingularization of a 1-form without performing blow-ups. Noether formula; intersection multiplicities; singular differential 1-forms; quadratic transformations; desingularization DOI: 10.1016/0022-4049(94)90012-4 Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Singularities of holomorphic vector fields and foliations Intersections of 1-forms and valuations in a local regular surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, the author studies the singularities of pairs in arbitrary characteristic via jet schemes. The main result is to establish the correspondence between cylinder valuations and divisorial valuations on a smooth variety over a perfect field of arbitrary characteristic. This extends a result of \textit{L. Ein} et al. [Compos. Math. 140, No. 5, 1229--1244 (2004; Zbl 1060.14004)] in characteristic $0$. The author further establishes Mustaţă's log canonical threshold formula in positive characteristic avoiding the use of log resolutions. As a consequence, the author obtains a comparison theorem via reduction modulo $p$ and the inversion of adjunction in positive characteristic.
Let $k$ be a perfect field $k$ of arbitrary characteristic. Let $X$ be a smooth integral variety of dimension $n$ over $k$. A divisorial valuation on $k(X)$ is of the form $\nu = q \cdot \operatorname{ord}_E:k(X)^*\rightarrow \mathbb{Z}$ where $E$ is a divisor over $X$ and $q$ is a positive integer number. The log discrepancy of $\nu$ is defined to be $q \cdot (1+ \operatorname{ord}_E (K_{X'/X}))$, where $K_{X'/X}$ is the relative canonical divisor. These numbers determine the log canonical threshold $\operatorname{lct}(X, Y )$ of a pair $(X, Y)$, where $Y$ is a closed subscheme of $X$.
Given $m \geq 0$, one defines the $m$th order jet scheme $X_m$ and the space of arcs $X_{\infty}$. A subset $C$ of $X_{\infty}$ is called a cylinder if it is the inverse images of a constructible subset in $X_m$ by the canonical projections $\psi_m : X_{\infty}\rightarrow X_m$. For every closed irreducible nonempty cylinder $C \subset X_{\infty}$ which does not dominate $X$, one defines a cylinder valuation $\operatorname{ord}_C : k(X)^*\rightarrow \mathbb{Z}$ by taking the order of vanishing along the generic point of $C$. It is easy to see that every divisorial valuation is a cylinder valuation.
Suppose the subscheme $Y$ is defined by a non-zero ideal sheaf $\mathfrak{a}\subseteq \mathscr{O}_X$. For every $p \geq 0$, the contact locus of order $\geq p$ of $Y$ is the closed cylinder $\mathrm{Cont}^{\geq p}(Y )=\{\gamma\in X_{\infty}\mid \operatorname{ord}_{\gamma}(\mathfrak{a}) \geq p \}$. If $C$ is an irreducible component of $\mathrm{Cont}^{\geq p}(Y)$, then $C$ is a cylinder. In this case, the valuation $\operatorname{ord}_C$ is called a contact valuation. Directly from the definition, the implication among these three valuations can be described as follows
$$\text{ contact valuation }\Longrightarrow \text{ cylinder valuation }\Longleftarrow \text{ divisorial valuation}.$$
When the ground field is of characteristic zero, Ein et al. [loc. cit.] showed that these three classes of valuations actually coincide, by showing that:
(a) Every contact valuation is a divisorial valuation;
(b) Every cylinder valuation is a contact valuation.
Through this correspondence, one can relate the codimension of the cylinder to the log discrepancy of the divisorial valuation. This yields a quick proof of Mustaţă's log canonical threshold formula.
In this paper, the author shows by induction on the codimension of cylinders and by only using the Change of Variable formula for blow-ups along smooth centers that the
correspondence between divisorial valuations and cylinders holds in arbitrary characteristic (Theorem A), i.e.,
$$\text{ cylinder valuation }\Longleftrightarrow \text{ divisorial valuation}$$
It enables the author to prove the following log canonical threshold formula without using log resolutions (Theorem B)
$$\operatorname{lct}(X,Y)=\inf_{C\subset X_{\infty}}\frac{\operatorname{codim} C}{\operatorname{ord}_C(Y)}=\inf_{m\geq 0}\frac{\operatorname{codim}(Y_m,X_m)}{m+1}$$
where $C$ varies over the irreducible closed cylinders which do not dominate $X$. Multiplier ideals, Arcs and motivic integration, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Positive characteristic ground fields in algebraic geometry Log canonical thresholds 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 Equations are obtained that are satisfied by the vectors of the tangent space to the variety \(X_{22}\) of complete pairs of zero-dimensional subschemes of length 2 of a smooth three-dimensional projective algebraic variety at the most special point of the variety \(X_{22}\). It is proved that the system of equations obtained is complete and the variety \(X_{22}\) is singular. Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry The variety of complete pairs of two-point spaces of a smooth three-dimensional variety is singular. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 work is an interesting review on singularities of normal surfaces comparing three points of view: analytical, topological and smoothing (smoothing a surface means to deform it to a smooth surface, remark that smoothing is not always possible), with a particular attention to the geometric genus. The author considers many particular cases of surface singularities and gives the relation between topological invariants like the link, the Milnor number, geometric genus, deformations or simultaneous resolution of singularities. He illustrates the paper by many well chosen examples. Particular focus is given on the case of quasi-homogeneous surface singularities. This note contains a considerable number of open questions and conjectures. The following two citations are related to this topics: \textit{W. Ebeling, S. M. Gusein-Zade} [Abh. Math. Semin. Univ. Hamb. 74, 175--179 (2004; Zbl 1070.14004)]; \textit{M. Morales} [Bull. Soc. Math. Fr. 112, 325--341 (1984; Zbl 0564.32006)]. link, geometric genus, deformation, smoothing Némethi, András, Invariants of normal surface singularities. Real and complex singularities, Contemp. Math. 354, 161-208, (2004), Amer. Math. Soc., Providence, RI Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) Invariants of normal surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the paper under review the following theorem is proved:
Theorem 1.2. Let \(\varphi: (X,U_{\alpha}, D_{\alpha})_{\alpha\in I}\to (Y,V_{\alpha}, E_{\alpha})_{\alpha\in I} \) be a locally toroidal morphism of nonsingular varieties. Then there exists a commutative diagram,
\[
\begin{tikzcd}
\widetilde{X} \ar[r, "\widetilde{\varphi}"] \ar[d, "\lambda" '] & \widetilde{Y}\ar[d, "\pi"] \\
X \ar[r, "\varphi"] & Y
\end{tikzcd}
\]
such that \(\lambda: \widetilde{X}\to X\) and \(\pi: \widetilde{Y}\to Y\) are sequences of blowups at non singular centers, \(\widetilde{X}\) and \(\widetilde{Y}\) are nonsingular, and there exists a simple normal crossing divisor \(\widetilde{E}\) on \(\widetilde{Y}\) such that \(\widetilde{D}:=\varphi^{-1}(\widetilde{E})\) is a simple normal crossings divisor on \(\widetilde{X}\), and \(\widetilde{\varphi}\) is toroidal with respect to \(\widetilde{E}\) and \(\widetilde{D}\).
Combining with results in [\textit{S. D. Cutkosky}, Local monomialization and factorization of morphisms. Paris: Société Mathématique de France (1999; Zbl 0941.14001)], the next a corollary follows:
Corollary 1.3. Suppose that \(\varphi: X\to Y\) is a generically finite morphism of nonsingular proper \(K\)-varieties. Then there exists a commutative diagram,
\[
\begin{tikzcd} X \ar[d, "\varphi"] & \overline{X} \ar[d, "\overline{\varphi}"] \ar[l, "\lambda" '] & \widetilde{X} \ar[l, "\Lambda" '] \ar[d, "\widetilde{\varphi}"] \\
Y & \overline{Y} \ar[l, "\pi"] & \widetilde{Y} \ar[l, "\Pi"]
\end{tikzcd}
\]
where \(\lambda: \overline{X} \to X\) and \(\pi: \overline{Y} \to Y\) are locally sequences of blowups with nonsingular centers, \(\overline{\varphi}: \overline{X} \to \overline{Y}\) is a locally toroidal morphism of complete varieties, \(\Lambda: \widetilde{X}\to \overline{X}\) and \(\Pi: \widetilde{Y}\to \overline{Y}\) are sequences of (global) blowups with nonsingular centers, and \(\widetilde{\varphi}: \widetilde{X}\to \widetilde{Y}\) is toroidal. toroidalization, resolution of morphisms, toroidal morphisms Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects), Local structure of morphisms in algebraic geometry: étale, flat, etc., Singularities in algebraic geometry Toroidalization of locally toroidal morphisms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The automorphism group \(\Aut X\) of a weighted homogeneous normal surface singularity \(X\) has a maximal reductive algebraic subgroup \(G\) which contains every reductive algebraic subgroup of \(\Aut X\) up to conjugation. In all cases except the cyclic quotient singularities the connected component \(G_1\) of the unit equals \(\mathbb{C}^*\). The induced action of \(G\) on the minimal good resolution of \(X\) embeds the finite group \(G/G_1\) into the automorphism group of the central curve \(E_0\) of the exceptional divisor. We describe \(G/G_1\) as a subgroup of \(\Aut E_0\) in case \(E_0\) is rational as well as for simple elliptic singularities. Moreover, sufficient conditions for \(G\) to be a direct product \(G_1\times G/G_1\) are presented. Finally, it is shown that \(G/G_1\) acts faithfully on the integral homology of the link of \(X\). automorphism group; weighted homogeneous normal surface singularity; central curve; integral homology Müller, G. -- Symmetries of surface singularities. In preparation. Automorphisms of surfaces and higher-dimensional varieties, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Local complex singularities, Global theory and resolution of singularities (algebro-geometric aspects), Automorphisms of curves Symmetries of surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This short communication gives some inequalites relating the multiplicity of a germ of an effective 1-cycle on a germ of a nonsingular surface, the full multiplicity of the 1-cycle with respect to the order and the discrepancies of a prime divisor over a point with respect to the ``lower storeys'' of a resolution of the surface. The proof uses the oriented graph associated to the resolution and the inequalities are rewritten in this language, they are application of inequalities of [\textit{A. Pukhlikov}, Birationally rigid varieties. Mathematical Surveys and Monographs 190. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1297.14001)]. blow-up; germ of a surface; prime divisor; multiplicity of a 1-cycle; oriented graph Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Multiplicity theory and related topics Two inequalities for a sequence of blow-ups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A necessary condition for the existence of torus-equivariant crepant resolutions of Gorenstein toric singularities can be derived by making use of a variant of the classical upper bound theorem which is valid for simplicial balls. toric singularities; Gorenstein singularities; lattice polytope Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Singularities in algebraic geometry, Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.), \(n\)-dimensional polytopes Crepant resolutions of Gorenstein toric singularities and upper bound theorem | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For part I of this paper see Mat. Sb. 180, No. 2, 226--243 (1989; Zbl 0674.14024)].
The present paper contains a proof of the conjecture given in Part I (loc. cit.) of this paper and formulated at the end of the previous review. intersection diagram; Lobachevsky space; anticanonical divisor; desingularisation; Picard lattice; log-Del Pezzo surface Nikulin, V. V.: Del Pezzo surfaces with log-terminal singularities. II, Math. USSR izv. 33, 355-372 (1989) Singularities of surfaces or higher-dimensional varieties, \(K3\) surfaces and Enriques surfaces, Transcendental methods of algebraic geometry (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Picard groups Del Pezzo surfaces with log-terminal singularities. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In these decades Hodge theory has been extended to the germs of isolated singular points of complex spaces by several people, and continuation theorems for holomorphic differential forms across exceptional sets have been known as application.
This article first gives a nice survey on this topic, collecting the results obtained by \textit{D. v. Straten} and \textit{J. Steenbrink} [Abh. Math. Semin. Univ. Hamburg 55, 97-109 (1985; Zbl 0584.32018)'), \textit{V. Navarro Aznar} [Systèmes différentiels et singularités, Colloq. Luminý, Astérisque 130, 272-307 (1985; Zbl 0599.14007)], \textit{Kersken} (Habilitationsschrift, Bochum 1987) and the reviewer [Publ. Res. Inst. Math. Sci. 24, No.2, 253-263 (1988; Zbl 0653.32012)], and extend them to the germs of non-isolated singular points. The result has an immediate application to the well known problem of Zariski and Lipman, as the author remarks in the introduction. The idea of the proof is very natural, but the reader is required to be acquaintd with basic techniques in algebraic geometry. Hodge theory; continuation theorems for holomorphic differential forms; germs of non-isolated singular points Jörder, C. : A weak version of the Lipman-Zariski conjecture. arXiv:1311.5141 [math.AG], November 2013 Global theory and resolution of singularities (algebro-geometric aspects), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Continuation of analytic objects in several complex variables, Singularities in algebraic geometry, Germs of analytic sets, local parametrization, Modifications; resolution of singularities (complex-analytic aspects) Extendability of differential forms on non-isolated singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field of characteristic 0. We give a brief survey on multiplicity-2 structures on varieties. Let \(Z\) be a reduced irreducible nonsingular \((n-1)\)-dimensional variety such that \(2Z= X\cap F\), where \(X\) is a normal \(n\)-fold with canonical singularities, \(F\) is an \((N-1)\)-fold in \(\mathbb{P}^N\), such that \(Z\cap\text{Sing}(X)\neq\emptyset\). Assume that Sing\((X)\) is equidimensional and \(\text{codim}_X(\text{Sing}(X))= 3\). We study the singularities of \(X\) through which \(Z\) passes. We also consider Fano cones. We discuss the construction of some vector bundles and the resolution property of a variety.
For part I, see [Math. Nachr. 165, 133--158 (1994; Zbl 0860.14037)]. \(n\)-fold; singularity; intersection Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, \(3\)-folds, \(4\)-folds, Fano varieties, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Hypersurfaces and algebraic geometry, Complex surface and hypersurface singularities Smooth double subvarieties on singular varieties. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field of characteristic zero and let \(A\) be a finitely generated \(k\)-algebra. The Nori-Hilbert scheme of \(A\), \(\mathrm{Hilb}^n_A\), parameterizes left ideals of codimension \(n\) in \(A\). It is well known that \(\mathrm{Hilb}^n_A\) is smooth when \(A\) is formally smooth. {
}In this paper we will study \(\mathrm{Hilb}^n_A\) for 2-Calabi-Yau algebras. Important examples include the group algebra of the fundamental group of a compact orientable surface of genus \(g\), and preprojective algebras. For the former, we show that the Nori-Hilbert scheme is smooth only for \(n=1\), while for the latter we show that a component of \(\mathrm{Hilb}^n_A\) containing a simple representation is smooth if and only if it only contains simple representations. Under certain conditions, we generalize this last statement to arbitrary 2-Calabi-Yau algebras. representation theory; Calabi-Yau algebras; Nori-Hilbert scheme Parametrization (Chow and Hilbert schemes), Noncommutative algebraic geometry, Representations of quivers and partially ordered sets, (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) The Nori-Hilbert scheme is not smooth for 2-Calabi-Yau algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0623.00011.]
We present some local and global results concerning the desingularization of vector fields by means of blowing-ups of the ambient space (supposed of dimension three). The local reduction has been made in an earlier work and here we present with complete proofs the globalization of the local algorithm for a special kind of vector fields: those which have a ``true'' strict tangent space or directrix. desingularization of vector fields; blowing-ups Cano, F.: Desingularization of plane vector fields. Lecture notes in mathematics 1259 (1987) Global theory and resolution of singularities (algebro-geometric aspects), \(3\)-folds, Singularities in algebraic geometry Local and global results on the desingularization of three-dimensional vector fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper deals with minimal models of quotient singularities by finite subgroups of \(\mathrm{SL}_n(\mathbb C)\). Let us recall that crepant resolutions have important roles in algebraic geometry. In this work, the author studies the existence of crepant resolutions of quotient singularities. He proves that a quotient singularity by a finite subgroup \(G\) has a crepant resolution if \(G\) is generated by junior elements and generalizes a result of Verbitsky. Then, he explains how to compute the corresponding Cox ring. Finally, he investigates the problem of smoothness of minimal models of some quotient singularities. quotient singularities; finite subgroups of \(\mathrm{SL}_n(\mathbb C)\); Cox ring; projective symplectic resolutions; crepant resolutions Yamagishi, R.: On smoothness of minimal models of quotient singularities by finite subgroups of \(SLn(C)\). (2016) Global theory of complex singularities; cohomological properties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) On smoothness of minimal models of quotient singularities by finite subgroups of \(SL_n(\mathbb{C})\) | 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 The McKay correspondence is a bijection involving certain simple Lie Algebras and irreducible representations of finite subgroups of \(\text{SL}(2,\mathbb C)\), observed by J. McKay in 1979. This has an interpretation in terms of the exceptional divisors of the minimal resolution of the (two-dimensional) singularity \(\mathbb C^2/G\). Since the appearance of \textit{J. McKay's} brief original paper [in: Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 183--186 (1980; Zbl 0451.05026)] many works were produced attempting to geometrically explain this process, and to generalize it. The present expository article primarily deals with a three-dimensional generalization of the original McKay correspondence [see \textit{Y. Ito} and \textit{M. Reid}, in: Higher dimensional complex varieties. Proc. Int. Conf., Trento 1994, 221--240 (1996; Zbl 0894.14024)]. Namely, if \(G\) is a finite subgroup of \(\text{SL}(3,\mathbb C)\), \(X=\mathbb C^3/G\) and \(p:Y \to X\) is a crepant resolution of \(X\) and \(S\) is the set of conjugacy classes of \(G\) then \(\chi (Y) = \text{card} (S)\), where \(\chi\) denotes Euler characteristic and crepant means: \(K_Y = p^*(K_X)\). For a while the validity of this equality (sometimes called ``Vafa's formula'') was a conjecture, arising from the work of researchers in Mathematical Physics. Other authors (Markushevich, Roan) also made contributions in this direction. Among other things, the present paper includes a discussion of the classification of the finite subgroups of \(\text{SL}(3,\mathbb C)\), a proof of existence of crepant resolutions for singularities of the form \(X=\mathbb C^3/G\), with \(G\) a finite subgroup of \(\text{SL}(3,\mathbb C)\) which is not of monomial type (based on the mentioned classification) and the notion of age of an element \(g\) of \(G\) as above, which plays an important role in the proposed proof of Vafa's formula. There are also some examples, as well as a summary of known results on McKay correspondences, both in the case of surfaces and three-folds. This paper is a good introduction to the subject. resolution of singularities; crepant resolution; group representation; Euler characteristic Ito, Y.: The McKay correspondence--a bridge from algebra to geometry. In: European Women in Mathematics (Malta, 2001), pp. 127-147. World Scientific Publishing, River Edge (2003) Singularities of surfaces or higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients) The McKay correspondence -- a bridge from algebra to geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0509.00008.]
This paper is a detailed and technical investigation of weak resolutions of deformations, culminating in the following theorem which provides a topological criterion for the existence of resolutions: Let \(f: V\to T\) be the germ of a flat deformation of the normal Gorenstein two dimensional singularity (V,p) with T a reduced analytic space. Then f has a weak simultaneous resolution if and only if each \(V_ t\) has a singularity \(p_ t\) with \((V_ t,p_ t)\) homeomorphic to (V,p). [This theorem essentially uses the result of \textit{W. Neumann}, Trans. Am. Math. Soc. 268, 299-343 (1981; Zbl 0546.57002) which asserts that the oriented homotopy type of V-p determines the topology of the pair (M,A) where \(g: M\to V\) is the minimal (good) resolution and A the exceptional set.] - On the way to the main theorem, a number of other results appear. For (V,p) a purely two-dimensional singularity, \(g: M\to V\) a resolution, and K the canonical divisor on M, \(S_ m\) denotes g(\({\mathcal O}(mK))\). The blow-up X of V at M is (for \(m\geq 3)\) shown to be the rational double point resolution of V. Moreover the canonical map \(S_ m\otimes S_ n\to S_{m+n}\) is shown to be onto for \(m\geq 2\) and \(n\geq 3\) (these assertions extend work of Shephard-Barron and Reid). With (V,p) normal and Gorenstein and g minimal then K is supported in the exceptional set A in M. So \(K\cdot K\) is defined, and is constant in the deformation \(f: V\to T\) if and only if f has a ''very weak'' simultaneous resolution; and if so then dim \(H^ 1(M_ t,{\mathcal O})\) is also constant. weak resolutions of deformations; normal Gorenstein two dimensional singularity; oriented homotopy type; simultaneous resolution H. B. LAUFER, Weak simultaneous resolution for deformations of Gorenstein surface singularities, Proc. Symp. Pure Math., 40 (1983), pp. 1-29. Zbl0568.14008 MR713236 Global theory and resolution of singularities (algebro-geometric aspects), Deformations of singularities, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Weak simultaneous resolution for deformations of Gorenstein surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Braverman, Finkelberg and Nakajima have recently given a mathematical construction of the Coulomb branches for a large class of \(3d\mathcal{N} = 4\) gauge theories, as algebraic varieties with Poisson structure. They conjecture that these varieties have symplectic singularities. We confirm this conjecture for all quiver gauge theories without loops or multiple edges, which in particular implies that the corresponding Coulomb branches have finitely many symplectic leaves and have rational Gorenstein singularities. We also give a criterion for proving that any particular Coulomb branch has symplectic singularities, and discuss the possible extension of our results to quivers with loops and/or multiple edges. Coulomb branch; affine Grassmannian; symplectic singularity; algebraic geometry; Poisson geometry; quiver gauge theory Yang-Mills and other gauge theories in quantum field theory, Representations of quivers and partially ordered sets, Poisson manifolds; Poisson groupoids and algebroids, Global theory and resolution of singularities (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Electromagnetic interaction; quantum electrodynamics, Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) Quiver gauge theories and symplectic singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Translated from the author's introduction: Binary polyhedral groups, \(\Gamma \subset \text{SL} (2, \mathbb{C})\), are associated with Kleinian singularities, \(\mathbb{C}^2/ \Gamma\), and McKay quivers. In this work we describe an invariant-theoretic interpretation of a differential-geometric construction of P. B. Kronheimer that explains the deformation of Kleinian singularities and their simultaneous resolution by means of families of representations of McKay quivers. In the construction of the resolution a new invariant-theoretic method, the linear modification of affine algebraic quotients, is developed. We discuss the basics of this method, give some applications to the representation of oriented CDW-graphs and finally connect it to the invariant theory of McKay quivers. Connections with the theory of abstract root systems and simple Lie algebras of corresponding CDW-type \(\Delta (\Gamma)\) also turn up in the representation theory of oriented CDW-graphs. Kleinian singularities; representations of McKay quivers; linear modification; invariant theory Homogeneous spaces and generalizations, Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry Linear modification of algebraic quotients, representations of the McKay quiver and Kleinian 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 [For the entire collection see Zbl 0509.00008.]
Let V, M be germs at \(0\in {\mathbb{C}}^ 2\) of curves with V irreducible, M smooth and \(M\neq V\). Let E be an irreducible component of the polar \(P_ M(V)\) of V with respect to M. The author proves the inequality \((V\cdot E)_ 0\neq(V\cdot M)_ 0(E\cdot M)_ 0\) between local intersection numbers. The author uses the method of \textit{M. Merle} [Invent. Math. 41, 103-111 (1977; Zbl 0371.14003)]. Then he applies the inequality to the equisingularity at \(\infty\) of a family of plane curves with one place at \(\infty\). polar curve; order of contact; plane curve singularity; Milnor number; local intersection numbers; equisingularity Ephraim R.: Special polars and curves with one place at infinity. Proceedings of the Symposium on Applied Mathematics, vol. 40, Part 1, pp. 353--359. American Mathematical Society, Providence (1983) Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Local complex singularities, Singularities in algebraic geometry Special polars and curves with one place at infinity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author gives an example of a complex-analytic surface singularity defined over \({\mathbb{Q}}(\sqrt{5})\) which is not equisingular to any singularity defined over \({\mathbb{Q}}\). - To construct this example he uses a configuration of 9 lines and 9 points in \({\mathbb{R}}^ 2\) which cannot be defined over \({\mathbb{Q}}\). The cone in \({\mathbb{C}}^ 3\) over the complexification of these lines is a surface S defined over \({\mathbb{Q}}(\sqrt{5}).\)
As the tangent cone of the surface S is reduced and as S does not have exceptional tangent lines, the author shows that any equisingular deformation of the surface S must induce an equisingular deformation of its tangent cone. Then any fiber of such a deformation cannot be defined over \({\mathbb{Q}}\). equisingular extension of tangent cone Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Global ground fields in algebraic geometry Un exemple de classe d'équisingularité irrationnelle. (An example of irrational equisingularity class) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field of characteristic 0. Let \(C\) be an irreducible nonsingular curve such that \({rC} = {S} \cap{F}\), \(r \in \mathbb{N} \), where \(S\) and \(F\) are two surfaces and all the singularities of \(F\) are of the form \(z^3 = x^s - y^s\), \(s\) prime, with \(\mathrm{gcd}(3, s) = 1\). We prove that \(C\) can never pass through such kind of singularities of a surface, unless \(r = 3a\), \({a} \in \mathbb{N} \). The case when the singularities of \(F\) are of the form \(z^3 = x^3s - y^3s\), \(s \in \mathbb{N} \), were studied in [\textit{M. R. Gonzalez-Dorrego}, Banach Cent. Publ. 108, 85--93 (2016; Zbl 1354.14007)]. Next, we study multiplicity-r structures on varieties for any positive integer \(r\). Let \(Z\) be a reduced irreducible nonsingular \(n- 1)\)-dimensional variety such that \({rZ} = {X} \cap F\), where \(X\) is a normal \(n\)-fold with certain type of singularities, \(F\) is a \((N - 1)\)-fold in \(\mathbb{P}^N \), such that \({Z} \cap \mathrm{Sing}(X) \neq \emptyset\). We study the singularities of \(X\) through which \(Z\) passes. Brieskorn singularities; fundamental cycle; maximal cycle; resolution of singularities Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, \(3\)-folds, \(4\)-folds, \(n\)-folds (\(n>4\)), Hypersurfaces and algebraic geometry On singular varieties with smooth subvarieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We show that any \(p\)-form on the smooth locus of a normal complex space extends to a resolution of singularities, possibly with logarithmic poles, as long as \(p \le \operatorname{codim}_X({X}_{\operatorname{sg}}) - 2\). A stronger version of this result, allowing no poles at all, is originally due to \textit{H. Flenner} [Invent. Math. 94, No. 2, 317--326 (1988; Zbl 0658.14009)]. Our proof, however, is not only completely different, but also shorter and technically simpler. We furthermore give examples to show that the statement fails in positive characteristic. Modifications; resolution of singularities (complex-analytic aspects), Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Local complex singularities, Transcendental methods of algebraic geometry (complex-analytic aspects), Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry A note on Flenner's extension theorem | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The purpose of this paper is to study the loci of arcs on a smooth variety defined by the order of contact with a fixed subscheme. More precisely, let consider us a non singular complex variety \(X\) and some sheaf of ideals \(\mathbf a\) defining a subscheme \(Y\subset X\). A log resolution of \(\mathbf a\) is a proper birational map \(\pi :X'\to X\), such that \(X'\) is non singular, \({\mathbf a}O_{X'}=O_{X'}(-\sum_i^t r_i E_i)\) is a normal crossing divisor and each component is smooth. Given a formal arc \(\gamma \) on \(X\), the order of contact \(\text{ord}_{\gamma }(Y)\) is well defined, the contact loci of order \(p\) will be:
\[
\text{Cont}^p(Y)=\{ \gamma \in X_\infty \mid \text{ord}_{\gamma }(Y)=p \}
\]
where \(X_\infty\) is the set of formal arcs on \(X\).
The main theorem describes the contact loci in terms of the log resolution, namely
\[
\text{Cont}^p(Y)=\sqcup_\nu \pi_\infty (\text{Cont}^\nu (E))
\]
where \(\nu \) runs over all t-uples of positive numbers such that \(\sum_i^tr_i=p\) and
\[
\text{Cont}^\nu (E)= \{ \gamma' \in X'_\infty \mid \text{ord}_{\gamma' }(E_i)=\nu_i \}.
\]
This is a Nash type theorem and it is related with theorem 2.4 of \textit{J. Denef} and \textit{F. Loeser} [Topology 41, 1031--1040 (2002; Zbl 1054.14003)].
As an application the authors give some results due to Mustata, without using motivic integration, for example: Let \(Y\subset X\) be a reduced and irreducible locally complete intersection subvariety. Then the arc space \(Y_l\) is irreducible for all \(l\geq 1\) if and only if \(Y\) has at worst rational singularities. log-canonical threshold; multiplier ideal Ein, Lawrence; Lazarsfeld, Robert; Mustaţǎ, Mircea, Contact loci in arc spaces, Compos. Math., 140, 5, 1229-1244, (2004) Singularities in algebraic geometry, Local complex singularities, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities Contact loci in arc spaces | 0 |
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