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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 Let \(Z=(Z_ 1,...,Z_ n)\) be a holomorphic vectorfield in a neighbourhood U of \(0\in {\mathbb{C}}^ n\) which has 0 as an isolated singularity. \({\mathcal F}_ z\) denotes the complex one-dimensional foliation with singularity at 0 defined by the integral curves of Z. A special case which should be kept in mind is the gradient vectorfield of a holomorphic function with isolated singularity at the origin.
The aim of the article is to define certain topological invariants of isolated singularities of holomorphic functions in the more general situation described above and to show that these invariants are indeed topological invariants of the foliation. E.g. if \(n\geq 2\), the Milnor number \(\mu (Z,0)=\dim_{{\mathbb{C}}} {\mathcal O}_{{\mathbb{C}}^ n,0}/(Z_ 1,...,Z_ n)\) is a topological invariant of \({\mathcal F}_ z\) as well as the index at 0 of the vectorfield \(Z|_ D\) where D is the intersection of a small ball with an irreducible complex curve V such that V-\(\{\) \(0\}\) is a leaf of \({\mathcal F}_ z.\)
In the second part the authors consider vectorfields Z in \({\mathbb{C}}^ 2\). They show that after finitely many quadratic transformations at singular points, the foliation \({\mathcal F}_ z\) is transformed into a foliation \(\tilde {\mathcal F}_ z\) with finitely many singularities of a very special kind which the authors call ''simple''. These simple singularities persist after further quadratic transformations and \(\tilde {\mathcal F}_ z\) is called a desingularization of \({\mathcal F}_ z\). A ''generalized curve'' is a vectorfield Z such that all simple singularities of \(\tilde {\mathcal F}_ z\) have nonvanishing eigenvalues (which is the case for the gradient vectorfield of a plane curve singularity). It is shown that two generalized curves have isomorphic desingularizations. From this the authors deduce that the algebraic multiplicity of a generalized curve is a topological invariant, which was of course well known for ''true'' plane curve singularities. holomorphic vectorfield; foliation; topological invariants of isolated singularities of holomorphic; functions; Milnor number; desingularization; algebraic multiplicity of a generalized curve; topological invariants of isolated singularities of holomorphic functions César Camacho, Alcides Lins Neto & Paulo Sad, ``Topological invariants and equidesingularization for holomorphic vector fields'', J. Differ. Geom.20 (1984) no. 1, p. 143-174 Complex singularities, Modifications; resolution of singularities (complex-analytic aspects), Deformations of complex singularities; vanishing cycles, Local complex singularities, Moduli, classification: analytic theory; relations with modular forms Topological invariants and equidesingularization for holomorphic 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 Habilitationsschrift gives an introduction to differential algebras over some coefficient field, differential graded algebras and modules, and De Rham cohomology. It proceeds to study modules of differential forms on complex spaces, criteria for regularity of an analytic local algebra [problem of Zariski-Lipman; cf. the author, Arch. Math. 51, No.5, 434-439 (1988; Zbl 0667.13010)] and singularities. In particular, the complex of residues associated to an analytic algebra A is constructed and the module of regular differential forms \(\Delta_ A\) defined. One then considers the module \(\Gamma_ A\) of differential forms that are extendable under a resolution of singularities and their connection with quadratically integrable forms. There is a natural homomorphism \(\Gamma_ A\to \Delta_ A\). Rational singularities are characterized by the property \(\Gamma^ n_ A=\Delta^ n_ A(n=\dim (A))\) and A Cohen- Macaulay. The theory of mixed Hodge structures is used to obtain results for isolated singularities over \({\mathbb{C}}\) and a correspondence principle allows to extend the results to arbitrary coefficient fields of characteristic zero. Here logarithmically extendable forms occur. The behaviour of the De Rham cohomology of an analytic algebra localized with respect to a not necessarily maximal prime ideal under passage to the residue field is studied and some special computations of De Rham cohomologies are made. The list of references contains 63 items. Bibliography; Zariski-Lipman problem. Kersken, Masumi; differential algebras; De Rham cohomology; regularity of an analytic local algebra; resolution of singularities; Rational singularities; mixed Hodge structures M. Kersken, Differentialformen in der algebraischen und analytischen Geometrie. Habilitationsschrift, Bochum 1987. Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Sheaves of differential operators and their modules, \(D\)-modules, Commutative rings of differential operators and their modules, Rings of differential operators (associative algebraic aspects), Modules of differentials Differentialformen in der algebraischen und analytischen Geometrie. (Differentiam forms in the algebraic and analytic geometry) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \({\mathfrak g}\) be a Lie algebra over an algebraically closed field \(k\) and \({\mathcal N}\) denote the associated nilpotent cone of \({\mathfrak g}\). Beginning with the investigation of commuting pairs of matrices, the nilpotent commuting variety \({\mathcal C}({\mathcal N}) := \{(x,y) \in {\mathcal N}\times{\mathcal N}~|~ [x,y] = 0\}\) has been long studied. For example, for the Lie algebra of a reductive algebraic group in zero or good characteristic, \textit{A. Premet} [Invent. Math. 154, No. 3, 653--683 (2003; Zbl 1068.17006)] showed that \({\mathcal C}({\mathcal N})\) is equidimensional and characterized its irreducible components.
In this paper, the authors consider the Witt Lie algebra \({\mathfrak g} := W_1\) under the assumption that \(p > 3\). Here \({\mathfrak g}\) admits the structure of a \(p\)-restricted Lie algebra and the nilpotent cone \({\mathcal N}\) happens to agree with the restricted nilpotent cone \({\mathcal N}_p := \{x \in {\mathfrak g}~|~ x^{[p]} = 0\}\). The authors show that \({\mathcal C}({\mathcal N})\) is reducible, equidimensional of dimension \(p\), and not normal. The irreducible components are explicitly described. A key ingredient in the proof is an explicit description of the centralizer of an arbitrary element of \({\mathfrak g}\). To obtain this latter description, the authors make use of the classification of the nilpotent orbits in \({\mathfrak g}\) under the action of its automorphism group that was obtained in previous work of the the first author with \textit{B. Shu} [Commun. Algebra 39, No. 9, 3232--3241 (2011; Zbl 1256.17002)].
The authors also consider the nilpotent commuting varieties for Borel subalgebras. The nilpotent cone for the negative Borel is only one dimensional, so the commuting variety is easily deduced. On the other hand, the problem is more complex for the positive Borel \({\mathfrak b}^+\). Similar to the full Lie algebra, the authors again show that \({\mathcal C}({\mathcal N}({\mathfrak b}^+))\) is reducible, equidimensional of dimension \(p\), and not normal. The irreducible components correspond to all but one of those from \({\mathcal C}({\mathcal N})\). Lastly, the authors observe that this may be applied to describe the spectrum of the cohomology ring of the second Frobenius kernel of the automorphism group of \({\mathfrak g}\). Witt algebra; Cartan type Lie algebra; nilpotent cone; restricted nilpotent cone; nilpotent commuting varieties; Borel subalgebra; cohomology of second Frobenius kernels Yao, Y-F; Chang, H., The nilpotent commuting variety of the Witt algebra, J Pure Appl Algebra, 218, 1783-1791, (2014) Coadjoint orbits; nilpotent varieties, Special varieties, Modular Lie (super)algebras, Cohomology theory for linear algebraic groups The nilpotent commuting variety of the Witt algebra | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is known that the ring of modular forms of the Picard modular group \(\Gamma_ K\subset U_ K\) of an imaginary quadratic field K which operates on the 1-ball \(B\subset {\mathbb{C}}^ 2\) is generated by three elements and that the values of certain quotients of these generators, i.e. modular functions, on so-called K-singular moduli \(\tau\in B\), i.e. fixed points of \(U_ K\), are algebraic.
As in the classical cases of the elliptic and Hilbert modular group these singular moduli correspond to ideals in orders of K (which in the classical cases at least furnish an algebraic equation with rational coefficients for the generators of the field of modular functions). Having this final result in mind the author first characterises all K- singular moduli by arithmetic data of K, secondly shows that a certain canonically given jacobian \(Jac(C_{\Phi}^{-1}{}_{(\tau)})\) is simple as long as \(\tau\in B\) is K-singular and finally expresses the number of K-singular moduli in terms of class-numbers of CM-extension, \(K\subset L\) of degree 3 following essentially the ideas of Hecke. elliptic modular group; Picard modular group; imaginary quadratic field; K-singular moduli; Hilbert modular group; jacobian; class-numbers of CM-extension Feustel, J, Eine klassenzahlformel für singuläre moduln der picardschen modulgruppen, Comp. Math, 76, 87-100, (1990) Picard groups, Algebraic moduli problems, moduli of vector bundles, Modular and Shimura varieties, Quadratic extensions, Picard schemes, higher Jacobians, Class numbers, class groups, discriminants Eine Klassenzahlformel für singuläre Moduln der Picardschen Modulgruppen. (A class number formula for singular moduli of Picard modular 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 \(\mathfrak G_0\) be the group of \(\mathbb{F}_q\)-rational points of a reductive connected algebraic group defined over a finite field \(\mathbb{F}_q\). Earlier [Characters of reductive groups over a finite field (Ann. Math. Stud. 107, 1984; Zbl 0556.20033)] the author classified the unipotent representations of \(\mathfrak G_0\) independently of \(q\). Let \(G\) be a simply connected almost simple algebraic group over \(\mathbb{C}\) with Lie algebra \(\mathfrak g\). The set of pairs \((\mathcal C,\mathcal F)\) for \(\mathcal C\) a nilpotent \(G\)-orbit in \(\mathfrak g\) and \(\mathcal F\) an irreducible \(G\)-equivariant local system on \(\mathcal C\) (up to isomorphism) is similar to the set of unipotent representations of \(\mathfrak G_0\). At one extreme, pairs \((\mathcal C,\mathcal F)\) from the Springer correspondence [\textit{T. A. Springer}, Invent. Math. 36, 173-207 (1976; Zbl 0374.20054)] are in bijective correspondence with the irreducible representations of the Weyl group. At the other extreme, there are the rare cuspidal pairs that the author has previously studied [Invent. Math. 75, 205-272 (1984; Zbl 0547.20032)]. A major theme of the paper under review is that these two classifications coalesce in the representation theory of \(\mathfrak G\), the group of rational points of a split adjoint simple algebraic group over a nonarchimedean local field.
An irreducible admissible representation of \(\mathfrak G\) is unipotent if its restriction to some parahoric subgroup contains a subspace on which the parahoric subgroup acts through a unipotent cuspidal representation of the ``reductive quotient'' of that parahoric subgroup (a reductive group over a finite field). (Note that replacement of \(\mathfrak G\) by \(\mathfrak G_0\) and parahoric by parabolic yields the notion of unipotent representation for \(\mathfrak G_0\).) An earlier conjecture of the author [Trans. Am. Math. Soc. 227, 623-653 (1983; Zbl 0526.22015)] is that the set of isomorphism classes of unipotent representations of \(\mathfrak G\) is in one-to-one correspondence with the set of triples \((s,y,{\mathcal V})\) modulo the natural action of \(G\). Here \(s\) is a semisimple element of \(G\), \(y\) is a nilpotent element of the Lie algebra \(\mathfrak g\) of \(\mathfrak G\) such that \(\text{Ad}(s)y=qy\) for \(q\) the number of elements of the residue field of the local field, and \(\mathcal V\) is an irreducible representation of the group of components of the simultaneous centralizer of \(s\) and \(y\) in \(G\), on which the center of \(G\) acts trivially. The present paper proves the conjecture by means of a program the author set out at the 1990 International Congress of Mathematicians [Proc. Int. Congr. Math., Tokyo 1990, 155-174 (1991; Zbl 0749.14010)]. The author already carried out several steps of his program [most recently in CMS Conf. Proc. 16, 217-275 (1995; Zbl 0841.22013)]. The first step in completing the program and establishing the conjecture in the current paper is the observation that the endomorphism algebra of a representation induced by a unipotent cuspidal representation of the reductive quotient of a parahoric subgroup is an affine Hecke algebra with presentation similar to that of \textit{N. Iwahori, H. Matsumoto} [Publ. Math., Inst. Hautes Étud. Sci. 25, 5-48 (1965; Zbl 0228.20015)] but with possibly unequal parameters. Recent work of \textit{A. Moy} and \textit{G. Prasad} [Invent. Math. 116, No. 1-3, 393-408 (1994; Zbl 0804.22008)] and \textit{L. Morris} [Invent. Math. 114, No. 1, 233-274 (1993; Zbl 0854.22022)] showed that these unipotent representations are parametrizable by the simple modules of the above endomorphism algebras.
The author shows further that given a subgroup of \(G\) that is the centralizer of some semisimple element and a unipotent class of the subgroup that carries a local cuspidal system, it is possible to construct explicitly an affine Hecke algebra (usually with unequal parameters). Those Hecke algebras arising from the geometry of \(G\) are the same as those that arise from the representation theory of \(\mathfrak G\). That reduces the problem to classification of the simple modules of the affine Hecke algebras that correspond to cuspidal local systems.
The author linearizes the affine Hecke algebras with respect to various points in the spectrum of the center to obtain graded Hecke algebras that he has previously studied [Publ. Math., Inst. Hautes Étud. Sci. 67, 145-202 (1988; Zbl 0699.22026) and J. Am. Math. Soc. 2, No. 3, 599-635 (1989; Zbl 0715.22020)]. The representation theory of those algebras yields the representation theory of the affine Hecke algebras in a way analogous to that in which the representation theory of a Lie group is recoverable from that of its Lie algebra. This thus reduces the problem at hand to classification of the simple modules of the linearized algebras. It turns out that the author's earlier 1995 paper, which used equivariant theory and perverse sheaves, provides the necessary representation theory of those linearized algebras.
The paper also classifies the unipotent representations of inner forms of \(\mathfrak G\), and the author conjectures that his methods should also apply to any form of \(\mathfrak G\) that becomes split after an unramified extension of the ground field. unipotent representations; simply connected almost simple algebraic groups; Borel subgroups; Weyl groups; cuspidal representations; irreducible representations; parahoric subgroups; equivariant \(K\)-theory; Hecke algebras Lusztig, G., Classification of unipotent representations of simple \textit{p}-adic groups, Int. Math. Res. Not. (IMRN), 11, 517-589, (1995) Representation theory for linear algebraic groups, Linear algebraic groups over finite fields, Linear algebraic groups over local fields and their integers, Simple, semisimple, reductive (super)algebras, Lie algebras of linear algebraic groups, Group schemes, Classical groups (algebro-geometric aspects) Classification of unipotent representations of simple \(p\)-adic groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The present book grew out of a series of lectures delivered by the two authors at the Summer School 1995 of the Graduiertenkolleg ``Geometry and nonlinear analysis'' at Humboldt University in Berlin. While the original lectures were designed to discuss some of the recent results on the geometry of moduli spaces of (semi-)stable coherent sheaves on an algebraic surface, the text at hand is a considerably elaborated and extended version of the initial notes. The outcome of the authors' rewarding and admirable effort at completing their lecture notes is now a book that serves several purposes at the same time. On the one hand, and in regard of its first part, it provides a textbook-like introduction to the theory of (semi-)stable coherent sheaves over arbitrary algebraic varieties and to their moduli spaces. On the other hand, mainly in view of its second part, the text has the character of both a research monograph and a comprehensive survey on some very recent results on those moduli spaces of (semi-)stable sheaves over (special) algebraic surfaces.
In both aspects, this book is rather unique in the existing literature on the classification theory of sheaves and vector bundles. Namely, for the first time in this central current area of research in algebraic geometry, a successful attempt has been undertaken to develop both the general, conceptual and methodical framework and the present state of knowledge in one of the most important special cases in a systematic, detailed, nearly complete and didactically processed presentation.
The text is divided into two major parts. After a careful introduction, which provides several motivations for studying sheaves on algebraic surfaces, in particular with a view to their significance in the differential geometry of four-dimensional manifolds and in gauge field theory (e.g., via Donaldson polynomials and Seiberg-Witten invariants), part I is devoted to the general theory of semi-stable sheaves and their moduli spaces. Chapter 1 introduces the basic concept of semi-stability for coherent sheaves over algebraic varieties, in the sense of D. Gieseker as well as in the (original) version of Mumford-Takemoto, and the fundamental material on Harder-Narasimhan filtrations, Jordan-Hölder filtrations, \(S\)-equivalence for semi-stable sheaves, and boundedness conditions. Flat families of sheaves, Grothendieck's Quot-scheme, the deformation theory of flags of sheaves, and Maruyama's openness-of-stability theorem are discussed in chapter 2, while chapter 3 deals with the most general form of the so-called Grauert-Mülich theorem and its application in establishing the boundedness of the set of semi-stable sheaves.
Moduli spaces for semi-stable sheaves, in their local and global aspects, is the subject of chapter 4. The authors discuss in detail C. Simpson's more recent approach to the construction of these moduli spaces, together with the related general facts from geometric invariant theory, and sketch the original construction by D. Gieseker and M. Maruyama likewise in an appendix. Furthermore, deformation theory is used to analyze the local structure of these moduli spaces, including dimension bounds and estimates for the expected dimension in the case of an algebraic surface. In another appendix the authors give an outlook to their own research contributions, in that they briefly describe moduli for ``decorated sheaves'' [cf. \textit{D. Huybrechts} and \textit{M. Lehn}, Int. J. Math. No. 2, 297-324 (1995; Zbl 0865.14004)]. This topic, though not systematically treated in the text, has recently found spectacular applications in conformal quantum field theory (e.g., in M. Thaddeus's proof of the famous Verlinde formula) and in non-abelian Seiberg-Witten theory.
The second part of the book, starting with chapter 5, mainly focuses on moduli spaces of semi-stable sheaves on algebraic surfaces. At first, the authors present various construction methods for stable vector bundle on surfaces, including Serre's correspondence between rank-2 vector bundles and codimension-2 subschemes, Maruyama's method of elementary transformations, and some illustrating examples. The geometry of moduli spaces of semi-stable sheaves on K3 surfaces is thoroughly explained in chapter 6, where in particular some very recent results by S. Mukai, A. Beauville, L. Göttsche-D. Huybrechts, K. O'Grady, J. Li, G. Ellingsrud-M. Lehn, and others are systematically compiled. Chapter 7 deals with the restriction of sheaves on surfaces to curves, focusing on the related work of H. Flenner, F. Bogomolov, and V. Mehta-A. Ramanathan in the 1980's. In chapter 8, the authors turn the attention to line bundles on moduli spaces and their Picard groups. The construction of determinantal line bundles and ampleness results for special line bundles on moduli spaces are presented by essentially following the approaches of J. Le Potier (1989) and J. Li (1993). As an application, the authors provide a profound comparison between the (algebraic) Gieseker-Maruyama moduli spaces of semi-stable vector bundles and the (analytic) Donaldson-Uhlenbeck compactification of the moduli spaces of Mumford-stable bundles. Chapter 9 is almost entirely devoted to K. O'Grady's recent work on the irreducibility and generic smoothness of moduli spaces for vector bundles on projective surfaces [cf. \textit{K. O'Grady}, Invent. Math. 123, No, 1, 141-207 (1996; Zbl 0869.14005)] and the related results by \textit{D. Gieseker} and \textit{J. Lie} [J. Am. Math. Soc. 9, 107-151 (1996; Zbl 0864.14005)].
Chapter 10, entitled ``Symplectic structures'', turns to differential forms on moduli spaces of stable sheaves on surfaces. After a lucid survey of the technical background material such as Atiyah classes, trace maps, cup products, the Kodaira-Spencer map, etc., the authors describe the tangent bundle of the smooth part of a moduli space by means of the universal family of vector bundles. Then, via the explicit construction of closed differential forms on moduli spaces, Mukai's theorem on the existence of a non-degenerate symplectic structure on the moduli space of stable sheaves on a K3 surface is derived.
The concluding chapter 11 deals with the birational properties of moduli spaces of semi-stable sheaves on surfaces. The main result presented here is a simplified proof of \textit{J. Li}'s recent theorem [Invent. Math. 115, No. 1, 1-40 (1994; Zbl 0799.14015)] stating that moduli spaces of semi-stable sheaves on surfaces of general type are also of general type. Other results on the birational type of such moduli spaces are surveyed in a brief sub-section, and the treatise concludes with two instructive examples showing how the Serre correspondence can be used to obtain information about the birational structure of moduli spaces of sheaves on a K3 surface. Actually, both examples are variations on two recent theorems due to T. Nakashima (1993) and K. O'Grady (1995), respectively, and their discussion is based upon an elegant combination of the results from chapter 8 and 10 in the book.
Altogether, the present text fascinates by comprehensiveness, rigor, profundity, up-to-dateness and methodical mastery. The bibliography comprises 263 references, most of which are really referred to in the course of the text. Each chapter comes with its own specific introduction and, always at the end, with a list of extra comments, hints to the original literature, and remarks on related topics, further developments and current research problems. The authors have successfully tried to keep the presentation of this highly advanced material as self-contained as possible, so that the text should be accessible for readers with a solid background in algebraic geometry. Active researchers in the field will appreciate this book as a valuable source and reference for their work. vector bundles on projective surfaces; stable coherent sheaves; moduli spaces; gauge field theory; Donaldson polynomials; Seiberg-Witten invariants; Grauert-Mülich theorem; semi-stable sheaves; geometric invariant theory; conformal quantum field theory; Verlinde formula; Seiberg-Witten theory; Picard groups; determinantal line bundles; Gieseker-Maruyama moduli spaces; Donaldson-Uhlenbeck compactification; differential forms on moduli spaces of stable sheaves; birational properties Hu D.~Huybrechts and M.~Lehn. \newblock \em Geometry of moduli spaces of sheaves, Vol. E31 of \em Aspects in Mathematics. \newblock Vieweg, 1997. Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Research exposition (monographs, survey articles) pertaining to algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli The geometry of moduli spaces of 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 The author shows that for any constants \(a<1/4\), and \(b,c\), there are at least \(y^{by^a+c}\) many irreducible components of the moduli space of regular surfaces of general type with \(K^2= c^2_1=y\) and fixed \(c_2\), where \(c_1\) and \(c_2\) are the Chern classes of the tangent bundle of the surface, and \(K\) is the canonical class. The same result holds true for the Hilbert scheme of surfaces in \(\mathbb{P}^4\) with \(K^2=y\) and fixed Hilbert polynomials. Similar results are given for threefolds. The idea of the proof is to look at projectively normal subvarieties of codimension two in the projective space with some very special resolution of the ideal sheaf. components of the moduli space; regular surfaces of general type; Chern classes; Hilbert scheme; surfaces; threefolds; codimension two M.-C. Chang, The number of components of Hilbert schemes. \textit{Internat. J. Math}. 7 (1996), 301-306. MR1395932 Zbl 0892.14006 Parametrization (Chow and Hilbert schemes), Low codimension problems in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The number of components of Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities An analog of the Procesi-Razmyslov theorem for the algebra of semi-invariants of representations of an arbitrary quiver with dimension vector \((2,2,\dots,2)\) is obtained. Procesi-Razmyslov theorem; representations of quivers; dimension vectors of representations; algebras of semi-invariants S. Fedotov, Semi-invariants of 2-representations of quivers, arXiv: 0909.4489. Representations of quivers and partially ordered sets, Trace rings and invariant theory (associative rings and algebras), Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Vector and tensor algebra, theory of invariants Semi-invariants of 2-representations of 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 We give a new criterion for when a resolution of a surface of general type with canonical singularities has big cotangent bundle and a new lower bound for the values of \(d\) for which there is a surface with big cotangent bundle that is deformation equivalent to a smooth hypersurface in \(\mathbb{P}^3\) of degree \(d\). big cotangent bundle; surfaces of general type; canonical singularities Global theory and resolution of singularities (algebro-geometric aspects) Resolutions of surfaces with big cotangent bundle and \(A_2\) 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 [Sel. Math., New Ser. 14, No. 1, 59-119 (2008; Zbl 1204.16008)], \textit{H. Derksen} et al. describe how to ``mutate'' at a vertex a pair \((Q,W)\) consisting of a quiver \(Q\) and a potential \(W\in(kQ)^\land/[(kQ)^\land,(kQ)^\land]\). This construction produces a new such pair \((Q',W')\). One peculiarity of the mutation process is that it is only defined if the vertex is not incident to a loop or two-cycle. Even if all vertices in \(Q\) have this property then this is not necessarily the case for \(Q'\). If the property of having no loops or two-cycles persists under iterated mutations, then we say that \((Q,W)\) (or \(W\)) is nondegenerate.
In this paper, we give a technical result (Theorem 3.1) which allows us to establish the nondegeneracy of potentials in some previously unknown or nonobvious cases. Our theorem implies, for example, that potentials derived from geometric helices on del Pezzo surfaces (see below) are nondegenerate. In this way, we recover part of the main result of \textit{T. Bridgeland} and \textit{D. Stern} [Adv. Math. 224, No. 4, 1672-1716 (2010; Zbl 1193.14022)]. nondegeneracy of potentials on quivers; McKay quivers De Völcsey, L. De Thanhoffer; Den Bergh, M. Van: Explicit models for some stable categories of maximal Cohen-Macaulay modules Representations of quivers and partially ordered sets, Rings arising from noncommutative algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Rational and ruled surfaces Some new examples of nondegenerate quiver potentials. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0509.00008.]
This paper is a short report on the study of b-functions for holomorphic functions. It contains some interesting examples calculated by the author. Let X be a complex manifold of dimension n. We denote by \(0_ X\) and \({\mathcal D}_ X\) the sheaf of holomorphic functions and holomorphic linear differential operators of finite order, respectively. We set \({\mathcal D}_ X[s]={\mathcal D}_ X\otimes_{{\mathbb{C}}}{\mathbb{C}}[s]\) the polynomial ring in s with coefficients in \({\mathcal D}_ X\). We define the b-function \(b(s)=b_{f,0}(s)\) of \(f\in 0_{X,0}\) at a point 0\(\in X\) to be the monic generator of the ideal \(\{\) b'(s)\(\in {\mathbb{C}}[s]\); there exists P(s)\(\in {\mathcal D}_ X[s]_ 0\) such that \(P(s,x,D_ x)f^{s+1}=b'(s)f^ s\}.\) The author guarantees the existence of b- functions and remarks on Kashiwara's theorem on the rationality of b- functions. Next, he states the relations with holonomic system theory and gives a formula in the case that f has an isolated singularity at 0. Moreover, he clears the connection with local monodromy and asymptotic expansion. Lastly, he gives some explicit examples. hypersurface isolated singularities; D-module; b-functions for holomorphic functions; holomorphic linear differential operators; rationality of b-functions; holonomic system; local monodromy T. Yano, ``\(b\)-functions and exponents of hypersurface isolated singularities'' in Singularities, Part 2 (Arcata, Calif., 1981) , Proc. Sympos. Pure Math. 40 , Amer. Math. Soc., Providence, 1983, 641--652. Local complex singularities, Differential forms in global analysis, Hyperfunctions, Holomorphic functions of several complex variables, Singularities of surfaces or higher-dimensional varieties b-functions and exponents of hypersurface 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, \(R = k[x, y]\) a polynomial ring in two variables, and \(I\) an ideal of \(R\) minimally generated by homogeneous forms \(h_1, h_2, h_3\) of the same degree \(d > 0\). Let \(C\) be the curve parametrized by \(h_1, h_2, h_3\) , we can assume that the curve \(C\) has degree \(d\). Let \(\psi\) the syzygy matrix of the ideal \(I\).
The two aspects, the curve C parametrized by the forms \(h_1, h_2, h_3\) and the syzygy matrix \(\psi\), are mediated by the Rees algebra \(R[It]\) of \(I\). \(R[It]\) becomes a standard bi-graded \(k\)-algebra if one sets \(\deg x = \deg y = (1, 0)\) and \(\deg t = (-d, 1)\), which gives \(\deg h_i t = (0, 1)\). The symmetric algebra \(\mathrm{Sym}(I)\), the Rees algebra \(R[It]\), and the ideal \(A\) of \(\mathrm{Sym}(I)\) that defines \(R[It]\) all are naturally equipped with two gradings. Let \(A_i\) be the \(S\)-submodule of \(A\) which consists of all elements homogeneous in \(x\) and \(y\) of degree \(i\), we can view \(A\) as a sum of the \(A_i\). The aim of this article is to study this ideal \(A\).
The purpose is more clear in the case \(d = 6\), the case of a sextic curve. The authors show that there is, essentially, a one-to-one correspondence between the bi-degrees of the defining equations of \(R[It]\) and the types of the singularities on or infinitely near the curve \(C\). bi-graded structures; duality; elimination theory; generalized zero of a matrix; generator degrees; Hilbert-Burch matrix; infinitely near singularities; Koszul complex; local cohomology; linkage; matrices of linear forms; Morley forms; parametrization; rational plane curve; rational plane sextic; Rees algebra; Sylvester form; symmetric algebra 10.1016/j.jalgebra.2016.08.014 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Plane and space curves, Singularities of curves, local rings, Rational and birational maps The bi-graded structure of symmetric algebras with applications to Rees rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a variety over a number field \(k\) with adele ring \(\mathbb{A}_k\). The left kernel of the Brauer-Manin pairing
\[
X(\mathbb{A}_k) \times \mathrm{Br}(X) \to \mathbb{Q}/\mathbb{Z}, ((x_v), \alpha) \mapsto \sum_v\mathrm{inv}_v(x_v^*\alpha)
\]
is called \(X(\mathbb{A}_k)^{\mathrm{Br}}\). By the Albert-Brauer-Hasse-Noether theorem, one has \(X(k) \subseteq X(\mathbb{A}_k)^{\mathrm{Br}}\).
\textit{A. N. Skorobogatov} [Invent. Math. 135, No. 2, 399--424 (1999; Zbl 0951.14013), Section 3] defined the étale Brauer-Manin obstruction
\[
X(\mathbb{A}_k)^{\mathrm{et,Br}} = \bigcap_{\text{\(G\) finite}}\bigcap_{f \in H^1(X,G)}\bigcup_{[\sigma] \in H^1(k,G)}f^\sigma(Y^\sigma(\mathbb{A}_k)^{\mathrm{Br}}).
\]
\textit{B. Poonen} [Prog. Math. 199, 307--311 (2001; Zbl 1079.14027)] constructed examples for which the Hasse principle fails, but the étale Brauer-Manin obstruction is non-empty, see also the articles by \textit{Y. Harpaz} and \textit{A. N. Skorobogatov} [Ann. Sci. Éc. Norm. Supér. (4) 47, No. 4, 765--778 (2014; Zbl 1308.14024)] and by \textit{J.-L. Colliot-Thélène} et al. [Math. Z. 282, No. 3--4, 799--817 (2016; Zbl 1365.14025)]. These examples are all fibrations over a base curve of genus \(\geq 1\). The author constructs examples with trivial Albanese variety where the étale Brauer-Manin obstruction does not explain the failure of the Hasse principle:
By a standard fibration, the author means a dominant and proper morphism of varieties with smooth and geometrically integral generic fibre.
Proposition 3.1. Let \(k\) be a real number field. Let \(Y\) be a smooth, projective, geometrically integral \(k\)-variety. Assume that there exists a standard fibration \(f: Y \to \mathbb{P}^1_k\) such that \(f(Y(k))\) is finite and non-empty. Then there exists a standard fibration in three-dimensional quadrics \(g: Q \to \mathbb{P}^1_k\) such that the fibre product \(X = Y \times_{\mathbb{P}^1_k} Q\) is a smooth, projective, geometrically integral \(k\)-variety for which \(X(k) = \emptyset\) and \(X(\mathbb{A}_k)^{\mathrm{et,Br}} \neq \emptyset\).
Proposition 3.2. Let \(k\) be any number field. Let \(Y\) be a smooth, projective, geometrically integral \(k\)-variety. Assume that there exists a standard fibration \(f: Y \to \mathbb{P}^1_k\) such that \(f(Y(k))\) is finite and non-empty. Then there exists a standard fibration in Chatelet surfaces \(g: C \to \mathbb{P}^1_k\) such that the fibre product \(X = Y \times_{\mathbb{P}^1_k} C\) is a smooth, projective, geometrically integral \(k\)-variety for which \(X(k) = \emptyset\) and \(X(\mathbb{A}_k)^{\mathrm{et,Br}} \neq \emptyset\).
Lemma 3.4. Let \(k\) be a perfect field and let \(f: Y \to X\) be a standard fibration. Assume that \(X\) has trivial Albanese variety, and that the same is true for at least one smooth fibre of \(f\). Then the Albanese variety of \(Y\) is trivial as well.
Theorem 3.5. Let \(k\) be a number field. There exists a smooth, projective, geometrically integral fourfold \(X\) over \(k\) such that \(X(k) = \emptyset\), \(X(\mathbb{A}_k)^{\mathrm{et,Br}} \neq \emptyset\) and \(\mathrm{Alb}(X)\) is trivial.
Theorem 4.1. Assume that the \(abc\) conjecture is true. Then there exists a number field \(k\) and a smooth, projective, geometrically integral and geometrically simply connected fourfold \(X\) over \(k\) such that \(X(k) = \emptyset\) and \(X(\mathbb{A}_k)^{\mathrm{et,Br}} \neq \emptyset\). Brauer groups of schemes; rational points; varieties over global fields Smeets, Arne, Insufficiency of the étale Brauer-Manin obstruction: towards a simply connected example, Amer. J. Math., 139, 2, 417-431, (2017) Brauer groups of schemes, Rational points, Varieties over global fields Insufficiency of the étale Brauer-Manin obstruction: towards a simply connected example | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a closed subvariety of a smooth variety over a perfect field \(k\), \(m\) the maximum value of the multiplicity \(m(x)\) of \(X\) at point \(x\), and \(M(X)\) the (closed) subset of \(X\) of points \(y\) where \(m(y)=m\). A possible problem is to find a sequence of monoidal transforms \(V=V_0, \ldots, V_r\), such that if \(X_i\) is the strict transform of \(X\) to \(V_i\), the centers satisfy \(C_i \subset M(X_i)\) and the maximum multiplicity of \(X_r\) is \(<m\). If this can be done, iterating the process we resolve the singularities of \(X\).
The problem has affirmative solution in characteristic zero and remains open, for \(\dim(X) > 3\), in positive characteristic. The present paper is a contribution to this open problem when \(X\) is a hypersurface in a smooth variety \(V\). As in other articles of the authors, the question is rephrased in terms of \textit{Rees algebras} (certain graded subagebras of the polynonial algebra \({\mathcal O}_V[T]\)).
In previous work the authors (and Ana Bravo) have shown that, in any characteristic, it is possible to ``improve'' the singularities of a Rees algebra \(\mathcal G\) (over a regular variety \(V\)). More precisely, let \(e(\mathcal G)\) denote the minimum value of \({\tau}_{\mathcal G,x}\), for \(x\) a singular point of \( \mathcal G\), (\(\tau\) is a version of a numerical invariant introduced by Hironaka). By descending induction on \(e(\mathcal G)\) it is possible to associate to \(\mathcal G\), locally in the etale topology: (a) sequences
\[
(1) \qquad V=V_0 \leftarrow \cdots \leftarrow V_s \, ,
\]
\[
(2) \qquad V'_0 \leftarrow \cdots \leftarrow V'_s \, ,
\]
of monoidal transforms with regular centers, (c) transversal smooth projections \(\beta _i:V_i \to V'_i\), \(\dim (V'_i)=\dim(V_i)-e\), for all \(i\), and (d) Rees algebras \({\mathcal R}_{i}\), such that \({\mathcal R}_s\) is monomial. The algebra \(\mathcal{R}_i\) is the \textit{elimination algebra of} \({\mathcal G}_i\) (the transform of \({\mathcal G}\) to \(V_i\)) with respect to \(\beta _i\), a concept thoroughly studied by Villamayor. Monomial algebras correspond to \textit{monomial ideals}, a particularly simple type of invertible sheaf, which can be resolved pretty easily essentially in a combinatorial way. In this case, one says that \(\mathcal G _s\) is in the \textit{monomial situation}. If the characteristic of the base field \(k\) is zero, the mentioned result leads to an extension of the sequence (1) which resolves \(\mathcal G\), but in positive characteristic there are difficulties. Trying to overcome these, in the present paper the authors introduce a concept of ``strong monomial situation''. If in (1) \(\mathcal G _s\) is in the strongly monomial situation, even in positive characteristic it is possible to extend the sequence (1) so that a resolution of \(\mathcal G\) is achieved. The authors give a numerical criterion to decide whether the strong monomial situation has been reached.
Both the notion of strong monomial situation and the mentioned criterion require some new auxiliary concepts, introduced and developed in this paper. For instance: (i) the notion of \textit{\(p\)-presentation} of a Rees algebra \(\mathcal G\) over \(V\) of dimension \(d\), relative to a transversal projection \(\beta : V \to V'\), \(\dim V' = d-1\), which is a way to describe \(\mathcal G\) (locally, in the etale topology) in terms of the elimination algebra \( \mathcal R\) of \(\mathcal G\) and a monic polynomial, whose degree is a power of \(p\), with coefficients defined on \(V'\); (ii) the notion of \textit{slope} of \(\mathcal G\) relative to a p-presentation; (iii) the notion of H-ord at a point of \(V\), defined in terms of slopes of suitable general presentations. The authors sometimes work under the assumption that \(e(\mathcal G)=1\), and announce that the general situation will be discussed in future papers.
If \(\dim X =2\), in other articles they showed that, even in positive characteristic, with the methods just reviewed a resolution of singularities is obtained. resolution of singularities; positive characteristic; differential operators; Rees algebras; monomial ideals. Benito, Angélica; Villamayor U., Orlando E., Monoidal transforms and invariants of singularities in positive characteristic, Compos. Math., 149, 8, 1267-1311, (2013) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Graded rings, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes Monoidal transforms and invariants of singularities in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We investigate the saturation rank of a finite group scheme defined over an algebraically closed field \(\Bbbk\) of positive characteristic \(p\). We begin by exploring the saturation rank for finite groups and infinitesimal group schemes. Special attention is given to reductive Lie algebras and the second Frobenius kernel of the algebraic group \(\mathrm{SL}_{n}\). saturation rank; finite groups; infinitesimal group schemes Modular Lie (super)algebras, Group schemes Saturation rank for finite group schemes: finite groups and infinitesimal group schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Main Theorem of the paper generalizes Th. Schneider's result on the transcendence of the values of the elliptic modular function \(j\) at algebraic non-CM-points to suitably normalized modular functions in several variables playing the same role as \(j\) for families of abelian varieties: At algebraic non-CM-points \(z\) in their bounded symmetric complex domain \(D\), the normalized quotient map \(J: D\to V\) into the corresponding Shimura variety \(V\) gives a non-algebraic image point \(J(z)\) of \(V\). The proof relies on three points:
1) By the theory of canonical models, it is sufficient to prove the following statement: If \(D\) parametrises the family \(A_z\) of abelian varieties and \(z\in D\) is algebraic but not a CM point, then \(A_z\) cannot be defined over a number field.
2) If \(A_z\) is defined over a number field and \(z\) an algebraic point of \(D\), there are additional linear relations over number fields between periods of \(A_z\). These relations imply by Wüstholz' analytic subgroup theorem that the endomorphism algebra of \(A_z\) is larger than generically expected for the family parametrised by \(D\).
3) This means that this \(A_z\) belongs to a smaller family of abelian varieties parametrised by a lower dimensional symmetric domain \(D'\) as well. By rationality properties of modular embeddings \(D'\to D\), the corresponding point \(\tau\in D'\) is algebraic as well. Therefore, the arguments 2) and 3) can be repeated to show that \(A_z\) belongs to a zero dimensional family, hence is of CM type.
The last paragraphs present some open problems and consequences for hypergeometric functions in one and several variables. For Hilbert modular functions, the Main Theorem takes a particularly simple form.
The paper is the major part of a joint work written with \textit{Paula Beazley Cohen}. She gives a short account on it in Sémin. Théor. Nombres Paris 1992-93 [``Propriétés transcendantes des fonctions automorphes'', Lond. Math. Soc. Lect. Note Ser. 215, 81--89 (1995; Zbl 0827.11044)] and a different and modernized version of it in a forthcoming paper [``Humbert surfaces and transcendence properties of automorphic functions'', Rocky Mt. J. Math. 26, No. 3, 987--1001 (1996; Zbl 0888.11030)] replacing the modular embeddings by more direct arguments. Such direct arguments were used by the first author also in a former paper [``On the transcendency of the values of the modular function at algebraic points'', Astérisque 209, 293--305 (1992; Zbl 0862.11046)]. complex multiplication; transcendence; modular functions in several variables; Shimura variety; hypergeometric functions; Hilbert modular functions Hironori Shiga and Jürgen Wolfart, Criteria for complex multiplication and transcendence properties of automorphic functions, J. Reine Angew. Math. 463 (1995), 1-25. Transcendence theory of elliptic and abelian functions, Complex multiplication and moduli of abelian varieties, Abelian varieties of dimension \(> 1\), Modular and automorphic functions, Algebraic theory of abelian varieties, Modular and Shimura varieties Criteria for complex multiplication and transcendence properties of automorphic 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 \(K\) be a finite extension of the rational function field \(\mathbb{Q}_p(t)\) in one indeterminate over the field \(\mathbb{Q}_p\) of \(p\)-adic numbers. The paper under review continues the study of central division \(K\)-algebras of dimensions prime to \(p\) presented in earlier papers by the author [see, for example, his articles in J. Ramanujan Math. Soc. 12, No. 1, 25-47 (1997; Zbl 0902.16021), ibid. 13, No. 2, 125-129 (1998; Zbl 0920.16008)].
The main result of the paper states that the considered algebras are cyclic in case they are of prime index \(q\neq p\). Under the hypothesis that \(K\) is the function field of a projective regular and excellent surface \(S\to\text{Spec}(\mathbb{Z}_p)\) of finite type and relative dimension one (where \(\mathbb{Z}_p\) is the ring of \(p\)-adic integers), it is also proved that if \(D\) is a central division \(K\)-algebra of exponent \(q\) and \(S\) has been blown up so that the ramification locus of \(D\) consists of nonsingular curves with normal crossings, then \(D\) has Schur index \(q\) if and only if there are no hot points.
The author's approach to the topic of his paper aims at proving as much as possible about Brauer classes over surfaces. He first reviews some material about Brauer groups, cyclic extensions, ramification, projective surfaces over \(\mathbb{Z}_p\), and develops a kind of a divisor theory by considering a certain cohomology group over a more general scheme \(X\). Next he studies the geometry of a Brauer class of order \(q\), taking as a basis of the analysis the special case where \(K\) contains a primitive \(q\)-th root of unity. Before proving the main results, the author also considers the behaviour of the ``residual'' classes. central division algebras; cyclic algebras; ramification; curve points; nodal points; Brauer groups; curves over local fields; \(p\)-adic curves; field extensions; algebraic function fields; curves over rings of integers of \(p\)-adic fields D. J. Saltman, ''Cyclic algebras over \(p\)-adic curves,'' J. Algebra, vol. 314, iss. 2, pp. 817-843, 2007. Finite-dimensional division rings, Curves over finite and local fields, Arithmetic ground fields for curves, Brauer groups of schemes, Skew fields, division rings, Algebras and orders, and their zeta functions, Algebraic functions and function fields in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Local ground fields in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Special surfaces Cyclic algebras over \(p\)-adic curves. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We provide a direct computation of the \(F\)-pure threshold of degree four homogeneous polynomials in two variables and, more generally, of certain homogeneous polynomials with four distinct roots. The computation depends on whether the cross ratio of the roots satisfies a specific Möbius transformation of a Legendre polynomial. We then make a connection between a long lasting open question, involving the relationship between the \(F\)-pure and the log canonical threshold, and roots of Legendre polynomials over \(\mathbb{F}_p\). \(F\)-pure threshold; Deuring polynomial; Legendre polynomial; singularities of curves; finite fields Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Computational aspects and applications of commutative rings, Curves over finite and local fields, Computational aspects of algebraic curves, Singularities in algebraic geometry, Elliptic curves over global fields Legendre polynomials roots and the \(F\)-pure threshold of bivariate forms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 book is the second edition of the author's [Geometric modular forms and elliptic curves, ibid. 361 p. (2000; Zbl 0960.11032)]. The contents (of the first edition) of the book has been described by the reviewer in Zbl 0960.11032. Here we only indicate the main changes done in the second edition.
A detailed description of Barsotti-Tate groups (including formal Lie groups) is added in Chapter 1. As an application, a down-to-earth description of formal deformation theory of elliptic curves is incorporated in Section 2.10.
In Chapter 3, two subsections (3.2.6 and 3.2.7) are added to facilitate good transition from horizontal/vertical control results in the earlier part of Section 3.2 for modular forms to ring/scheme theoretic control results on the side of Hecke algebra.
The newly added Section 4.3 contains Ribet's theorem of full image of modular \(p\)-adic Galois representation and its generalization to `big' \(\Lambda\)-adic Galois representations under middle assumptions (a new result of the author).
The newly added Section 5.3 discusses modularity of abelian \(\mathbb Q\)-varieties. Modularity of abelian \(\mathbb Q\)-varieties of \(\text{GL}(2)\)-type was predicted by Ribet, and finally proved in 2009 by \textit{Khare} and \textit{Winterberger} as a special case of modularity of strict compatible systems of odd two-dimensional Galois representations. The author gives a proof of special cases of the modularity directly based on the theorem of Wiles-Taylor-Diamond-Skinner (Theorem 5.2.1).
The bibliography has been extended and updated.
The book, addressed to graduate students and experts working in number theory and arithmetic-geometry, is a welcome addition to this beautiful and difficult subject. control theorems; Shimura-Taniyama-Weil conjecture; elliptic curve; modular curve; deformation rings; Hecke algebras; modular Galois representations; moduli spaces of elliptic curves; modular forms; Abelian \(\mathbb{Q}\)-curves Hida, H.: Geometric Modular Forms and Elliptic Curves, 2nd edn. World Scientific, Singapore (2012) Galois representations, Elliptic curves over global fields, Research exposition (monographs, survey articles) pertaining to number theory, \(p\)-adic theory, local fields, Holomorphic modular forms of integral weight, Local ground fields in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry Geometric modular forms and elliptic 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 \(S\) be a minimal surface of general type and let \({\mathcal M}(S)\) be the moduli space of surfaces of general type homeomorphic (by an orientation preserving homeomorphism) to \(S\). \({\mathcal M} (S)\) is a quasi projective variety by the well known theorem of Gieseker. Let \(k_S \in H^2 (S, \mathbb{Z})\) be the first Chern class of the canonical bundle of \(S\) and let \(r(S)\) its divisibility, i.e.
\[
r(S) = \max \{r \in \mathbb{N} \mid k_S = rc \text{ for some }c \in H^2 (S, \mathbb{Z})\}.
\]
Obviously if \(S' \in {\mathcal M}(S)\) is in the same connected component of \(S\) then there exists an orientation preserving diffeomorphism \(f : S'\to S\) such that \(f^* (k_S) = k_{S'}\) and \(r(S) = r(S')\). Catanese first proved that in general \({\mathcal M} (S)\) is not connected giving homeomorphic surfaces of general type with different divisibility \(r\). Similarly, using the fact that for surfaces with ``big monodromy'' \(r(S)\) is a differential invariant, Friedman, Morgan and Moishezon gave the first examples of surfaces of general type homeomorphic but not diffeomorphic. Moreover it is possible that \(r(S)\) is a differential invariant of minimal surfaces of general type. -- Define \({\mathcal M}_d (S) = \{S' \in {\mathcal M} (S) \mid r(S) = r(S')\}\), it is natural to ask whether \({\mathcal M}_d (S)\) is connected. In this paper we show that the answer is no, more precisely we prove:
Theorem A. For every \(k > 0\) there exists a simply connected minimal surface of general type \(S\) such that \({\mathcal M}_d (S)\) has at least \(k\) connected components.
From our proof it follows moreover that the \(k\) connected components have different dimension. Here we study a particular class of surfaces introduced by \textit{F. Catanese} [J. Differ. Geom. 19, 483-515 (1984; Zbl 0549.14012), Compos. Math. 61, 81-102 (1987; Zbl 0615.14021) and J. Differ. Geom. 24, 395-399 (1986; Zbl 0621.14014)] and called ``simple bihyperelliptic surfaces''. Denote \(X=\mathbb{P}^1\times\mathbb{P}^1\) and let \({\mathcal O}_X (a,b)\) be the line bundle on \(X\) whose sections are bihomogeneous polynomials of bidegree \(a,b\). A minimal surface of general type is said to be simple bihyperelliptic of type \((a,b) (n,m)\) if its canonical model is defined in \({\mathcal O}_X (a,b) \oplus {\mathcal O}_X (n,m)\) by the equation
\[
z^2 = f(x,y),\;w^2 = g(x,y)
\]
where \(f,g\) are bihomogeneous polynomials of respective bidegree \((2a, 2b)\), \((2n, 2m)\). Simple bihyperelliptic surfaces of type \((a,b)\), \((c,d)\) are simply connected if \(a,b, c,d> 0\). In the same notation as F. Catanese's paper in Compos. Math. (loc. cit.), if \(a > 2n\), \(m > 2b\) let \(\widehat N_{(a,b), (n, m)}\) be the subset of moduli space of simple bihyperelliptic surfaces of type \((a,b)\), \((n,m)\). Here we prove, as conjectured in that paper, the following
Theorem B. If \(a \geq \max (2n + 1, b + 2)\), \(m \geq \max (2b + 1, n + 2)\) then \(\widehat N = \widehat N_{(a,b), (n,m)}\) is a connected component of moduli space.
Theorem B is an easy consequence of the last cited paper (corollary 4.4 and lemma 9 that follows from the technical proposition 7). Theorem A is then an easy consequence of theorem B and Freedman's results on four-dimensional manifolds. simple bihyperelliptic surfaces; minimal surface of general type; Chern class; divisibility; homeomorphic but not diffeomorphic surfaces M Manetti, On some components of moduli space of surfaces of general type, Compositio Math. 92 (1994) 285 Surfaces of general type, Families, moduli, classification: algebraic theory, Rational and ruled surfaces, Differentiable mappings in differential topology On some components of moduli space of surfaces of general type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is the first in a series of three papers [Osaka J. Math. 40, No.4, 857--893 (2003; Zbl 1080.14516); Commun. Algebra 34, No.1, 89--106 (2006; Zbl 1086.14014)] concerning the surface \(T\times T\). Here, we study the degeneration of \(T\times T\) and the regeneration of its degenerated object. We also study the braid monodromy and its regeneration. homotopy groups; algebraic topology of manifolds; fundamental groups; covering spaces; degeneration; monodromy; moduli spaces; elliptic surfaces; curves; surfaces; singularities M. Amram and M. Teicher, On the degeneration, regeneration and braid monodromy of the surface \(T\times T,\) Acta Appl. Math. 75 (2003), 195--270. Fibrations, degenerations in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Fine and coarse moduli spaces, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Homotopy groups, general; sets of homotopy classes, Euclidean geometries (general) and generalizations On the degeneration, regeneration and braid monodromy of \(T\times T\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 well-written article constitutes a very accessible introduction to the Hodge conjecture for abelian varieties, with special emphasis on abelian varieties of Weil type and Mumford-Tate groups. It provides a welcome partial update of \textit{T. Shioda}'s survey article [in: Algebraic Varieties and Analytic Varieties, Proc. Symp., Tokyo 1981, Adv. Stud. Pure Math. 1, 55--68 (1983; Zbl 0527.14010)], which, strangely enough, is missing from the bibliography.
Notation is set and the conjecture explained in the first three sections: If \(X\) is a smooth complex projective variety, one denotes by \(B^p\) the set of rational classes of type \((p,p)\), by \(B^\bullet = \bigoplus B^p\) the graded ring of Hodge classes, by \(D^\bullet\) the graded subring generated by \(B^0 = \mathbb Q [X]\) and \(B^1\), and by \(\Psi_p\) the cycle map in codimension \(p\). Hodge classes not in \(D^\bullet\) are called exceptional. One has \(D^p \subset \text{Im} (\Psi_p) \subset B^p\), and the Hodge conjecture for \(X\) in codimension \(p\) is the equality \(\text{Im} (\Psi_p) = B^p\); it holds in particular if \(D^p = B^p\), i.e. if there are no exceptional Hodge classes.
After describing the Hodge structure on the cohomology, the author reviews (without proofs) what is known about the Hodge conjecture for an abelian variety \(X\): It holds for \(X\) generic (Mattuck), or for \(X\) isogenous to a product of elliptic curves (Tate), or for \(X\) simple of prime dimension (Tankeev, Ribet) because there are no exceptional Hodge classes. -- The author then explains in section 4 how Mumford's example of a simple abelian fourfold with exceptional Hodge classes lead Weil to study \(2n\)-dimensional abelian varieties with complex multiplication by an imaginary quadratic field \(K\) (called here abelian varieties of Weil type). Various results are recalled (without proofs): Weil's description of \(B^n\) for a general \(2n\)-dimensional abelian variety of Weil type with \(n \geq 2\), which shows that it does have exceptional Hodge classes, Murty and Hazama's related constructions, Moonen-Zarkhin's examples of abelian varieties with trivial endomorphism ring and exceptional Hodge classes, their theorem that simple abelian fourfolds with exceptional Hodge classes are of Weil type, and Schoen's proof of the Hodge conjecture for general abelian fourfolds of certain Weil types.
Section 5 is devoted to a detailed investigation of abelian varieties of Weil type and their moduli spaces (with proofs). -- Mumford-Tate groups for abelian varieties are defined in section 6, and their relationship with the Hodge conjecture explained. Weil's calculation of this group for general abelian varieties of Weil type, and the derivation of his above- mentioned theorem, are explained in detail. The last section explains Schoen's above-mentioned proof (with the correction of a minor error), and the author's related results on other abelian fourfolds of Weil type. Hodge conjecture for abelian varieties; abelian varieties of Weil type; Mumford-Tate groups; abelian variety of Weil type; abelian fourfolds; exceptional Hodge classes van Geemen B.: An introduction to the Hodge conjecture for abelian varieties, Algebraic cycles and Hodge theory, Torino 1993, Lecture Notes in Math., vol. 1594, pp. 233--252. Springer (1994) Transcendental methods, Hodge theory (algebro-geometric aspects), Complex multiplication and abelian varieties, Algebraic moduli of abelian varieties, classification, Complex multiplication and moduli of abelian varieties, Algebraic cycles, Picard schemes, higher Jacobians An introduction to the Hodge conjecture for abelian varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the given paper, the authors generalise a construction of mixed Beauville groups first given by \textit{I. Bauer, F. Catanese} and \textit{F. Grunewald} [in ``Geometric methods in algebra and number theory'', Prog. Math. 235, 1-42 (2005; Zbl 1094.14508)]. Furthermore, they provide a wealth of new examples of finite characteristically simple mixed Beauville groups. finite characteristically simple groups; mixed Beauville groups Fairbairn, BT; Pierro, E, New examples of mixed Beauville groups, J. Group Theory., 18, 761-792, (2015) Finite simple groups and their classification, Simple groups: alternating groups and groups of Lie type, Generators, relations, and presentations of groups, Conjugacy classes for groups, Surfaces of general type, Families, moduli, classification: algebraic theory, Compact Riemann surfaces and uniformization, General structure theorems for groups New examples of mixed Beauville 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 What the authors call \textsl{The Main Theorem} of this enlightening paper can be conventionally split into two parts. The first is concerned with a certain symmetric structure of the \(n\)th exterior power of a polynomial ring; the second with a general determinantal formula evoking Giambelli's formula in classical Schubert Calculus on Grassmann Schemes or Jacobi-Trudy formula in the theory of symmetric polynomials. In spite of being strongly related with classical and widely investigated subjects, the result is new and shed further light on the beautiful algebraic properties of exterior powers of a module.
To describe the main theorem of the paper in a more precise way, here is a piece of notation. Let \(\bigotimes^n_AA[X]\) and \(\bigwedge^n_AA[X]\) be, respectively, the tensor and exterior \(n\)th-power of a polynomial algebra in one indeterminate with coefficients in \(A\), a commutative ring with unit. Also, let \(R:=A[X_1,\dots, X_n]\) be the ring of polynomials in \(n\) indeterminates. Because of the natural identification between \(\bigotimes^n_AA[X]\) and \(R\), the former is naturally an \(S:=R^{\text{sym}}\)-module, where \(R^{\text{sym}}\) is the \(A\)-algebra of the symmetric polynomials in \(R\). The first part of the main theorem then says that there exists a unique \(S\)-module structure on \(\bigwedge^n_AA[X]\) such that the canonical projection \(\bigotimes^nA[X]\rightarrow \bigwedge^nA[X]\) is \(S\)-linear. The \textsl{symmetric structure} of \(\bigwedge^n_AA[X]\) is described in a very explicit way: it turns out that \(\bigwedge^nA[X]\) is a free \(S\)-module of rank \(1\) generated by \(\phi:=X^{n-1}\wedge X^{n-2}\wedge\ldots\wedge X^0\) (\(X^0=1_A\)). As a consequence, if \(f_1,\dots, f_{n}\) are \(n\) arbitrary elements of \(A[X]\), the \(n\)-vector \(f_1(X)\wedge\ldots\wedge f_n(X)\) must be an \(S\) multiple of \(\phi\) and at this point the second part of the main theorem comes into play.
It shows that such a multiple can be computed by means of a very general and beautiful determinantal formula (that alluded to in the title) involving the coefficients of the polynomials \(f_i\)s only. We omit to write down the general determinantal formula, as it appear in the paper, which would require some additional explanations, but we mention a remarkable particular case: when \(f_i=X^{h_i+n-i}\) (\(1\leq i\leq n\)), one gets \( X^{h_1+n-1}\wedge X^{h_2+n-2}\wedge\ldots\wedge X^{h_{n}}=s_{h_{1},\ldots,h_n}\cdot\phi \) where, if \(s_h\) is the complete symmetric polynomial of degree \(h\), then \(s_{h_1,h_2,\ldots, h_n}\) is the usual Schur-polynomial \(\det(s_{h_i+j-1})\), which can be interpreted as the classical Giambelli's formula of Schubert calculus on Grassmann schemes. Hence Laksov and Thorup's determinantal formula can be seen as the ultimate and most natural generalization of it.
As one may expect from the nature itself of the results, the topic of this paper is related with many different subjects in mathematics, such as combinatorics, representation theory, geometry\dots To emphasize such a wide interplay, the authors care to prove the main theorem using different techniques within different frameworks. The most combinatorial in character is certainly that proposed in Section~2, based on a Pieri type formula enjoyed by the action of complete symmetric polynomials on the natural basis of \(\bigwedge^n_AA[X]\). That of Section~3, instead, relies on the isomorphism between \(\bigwedge^n_AA[X]\) and the ring of alternating polynomials. Section 4 proposes another proof based on symmetrization: let \(\xi\) be the residue class of \(T\) modulo \(P=\prod(T-X_i)\) in the ring \(S[\xi]=S[T]/(P)\). Then \(\bigwedge^nS[\xi]\) is naturally an \(S\)-module and remarkably such a module structure coincides with the symmetric structure defined in Section~1 of the paper, described in the first part of this review. Within the same framework, Section~5 proposes a very short proof of the main theorem which has the nice feature of implying Jacobi-Trudy formula. Finally, last two sections are devoted to look at the main theorem using the divided difference operators as well as the theory of universal splitting algebras, related with the work of Grothendieck on the homology of flag schemes. determinantal formula; Schubert calculus; exterior algebras; Giambelli's formula; Grassmann schemes; symmetric structures; symmetric functions; symmetrizing operators; divided difference operators; intersection theory; universal splitting algebras Laksov, D. and Thorup, A., A determinantal formula for the exterior powers of the polynomial ring, Indiana Univ. Math. J. 56 (2007), 825--845. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations A determinantal formula for the exterior powers of the polynomial ring | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Brief comments on selected topics in computational algebraic geometry are given. One of the topics is an experimental investigation of the possible Betti numbers of smooth canonical curves of low genus. Gröbner bases; syzygies; resolution of singularities; monodromy; Brieskorn lattice; Tate resolution; cohomology of coherent sheaves; Beilinson monads; invariant rings; binary forms; Green's conjecture; construction of canonical curves Schreyer, F.O.: Some topics in computational algebraic geometry. In: Conference Proceedings of 'Advances in Algebra and Geometry, Hyderabad 2001, pp. 263--278 (2003) Computational aspects of algebraic curves, Singularities in algebraic geometry, Computational aspects in algebraic geometry, Syzygies, resolutions, complexes and commutative rings, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Some topics in computational algebraic geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathfrak g\) be a complex simple Lie algebra, let \(\mathcal O\) be the orbit of its highest weight vectors which is a smooth quasi-affine variety. Denote by \(\mathcal{R(O)}\) and \(\mathcal{D(O)}\) the algebras of regular functions and differential operators on \(\mathcal O\), respectively. The authors construct a subalgebra \({\mathcal A}\subset\mathcal{D(O)}\) which is \(\mathfrak g\)-stable, graded and maximal commutative. The algebra \(\mathcal A\) has non-positive grading and its finite dimensional component \({\mathcal A}_{-1}\) consisting of differential operators of the fourth order generates \(\mathcal A\) as an algebra and transforms as the adjoint representation. A \(\mathfrak g\)-equivariant graded algebra isomorphism between \(\mathcal{R(O)}\) and \(\mathcal A\) is described. The authors underline that elements of \(\mathcal A\) are called ``exotic'' operators since they lie outside the realm of familiar differential operators. In conclusion they compare their constructions with one of \textit{R. Brylinski, B. Konstant} [in: Lie theory and geometry. Prog. Math. 123, 65-96 (1994; Zbl 0878.58033)] and some others. The authors will explain in a subsequent paper how one can use the operators in \({\mathcal A}_{-1}\) to quantize \(\mathcal O\) and to construct an algebraic star product on \(\mathcal{R(O)}\). simple Lie algebras; nilpotent orbits; adjoint orbits; pseudo-differential operators; Weyl quantization; Killing forms; Cartan involutions; Joseph ideals; Howe pairs; star products A. Astashkevich and R. Brylinski, Exotic differential operators on complex minimal nilpotent orbits, in Advances in geometry, 19-51, Progr. Math., 172, Birkhäuser, Boston, Boston, MA, 1999. Rings of differential operators (associative algebraic aspects), Geometric quantization, Group actions on varieties or schemes (quotients), Geometry and quantization, symplectic methods, Simple, semisimple, reductive (super)algebras, Semisimple Lie groups and their representations Exotic differential operators on complex minimal nilpotent orbits | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Translation from Serdica 7, 195-206 (1981; Zbl 0494.46054). Banach algebras of differentiable functions; homogeneous algebras of functions; classification up to a global isomorphism Banach algebras of differentiable or analytic functions, \(H^p\)-spaces, Banach algebras of continuous functions, function algebras, Families, moduli of curves (analytic) Isomorphisms of homogeneous function algebras on the torus | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author gives a definition of regular topological triviality for a germ of functions and proves this property for a wide class of germs. For example, all unimodal and bimodal families of singularities in V. I. Arnold's classification are regular topologically trivial. topological classification; germ of functions; singularities Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) On topological equivalence of germs of 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 This survey of 51 pages and 161 references contains some fundamental discussions and recent characterisations about finite solvable groups. It is impossible to list here all the details, which go from some old classical conjectures to new and intriguing perspectives of study. The main motivation of this survey is due to the results obtained by Professor Grunewald and his coauthors in the last decades. There are so many deep ideas, open questions, and relations with various branches of pure mathematics that I think that the best way to review this contribution is to refer directly to the introduction from page 1013 to page 1014. From these two pages, in my humble opinion, it is clear that the quantity and the quality of the mathematics, which is contained in the survey, will offer a valid source of inspiration for the research on the topic of the next years. finite solvable groups; solvable radical; finite simple groups; algebraic varieties; finite fields Grunewald F., Kunyavskiĭ B., Plotkin E., Characterization of solvable groups and solvable radical, Internat. J. Algebra Comput., 2013, 23(5), 1011--1062 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks, Finite nilpotent groups, \(p\)-groups, Special subgroups (Frattini, Fitting, etc.), Finite simple groups and their classification, Simple groups: alternating groups and groups of Lie type, Arithmetic and combinatorial problems involving abstract finite groups, Solvable, nilpotent (super)algebras, Finite ground fields in algebraic geometry Characterization of solvable groups and solvable radical. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 short survey of the most significant results concerning the number of rational points of algebraic curves over finite fields.
Let \(p\) be a prime number, \(X\) a nonsingular, projective, geometrically irreducible curve of genus \(g\) over a finite field \({\mathbb F}_{q}\) with \(q=p^{ \nu}\) elements and \({\#}X({ \mathbb F}_{q})\) the number of \({ \mathbb F}_{q}\)-\textit{rational points} on \(X\). Firstly the author demonstrates several classical results of P. Fermat, L. Euler, C. F. Gauss, J. Lagrange, A. M. Legendre, C. Jacobi et al. related to the number of rational points on an projective curve of a special form defined over \({\mathbb F}_{p}\). The author next discusses a conjecture of E. Artin (1924) on the complex zeroes of the \textit{zeta function} \(Z(X,q,t)\) of \(X\) and then presents the well-known \textit{Hasse-Weil} bound
\[
|{ \#}X({ \mathbb F}_{q})-(q+1)| \leq \lfloor 2g \sqrt{q} \rfloor
\]
for \({\#}X({\mathbb F}_{q})\). Thereafter the author describes a series of distinct proofs of the above bound (H. Hasse, A. Weil, S. A. Stepanov, E. Bombieri et al.) and then demonstrates several its refinements (J. P. Serre, K. Lauter et al.).
Let \(N_{q}(g)\) denote the maximal number of \({\mathbb F}_{q}\)-rational points that a curve \(X\) of genus \(g\) can have. The author discusses several examples of \textit{optimal curves} \(X\) of genus \(g\) (for which \({\#X}({\mathbb F}_{q})=N_{q}(g)\)) and then describes a particular family of optimal curves, the so-called \textit{maximal curves}, whose number of rational points attains the upper Hasse-Weil bound. A distinguished example here is the \textit{Hermitian curve} which is intrinsically determined by its genus and number of rational points. Also, the author shows that there are two important families of optimal curves, namely the \textit{Suzuki curves} and the \textit{Ree curves} with the property that each curve in these families is intrinsically determined by the data: (1) the genus, (2) the number of rational points and (3) the automorphism group.
For applications to coding theory, the key matter is to find a family of algebraic curves \((X_{g})\) (indexed be its genus and defined over a fixed field \({ \mathbb F}_{q}\)) such that
\[
A(q)= \limsup_{g} \frac{N_{q}(g)}{g}
\]
be a large as possible. It was shown by Y. Ihara (with using \textit{supersingular points} on a family of modular curves \((X_{g})\)) that when \(q\) is a square then \(A(q)= \sqrt{q}-1\). Later the same result was obtained by A. Garcia and H. Stichtenoth via modular curves of a different shape defined by ``explicit equations''. The author briefly describes the Goppa construction of so-called \textit{geometric Goppa codes} and then demonstrates the Tsfasman-Vläduţ-Zink result on existence of a family \((C_{i})\) of asymptotically long linear codes (coming from a family of modular curves of growing genus with many \({\mathbb F}_{q}\)-rational points) whose parameters \((R, \delta)\) satisfy
\[
R+ \delta=1-1/( \sqrt{q}-1).
\]
It should be pointed that the last result improves the well-known in coding theory \textit{Gilbert-Varshamov bound}.
The survey is completed with a discussion of Stör-Voloch theory concerning \textit{Weierstrass points} of an algebraic curve and \textit{Frobenius orders}. finite fields; algebraic curves; Riemann-Roch theorem; number of rational points of an algebraic curves over a finite field; Riemann hypothesis; Hasse-Weil bound; asymptotic problems; zeta-functions and linear systems; a characterization of the Suzuki curve; maximal curves; Hermitian curve; Weierstrass points Torres F.: Algebraic curves with many points over finite fields. In: Martínez-Moro, E., Munuera, C., Ruano, D. (eds) Advances in Algebraic Geometry Codes, pp. 221--256. World Scientific Publishing Company, Singapore (2008) Local ground fields in algebraic geometry, Complex multiplication and moduli of abelian varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Applications to coding theory and cryptography of arithmetic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory, Research exposition (monographs, survey articles) pertaining to number theory Algebraic curves with many points over finite fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The study of the topology of the moduli spaces \(\mathcal M(G)\) of \(G\)-Higgs bundles over a compact Riemann surface \(X\) (hence also of the corresponding representation variety \(\mathcal R(G)\) obtained through non-abelian Hodge correspondence) has mostly been done for the cases where \(G\) is a complex Lie group, especially \(\mathrm{SL}(n,\mathbb C)\).
For real groups, the topology of \(\mathcal M(G)\) is basically unknown, with the exception of the most basic topological invariant: the number of connected components. A lot of research on \(\pi_0(\mathcal M(G))\) has been carried out in the last two to three decades, and the cases of most classical groups have been handled.
This study has been mostly done by a case-by-case approach, with two exceptions being the following. On one hand, the case where the group \(G\) is a split real form, which has been intrinsically studied by \textit{N. Hitchin} in [Topology 31, No. 3, 449--473 (1992; Zbl 0769.32008)], and where it was proved that \(\mathcal M(G)\) has special connected components -- now known as Hitchin components -- with special properties, and which most of the times are not labelled by basic topological invariants of the bundles themselves. On the other hand, the case where \(G\) is any complex Lie group has been dealt in [\textit{O. García-Prada} and \textit{A. Oliveira}, Asian J. Math. 21, No. 5, 791--810 (2017; Zbl 1387.14048)], and proved that the connected components of \(\mathcal M(G)\) are exactly labelled by topological types.
The paper under review is gives an intrinsic study of another important case: \(G\) is a connected, non-compact, simple Lie group with finite centre of Hermitian type (i.e. such that its maximal compact subgroup has a centre isomorphic, up to torsion, to \(\mathrm{U}(1)\)). From the cases which have been studied so far, it is expectable that \(G\)-Higgs bundles for such \(G\) (thus also \(\mathcal M(G)\)) have special features, such as the existence of a topological invariant -- known as the Toledo invariant -- bounded by a so-called Milnor-Wood inequality. These features have been identified for several such \(G\), but its intrinsic (i.e. independent of the particular choice of \(G\)) definitions and proofs have never been obtained within the Higgs bundle context. This is one of the main contributions of the present paper. Another particular phenomenon with has been detected for Hermitian groups is a so-called Cayley correspondence, yielding an isomorphism between the subspace of \(\mathcal M(G)\) consisting of Higgs bundles with maximal Toledo invariant and the moduli space of \(K^2\)-twisted \(G^*\)-Higgs bundles, where \(G^*\) is a Lie group, called the Cayley partner of \(G\), and \(K\) is the canonical line bundle of \(X\). To be more precise, this correspondence occurs exactly when \(G\) is of tube type, and the intrinsic proof of such Cayley correspondence is also given by the authors in this paper. This correspondence is many times an important tool to prove that \(\mathcal M(G)\) has extra connected components, not a priori distinguished by basic topological types, similarly to the split case mentioned above. If \(G\) is still Hermitian but not of tube type, there the authors also provide an intrinsic proof of a rigidity result (also identified in the particular cases previously studied) for Higgs bundles with maximal Toledo invariant. Higgs bundles, Hermitian groups; Toledo invariant; Cayley correspondence Biquard, O.; García-Prada, O.; Rubio, R., Higgs bundles, Toledo invariant and the Cayley correspondence, J. Topol., 10, 795-826, (2017) Vector bundles on curves and their moduli, Applications of global analysis to structures on manifolds, Moduli problems for topological structures Higgs bundles, the Toledo invariant and the Cayley 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 \(Z\) be an affine variety and \(E\) a vector bundle over \(Z\). The algebras of differential operators on \(E\) and \(Z\) become graded algebras (denoted \(\text{gr} {\mathcal D}_E (Z)\) and \(\text{gr} {\mathcal D} (Z)\), respectively) using the filtrations \({\mathcal D}^n_E (Z)\) and \({\mathcal D}^n (Z)\) of sets of differential operators of degree at most a given degree \(n\). Properties of these algebras are well known if \(Z\) is smooth (for instance, \({\mathcal D} (Z)\) is a finitely generated domain). However, the corresponding properties in the singular case are open.
The paper comments a wealth of results by the author together with recent work by other mathematicians. In particular, the case of quotient singularities is considered. If \(X\) is a smooth affine \(G\)-variety with \(G\) reductive, then \(\text{gr} {\mathcal D} (X//G)\) is conjectured to be finitely generated and \({\mathcal D} (X//G)\) to be simple.
Special cases in which the conjectures hold are extracted. This is related to another question concerned with properties of algebras \(\text{gr} {\mathcal D}_{\mathcal E} (X//G)\) where \({\mathcal E}\) is the sheaf of \({\mathcal O}_{X//G}\)-modules corresponding to the \(G\)-invariant sections of a \(G\)-vector bundle \(E\) over \(X\). In this case reduction to the cases of the form \(X=V\) and \(E=\Theta_W\): \(V\times W\to V\), where \(V\) and \(W\) are \(G\)-modules is possible. Let \(P\in {\mathcal D}^n (V)^G \cong {\mathcal D}^n ({\mathcal O} (V))^G\). Then we have an element \((\pi_V)^*(P)\in {\mathcal D}_n (V//G)\). If \((\pi_V)^* ({\mathcal D}^n (V)^G)= {\mathcal D}_n (V//G)\), then \((\pi_V)^*\) is said graded surjective. Necessary and sufficient conditions for \((\pi_V)^*\) to be graded surjective are studied together with a number of related questions: For example, is it possible for \((\pi_V)^*\) to be surjective but not graded surjective? Partial answers and special cases are again presented.
Finally, an analogue of the last mentioned question with differential operators on adjoint representations is considered. This is connected to a construction presented by Harish-Chandra in the 50's. algebras of differential operators; quotient singularities ------, Invariant differential operators, Proceedings of the International Congress of Mathematicians (Zürich, 1994), vol. 1, pp. 333--341, Birkhäuser, Basel, 1995. Commutative rings of differential operators and their modules, Homogeneous spaces and generalizations, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Invariant differential operators | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(C\) be a geometrically irreducible smooth projective curve of genus \(g\) over a field \(k\) with absolute Galois group \(G_k\). There is a short exact sequence of algebraic fundamental groups:
\[
1\rightarrow \pi_{1}(C_{\bar{k}^{sep}},\bar{x})\rightarrow \pi_{1}(C,\bar{x})\rightarrow G_k\rightarrow
1.
\]
Each \(k\)-rational point \(x\) of \(C\) induces a section \(s_x\) of \(\pi_1(C,\bar{x})\rightarrow G_k\). Grothendieck's section conjecture states that if \(g\geq 2\) and \(k\) is a finitely generated infinite field, then there is a bijection between the set of \(k\)-rational points and the set of conjugacy classes of sections of \(\pi_1(C,\bar{x})\rightarrow G_k\) via the association \(s\rightarrow [s_x]\).
Hain proves in [\textit{R. Hain}, J. Am. Math. Soc. 24, No. 3, 709--769 (2011; Zbl 1225.14016)] that if \(g\geq5\), \(\operatorname{char}(k)=0\) and the image of the \(\ell\)-adic cyclotomic character \(\chi_\ell:\,G_k\rightarrow \mathbb{Z}_{\ell}^{\times}\) is infinite, the section conjecture holds for the restriction of the universal curve \(\mathcal{C}\rightarrow \mathcal{M}_{g/k}\) to its generic point \(\operatorname{Spec}k(\mathcal{M}_{g})\). In this paper, this result is extended to positive characteristic and \(g\geq4\).
The key ingredient that allows to use Hain's method in positive characteristic is the comparison of algebraic fundamental groups of a certain finite étale cover of \(\mathcal{M}_{g,n}\). universal curve; rational points; fundamental groups; exact sequence; moduli stack of curves; monodromy; finite étale cover Arithmetic ground fields (finite, local, global) and families or fibrations, Stacks and moduli problems, Families, moduli of curves (algebraic), Coverings of curves, fundamental group Rational points of universal curves in positive characteristics | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 contains the construction of finite-zone solutions of the Landau-Lifshits equation \(\vec s_ t=\vec s\times \vec s_{xx}+\vec s\times J\vec s,\quad | \vec s| =1,\) \(J=diag (J_ 1,J_ 2,J_ 3)\) in the case of ''one-axial anisotropy'': \(J_ 1=J_ 2>J_ 3\). A central feature of the approach is the synthesis of the theory of finite- zone integration with general ideas of the method of the matrix Riemann problem. As the result the solutions in terms of Riemann theta-functions are obtained.
Remark that recently one of the authors took forward the approach and settled the anisotropic case \(J_ 1\neq J_ 2\neq J_ 3\) integration [\textit{A. I. Bobenko}, Real algebraic-geometric solutions of the Landau- Lifshitz equation in terms of Prym theta-functions, Funkts. Anal. Prilozh. 19, No.1, 6-19 (1985)]. method of the inverse problem; completely integrable nonlinear equations; finite-zone solutions; Landau-Lifshits equation; matrix Riemann problem; Riemann theta-functions R. F. Bikbaev and A. I. Bobenko, ''On finite-gap integration of the Landau-Lifshitz equation. X-Y-Z case,'' Preprint LOMI E-8-83, Leningrad (1983). Partial differential equations of mathematical physics and other areas of application, Theta functions and abelian varieties, Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, Inverse problems for PDEs On finite-zone integration of the Landau-Lifshits equation | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 following is the quote of the author's abstract: ``For the algebraic sphere \(S^ 2\) defined as the zero set of the equation \(x^ 2_ 1+x^ 2_ 2+x^ 2_ 3=1\) in \({\mathbb{C}}^ 3\), we see that there are polynomial maps \(f_ n:\quad S^ 2\to S^ 2\) of each degree. For the algebraic sphere \({\mathbb{S}}^ 2\) defined as the suspension of \(S^ 1={\mathbb{C}}-\{0\}\) in the category \(AFF_{{\mathbb{C}}}\) of based affine schemes of countable type over \({\mathbb{C}}\), the set \(Hom_{AFF_{{\mathbb{C}}}}({\mathbb{S}}^ 2,{\mathbb{S}}^ 2)\) modulo homotopy is seen, using standard definitions adapted to our algebraic situation, to be \({\mathbb{Z}}\times {\mathbb{Z}}/\psi\) where \(\psi\) collapses (\({\mathbb{Z}}\times 0)\cup (0\times {\mathbb{Z}})\) to a point.''
Note that the first of these statements is argued by exhibiting concrete polynomials. The second one depends on the ``categories of based affine schemes of countable type over a field'' previously defined by the author and the category of ``ind affine schemes'' using which he defines the suspension and the loop spaces for which the author refers to his previous work in Cah. Topologie Geom. Differ. 23, 291-316 (1982; Zbl 0499.18004) and Quaest. Math. 6, 49-66 (1983; Zbl 0519.14013).
No connection with other work on homotopy theory of algebraic varieties are indicated. algebraic sphere; affine schemes of countable type; homotopy theory of algebraic varieties Homotopy theory and fundamental groups in algebraic geometry, Homotopy groups of spheres, Structure of families (Picard-Lefschetz, monodromy, etc.) Algebraic \(\pi _ 2(S^ 2)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(F\) be an infinite field of characteristic different from 2 and let \(V=M_n(F)\oplus M_n(F)\) be the direct sum of two matrix algebras with the action of the general linear group \(\text{GL}_n=\text{GL}_n(F)\) by simultaneous conjugation. Since the center of \(\text{GL}_n\) acts trivially, one has an action of \(\text{PGL}_n\). It is known that the invariant fields \(F(V)^{\text{PSp}_n}\) and \(F(V)^{\text{PO}_n}\) of the projective symplectic and orthogonal groups, are the centers of the generic division algebras of degree \(n\) with symplectic and orthogonal involution, respectively. One of the main problems for these fields is whether they are rational over \(F\).
By the results of \textit{D. J. Saltman} [J. Algebra 258, No. 2, 507-534 (2002; Zbl 1099.13013)], \(F(V)^{\text{PO}_n}\) is stably rational for \(n\) odd. For \(n\) even \(F(V)^{\text{PSp}_n}\) and \(F(V)^{\text{PO}_n}\) are stably isomorphic and \(F(V)^{\text{PSp}_n}\) is stably isomorphic to the invariants of a lattice over the Weyl group \(W\) of \(\text{PSp}_n\), where \(W\) is the wreath product of the group \(\mathbb{Z}/2\mathbb{Z}\) of order 2 by \(S_m\), the symmetric group of degree \(m=n/2\). Also, if \(n\) is not divisible by 8, then \(F(V)^{\text{PSp}_n}\) is stably rational.
The main result of the paper under review is to make the next step and to show that \(F(V)^{\text{PSp}_n}\) is stably rational when \(n=8s\) for some odd \(s\). For the proof, by the results of \textit{P. I. Katsylo} [Math. Notes 48, No. 2, 751-753 (1990); translation from Mat. Zametki 48, No. 2, 49-52 (1990; Zbl 0729.14034)] and \textit{A. Schofield} [J. Algebra 147, No. 2, 345-349 (1992; Zbl 0785.14030)], combined with the results of Saltman, it is sufficient to consider the case \(n=8\) only. In order to handle this highly nontrivial case, the author replaces the problem for stable rationality of the center of the corresponding generic division algebra with involution with a question of rationality, referred also as a lattice invariant problem, in the special case of the Noether setting for a suitable finite group \(G\). rationality; flasque classes; generic algebras; symplectic groups; orthogonal groups; stably rational field extensions; Noether settings; division rings of generic matrices; fields of invariants E. Beneish, Centers of generic algebras with involution, J. Algebra 294 (2005), no. 1, 41--50. Trace rings and invariant theory (associative rings and algebras), Transcendental field extensions, Finite-dimensional division rings, Integral representations of finite groups, Representations of finite symmetric groups, Geometric invariant theory Centers of generic algebras with involution. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0611.00005.]
The paper being reviewed contains a summary of the known (as of summer 1985) results on that part of the arithmetic of function fields that is concerned with the 'cyclotomic' construction of abelian extensions for \({\mathbb{F}}_ r(T)\), \(r=p^ n\). Moreover, it contains comparisons of these results with structure coming from classical cyclotomic theory.
The results in the paper center around the interplay between the author's L-functions and class numbers of the cyclotomic function fields. In particular, it is explained how the ''special polynomials'' arising from these zeta-functions at negative integers are related to unit-root pieces of characteristic-zero L-series associated to Teichmüller characters. This result points in a number of directions: One can use this idea to establish the Artin conjecture in the author's context for function fields; and one is lead to try to do something analogous for number fields. In the paper being reviewed, one approach for an analog of the special polynomials is suggested although nothing was established.
However, the author, together with \textit{W. Sinnott}, was able to find an analog in the theory of Artin L-series of number fields through another approach: In ''Special values of Artin L-series'', Math. Ann. 275, 529-537 (1986), there is associated to an Artin L-series a certain canonical power-series with algebraic (over \({\mathbb{Q}})\) coefficients and which plays some (at least) of the roles played by the above special polynomials. In particular, this power-series ''is'' the canonical p-adic measure on \({\mathbb{Z}}_ p\), for all primes p, associated to the given L-series once one knows the general Main conjecture of Iwasawa theory. Moreover, in the paper being reviewed, the relationship between the characteristic polynomials of Frobenius (acting on homology) and the special polynomials is made explicit. In a similar fashion, the power-series constructed for number fields can be thought of as the ''characteristic power series of Frobenius'', where ''the Frobenius'' is the generator ''1'' of \(\prod {\mathbb{Z}}_ p={\hat {\mathbb{Z}}}.\)
The second idea that one is lead to is a certain cyclicity criterion for the p-class groups of the cyclotomic function fields. Actually, there are discussed two separate criteria: The first one uses the above relationship with unit-root polynomials. The second one, which can only work for ''even'' components, uses the Bernoulli-Carlitz numbers and is based on an analog of the classical Kummer homomorphism due to S. Okada. It is shown that this second criterion implies the first. What is remarkable is that for components associated to \(i=r^ m-1\), one can actually establish this second criterion. Finally, in ''The \(\Gamma\)- function in the arithmetic of function fields'' (to appear in Duke Math. J., Feb. 1988), the author shows that these criteria lead to a ''Kummer- Vandiver'' phenomenon for function fields that one can check in many cases by computer.
Lastly, it should be mentioned that the experimental evidence relating \(\Gamma_ i\) and \(\beta\) (i) is now understood. This is contained in a paper by \textit{E.-U. Gekeler} to appear in J. Number Theory. However, surprisingly, it (and other results of interest) is also established - from a different point of view - in an old paper by \textit{H. L. Lee}, ''Power sums of polynomials in a Galois field'', Duke Math. J. 10, 277-292 (1943). division points of Drinfeld modules; arithmetic of function fields; class numbers; cyclotomic function fields; zeta-functions; Teichmüller characters; Artin conjecture; Artin L-series; p-adic measure; Main conjecture of Iwasawa theory; Frobenius; p-class groups; Bernoulli- Carlitz numbers Goss, D.: Analogies between global fields. Canad. math. Soc. conf. Proc. 7, 83-114 (1987) Arithmetic theory of algebraic function fields, Finite ground fields in algebraic geometry, Fibonacci and Lucas numbers and polynomials and generalizations, Algebraic functions and function fields in algebraic geometry, Iwasawa theory, Cyclotomic extensions, Zeta functions and \(L\)-functions of number fields Analogies between global fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The present review uses heavily excerpts of the excellent foreword to the book by Professor Kazuhiro Konno. As explained in the foreword:
``In the academic year 1967, Kunihiko Kodaira gave a course of lectures at the University of Tokyo on the theory of complex algebraic surfaces. The lecture notes were published in 1968 as Volume 20 in the series of Seminary Notes by the University of Tokyo. This was the copy of the handwritten manuscript in Japanese by Shigeho Yamashima, based on his beautiful notes reflecting faithfully the atmosphere of Kodaira's lectures. The present book is an English translation of that volume with slight modifications, correcting typos, etc. of the Japanese version.
The readers are expected to have only the elementary prerequisites on complex manifolds as background.
The book consists of two parts: Chaps. 1 and 2.
After stating the goal of the lecture in the Introduction, in Chap. 1, basic facts on algebraic surfaces are reviewed, touching upon divisors, linear systems, intersection theory, and the Riemann-Roch theorem. It provides an elegant introduction to the theory of algebraic surfaces covering some classical materials whose modern proofs first appeared in Kodaira's papers. Among others, one can find a concise analytic proof of Gorenstein's theorem for curves on a non-singular surface, which is a detailed explanation of the one given in Appendix I to [\textit{K. Kodaira}, Ann. Math. (2) 71, 111--152 (1960; Zbl 0098.13004)]. Another highlight is the elementary proof of Noether's formula for the arithmetic genus of an algebraic surface. Nowadays, the formula is known and treated as a special case of Hirzebruch's Riemann-Roch theorem. Kodaira's approach is based on the standard fact that, via generic projections, every algebraic surface can be obtained as the normalization of a surface with only ordinary singularities in the projective 3-space. However, unlike the other modern proofs, the argument does not rely on general facts, such as Porteus' formula, which requires a separate treatment. It is self-contained and follows a classical line, using Lefschetz pencils, much more in the style of Noether's original proof. The second part, Chapter 2, discusses the behaviour of the pluricanonical maps of algebraic surfaces of general type, as an application of the general theory provided in Chap. 1. It gives a detailed account of the paper Pluricanonical Systems on Algebraic Surfaces of General Type, [\textit{K. Kodaira}, J. Math. Soc. Japan 20, 170--192 (1968; Zbl 0157.27704)]''.
In the introduction it is explained the the main goal of the lectures is to prove that for minimal surfaces of general type the pluricanonical map \(\phi_{mK}\) is a birational holomorphic map for \(m\geq 6\) and this is done in chapter 2.
Of course subsequent work of many authors among which Kodaira himself and the epochal paper of \textit{E. Bombieri} [Publ. Math., Inst. Hautes Étud. Sci. 42, 171--219 (1972; Zbl 0259.14005)] and the use of new methods like \textit{I. Reider}'s theorem [Ann. Math. (2) 127, No. 2, 309--316 (1988; Zbl 0663.14010)] supplanted this result.
As such one could think that the present book is only interesting from a historical viewpoint and in a sense this is partly true about Chapter 2, although the level of detail of its analytical approach still renders it interesting.
However the clarity and detail of the definitions and proofs of preparatory facts given in Chapter 1 (called ``Fundamentals of algebraic surfaces'') and the analytic proof of Mumford's vanishing theorem given in Chapter 2 make this book very interesting for anyone wanting to learn working hands-on with complex surfaces. The definitions and proofs are extremely detailed and beautifully accompanied by examples and illustrations.
As said above the readers are expected to have only elementary prerequisites on complex manifolds and as such this book (specially Chapter 1) can be very useful for a graduate student or a non-expert starting to work on algebraic geometry. complex algebraic surfaces; surfaces of general type; pluricanonical maps; intersection theory; singularities; complex manifolds Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Families, moduli, classification: algebraic theory, Special surfaces, Rational and ruled surfaces, Elliptic surfaces, elliptic or Calabi-Yau fibrations, \(K3\) surfaces and Enriques surfaces, Surfaces of general type, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Compact complex surfaces, Transcendental methods of algebraic geometry (complex-analytic aspects), Research exposition (monographs, survey articles) pertaining to algebraic geometry, History of algebraic geometry, Collected or selected works; reprintings or translations of classics Theory of algebraic surfaces. Translated from the Japanese by Kazuhiro Konno. Notes taken by Shigeho Yamashima | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth projective complex variety, \(B\) be a line bundle on \(X\), and \(L_d:=dA+P\), where \(A\) is an ample line bundle and \(P\) is an arbitrary line bundle on \(X\). Assume that \(d>0\) is a sufficiently large integer; so in particular, \(L_d\) is very ample. The Koszul cohomology \(K_{p,q}(X, B; L_d)\) is the space of the \(p\)-th syzygies of \(B\) with respect to \(L_d\) of weight \(q\). It is a natural problem to study vanishing and nonvanishing of \(K_{p,q}(X, B; L_d)\). The asymptotic behaviors are largely understood thanks to results of \textit{L. Ein} and \textit{R. Lazarsfeld} [Invent. Math. 190, No. 3, 603--646 (2012; Zbl 1262.13018)] and Park [Comm. Amer. Math. Soc. 2, 133--148 (2022)]. When \(q=1\), \textit{D. H. Yang} [Res. Math. Sci. 1, Paper No. 10, 6 p. (2014; Zbl 1349.14088)] proved that there is an integer \(c \geq 0\) (not depending on \(d\)) such that \(K_{p,1}(X, B; L_d) = 0\) for \(0 \leq p \leq c\) but \(K_{c+1, 1}(X, B; L_d) \neq 0\). There should be geometric explanation for \(c\) in terms of \(X\) and \(B\). Along this line, \textit{L. Ein} et al. [Algebra Number Theory 10, No. 9, 1965--1981 (2016; Zbl 1351.14005)] showed that if \(B\) is \(p\)-jet very ample, then \(K_{p,1}(X, B; L_d)=0\). The first main result of the paper under review concerns the converse: \emph{If \(B\) is not \(p\)-very ample, then \(K_{p,1}(X, B; L_d) \neq 0\).} Notice that if \(B\) is \(p\)-jet very ample, then \(B\) is \(p\)-very ample. But the converse may not be true when \(p \geq 2\). As an application of the first main theorem, the author shows that the vanishing of certain Koszul cohomology gives a lower bound of the covering gonality and the degree of irrationality of \(X\). The second main result of the paper is the following: \emph{If \(\dim X = 2\) and \(0 \leq p \leq 3\), then \(K_{p,1}(X, B; L_d) = 0\) if and only if \(B\) is \(p\)-very ample.} For this result, the author utilizes the Bridgeland-King-Reid-Haiman correspondence for tautological bundles on the Hilbert scheme of points. asymptotic syzygies; higher order embedding; Hilbert scheme; derived McKay correspondence Divisors, linear systems, invertible sheaves, Syzygies, resolutions, complexes and commutative rings Asymptotic syzygies and higher order embeddings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper deals with the relation between the local fundamental group of a normal surface singularity and information about the type of singularities. Let \((A,{\mathfrak m})\) be a complete normal local domain with residue field \(k\) algebraically closed and of characteristic zero. Mumford and Fenner have shown that \(S=\text{spec} (A)\) is smooth if and only if the algebraic fundamental group \(\pi_1 (S-{\mathfrak m})\) is trivial. In the characteristic \(p>0\) case, Artin has asked if the following generalization holds: \(S\) has a finite local fundamental group if and only if \(S\) has a smooth cover.
In this paper, the authors prove the above statement for normal Brieskorn singularities, of the form \(x^a+ y^b+ z^c\), in characteristic \(p>3\). They show, for this case, that \(\pi_1 (S- {\mathfrak m})=0\) if and only if \(S\) has a purely inseparable cover. This implies Artin's theorem for these examples, since one can apply the result to the universal cover of a singularity with finite local fundamental group. To prove their result, the authors give a presentation of the local fundamental group (for any normal surface singularity with finite local fundamental group) in terms of the intersection diagram of a resolution of singularities with normal crossings. They then make explicit calculations for the case of Brieskorn singularities. characteristic \(p\); local fundamental group; normal surface singularity; type of singularities; normal Brieskorn singularities Steven Dale Cutkosky and Hema Srinivasan, Local fundamental groups of surface singularities in characteristic \?, Comment. Math. Helv. 68 (1993), no. 2, 319 -- 332. Singularities of surfaces or higher-dimensional varieties, Homotopy theory and fundamental groups in algebraic geometry, Singularities in algebraic geometry Local fundamental groups of surface singularities in characteristic \(p\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth projective curve, \(G\) a simple and simply connected complex Lie group, \({\mathcal M}_X(G)\) the moduli stack parametrizing principal \(G\)-bundles over \(X\) and \({\mathcal L}_G\) the ample generator of the Picard group of \({\mathcal M}_X(G)\). It is known (see, for example, \textit{G. Faltings} [J. Algebr. Geom. 18, No. 2, 309--369 (2009; Zbl 1161.14025)] or \textit{C. Sorger} [Ann. Sci. Éc. Norm. Supér. (4) 32, No. 1, 127--133 (1999; Zbl 0969.14016)]) that \(\text{dim}\, \text{H}^0({\mathcal M}_X(E_8),{\mathcal L}_{E_8}) = 1\), hence \({\mathcal M}_X(E_8)\) carries a natural divisor \(\Delta\). A geometric interpretation of this divisor is, however, not available.
In the paper under review, the authors study the pullback of \(\Delta\) under the morphisms \({\tilde \phi} : {\mathcal M}_X(P) \rightarrow {\mathcal M}_X(E_8)\) induced by the group homomorphisms \(\phi : P \rightarrow E_8\), where \(P\) is connected, simply connected and semisimple, and \(d\phi : \text{Lie}(P) \rightarrow \text{Lie}(E_8)\) embeds \(\text{Lie}(P)\) as a subalgebra of maximal rank 8 of \(\mathfrak{e}_8\). Actually, \(P\) must be one of the groups \(\text{Spin}(16)\), \(\text{SL}(9)\), \(\text{SL}(5)\times \text{SL}(5)\), \(\text{SL}(3)\times E_6\), \(\text{SL}(2)\times E_7\) and the kernel \(N\) of \(\phi\) is \({\mathbb Z}/2{\mathbb Z}\), \({\mathbb Z}/3{\mathbb Z}\), \({\mathbb Z}/5{\mathbb Z}\), \({\mathbb Z}/3{\mathbb Z}\) and \({\mathbb Z}/2{\mathbb Z}\), respectively. There exists a \textit{canonical} \({\mathcal M}_X(N)\)-linearization of \({\mathcal L}_P\) induced from the isomorphism \({\tilde \phi}^{\ast} {\mathcal L}_{E_8} \simeq {\mathcal L}_P\).
The main result of the paper asserts that the induced map
\[
{\phi}_P : \text{H}^0({\mathcal M}_X(E_8),{\mathcal L}_{E_8}) \rightarrow \text{H}^0({\mathcal M}_X(P),{\mathcal L}_P)
\]
is non-zero and its image coincides with the \({\mathcal M}_X(N)\)-invariant subspace of the space \(\text{H}^0({\mathcal M}_X(P),{\mathcal L}_P)\). The proof uses a result of \textit{P. Belkale} [J. Differ. Geom. 82, No. 2, 445--465 (2009; Zbl 1193.14013)] which implies that, in the case under consideration, \({\phi}_P\) has constant rank when the curve \(X\) varies in a family of smooth curves, and the identification of \(\text{H}^0({\mathcal M}_X(G), {\mathcal L}_G)\) with the space of \textit{conformal blocks} associated to \(X\) (with one marked point labelled with the zero weight) and to \(\text{Lie}(G)\).
The authors also show that if \((A,B)\) is one of the pairs \((\text{SL}(5), \text{SL}(5))\), \((\text{SL}(3),E_6)\), \((\text{SL}(2),E_7)\) and if one considers \textit{any} \({\mathcal M}_X(N)\)-linearization of \({\mathcal L}_{A\times B}\) then a non-zero element of the 1-dimensional \({\mathcal M}_X(N)\)-invariant subspace of:
\[
\text{H}^0({\mathcal M}_X(A\times B),{\mathcal L}_{A\times B}) = \text{H}^0({\mathcal M}_X(A),{\mathcal L}_A) \otimes \text{H}^0({\mathcal M}_X(B),{\mathcal L}_B)
\]
induces an isomorphism:
\[
\text{H}^0({\mathcal M}_X(A),{\mathcal L}_A)^{\ast}\overset{\sim}\longrightarrow \text{H}^0({\mathcal M}_X(B),{\mathcal L}_B)\, .
\]
A similar isomorphism is obtained for the pair \((\text{Spin}(8),\text{Spin}(8))\). The proof of this result uses the representation theory of Heisenberg groups. principal bundle; projective curve; moduli stack; generalized theta function; Strange Duality; conformal embedding of Lie algebras; space of conformal blocks Boysal, A., Pauly, C.: Strange duality for Verlinde spaces of exceptional groups at level one. Int. Math. Res. Not. (2009). 10.1093/imrn/rnp151 Stacks and moduli problems, Vector bundles on curves and their moduli, Holomorphic bundles and generalizations, Exceptional groups, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Strange duality for Verlinde spaces of exceptional groups at level one | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X_ g\) be the space of binary forms of degree \(g\), and let \(X_{p,g}\) be the subspace of forms having a root of multiplicity \(p\). \(X_ g\) can be identified with \(\text{Spec} R_ g\), where \(R_ g=\text{Sym} (S_ gV)\) for a fixed two-dimensional vector space \(V\) over \(\mathbb{C}\). Let \(J_{p,g}\) be the ideal of polynomials in \(R_ g\) vanishing on \(X_{p,g}\). For \(p=2\) it is well known that \(J_{p,g}\) is generated by one element of degree \(2g-2\), namely the discriminant. In this paper a formula for the dimensions of the graded pieces of \(J_{p,g}\) in the general case is derived. If \(g-1=(p-1) h+1\), \(0 \leq 1<p-1\), it is conjectured that \(J_{p,g}\) is generated by its elements of degrees \(2h\), \(2h+1\), and \(2h+2\). Hilbert functions of multiplicity ideals; binary forms Weyman, J.: On Hilbert functions of multiplicity ideals. J. algebra 161, 358-369 (1993) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Relevant commutative algebra, General ternary and quaternary quadratic forms; forms of more than two variables, General binary quadratic forms On the Hilbert functions of multiplicity ideals | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a prime \(p\), the theory of \(p\)-adic representations of \(p\)-adic reductive groups, and in particular, the \(p\)-adic Langlands correspondence, is much more involved than the theory of \(\ell\)-adic and complex representations of \(p\)-adic groups, where \(\ell\neq p\) is another prime. The ultimate goal of the \(p\)-adic Langlands program [\textit{C. Breuil}, in: Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19--27, 2010. Vol. II: Invited lectures. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency. 203--230 (2011; Zbl 1368.11123)] would be to find the correspondence between the \(p\)-adic Galois representations and the norm-preserving \(p\)-adic Banach space representations of the \(p\)-adic general linear group, which is compatible with the modulo \(p\) Langlands correspondence under restriction modulo \(p\). It should also be compatible with the cohomology in the sense that the correspondence may be realized in the cohomology of certain Shimura varieties. One of the motivations for the \(p\)-adic Langlands program is to study the \(p\)-adic Hodge theory in terms of continuous or locally analytic \(p\)-adic representations of the \(p\)-adic general linear group. However, little is known beyond the case of the general linear group \(\mathrm{GL}_2(\mathbb{Q}_p)\) of rank one over the field \(\mathbb{Q}_p\) of \(p\)-adic numbers.
Let \(L\) be a finite extension of \(\mathbb{Q}_p\) and \(G_L\) the absolute Galois group of \(L\). Let \(\rho_p\) be a de Rham representation of \(G_L\) in an \(n\)-dimensional vector space over \(E\), where \(E\) is a finite extension of \(\mathbb{Q}_p\). Suppose that the Hodge-Tate weights of \(\rho_p\) are distinct. One may attach to such \(\rho_p\), a Deligne--Fontaine module \(D_{\mathrm{dR}}(\rho_p)\). It is a vector space over \(E\) of dimension \(n\) with a natural action of the Weil--Deligne group of \(L\) and certain decreasing filtration. On the other hand, according to the expected hypothetical \(p\)-adic Langlands program, one should be able to attach to \(\rho_p\) at least one admissible locally analytic representation of \(\mathrm{GL}_n(L)\) over \(E\), denoted \(\Pi^{\mathrm{an}}(\rho_p)^?\) [\textit{A. Caraiani} et al., Camb. J. Math. 4, No. 2, 197--287 (2016; Zbl 1403.11073)].
In what follows, let \(L=\mathbb{Q}_p\), for simplicity. It is expected, and in some cases proved in [loc. cit.], that \(\Pi^{\mathrm{an}}(\rho_p)^?\) contains a locally algebraic representation \(\mathrm{Alg}\otimes_E \Pi^\infty\), where \(\mathrm{Alg}\) is the irreducible algebraic representation of \(\mathrm{GL}_n\) of highest weight given in terms of Hodge-Tate weights of \(\rho_p\), and \(\Pi^\infty\) is the smooth representation of \(\mathrm{GL}_n(\mathbb{Q}_p)\) over \(E\) corresponding under the local Langlands correspondence [\textit{C. Breuil} and \textit{P. Schneider}, J. Reine Angew. Math. 610, 149--180 (2007; Zbl 1180.11036); \textit{M. Harris} and \textit{R. Taylor}, The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich. Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027); \textit{G. Henniart}, Invent. Math. 139, No. 2, 439--455 (2000; Zbl 1048.11092); \textit{P. Scholze}, Invent. Math. 192, No. 3, 663--715 (2013; Zbl 1305.22025)] to the representation of the Weil-Deligne group of \(\mathbb{Q}_p\) in the Deligne-Fontaine module \(D_{\mathrm{dR}}(\rho_p)\).
As the main motivation for the present paper, the author states a conjecture regarding representations appearing in \(\Pi^{\mathrm{an}}(\rho_p)^?\). It is concerned with representations not considered in the earlier conjecture of the author [Math. Ann. 361, No. 3-4, 741-785 (2015; Zbl 1378.11060)], which considers the locally analytic socle representations in the case of \(\mathrm{GL}_n(L)\), and has been proved in the crystalline case subject to certain conditions by \textit{C. Breuil} et al. [Publ. Math., Inst. Hautes Étud. Sci. 130, 299--412 (2019; Zbl 1454.14120)].
More precisely, let \(\underline{k}=(k_1,\dots ,k_n)\) be an \(n\)-tuple of integers such that \(k_1>\dots >k_n\). Let \(\mathrm{WD}\) be a representation of the Weil-Deligne group of \(\mathbb{Q}_p\) such that its restriction to the Weil group of \(\mathbb{Q}_p\) is semi-simple and of multiplicity one. These data give rise to the locally algebraic representation \(\mathrm{Alg}\otimes_E \Pi^\infty\) as above. Moreover, assume that there exists at least one \(n\)-dimensional de Rham representation \(\rho_p\) over \(E\) of the absolute Galois group \(G_{\mathbb{Q}_p}\) of \(\mathbb{Q}_p\) such that the Hodge-Tate weights of \(\rho_p\) are \(\underline{k}\) and \(\mathrm{WD}\) is isomorphic to the representation of the Weil-Deligne group in the Deligne-Fontaine module \(D_{\mathrm{dR}}(\rho_p)\). Without this assumption the conjecture would be trivially false.
The conjecture claims, that for any \(\underline{k}\) and \(\mathrm{WD}\) as above, and for any \(j\in\{1,\dots ,n-1\}\), there exists an admissible finite-length locally analytic representation \(\Pi^j(\underline{k},\mathrm{WD})\) of \(\mathrm{GL}_n(\mathbb{Q}_p)\) over \(E\), and an isomorphism
\[
\mathcal{R}^j:\wedge_E^j\mathrm{WD}\to \mathrm{Ext}_{\mathrm{GL}_n(\mathbb{Q}_p)}^1 \left( \Pi^j(\underline{k},\mathrm{WD}),\mathrm{Alg}\otimes_E \Pi^\infty \right)
\]
of vector spaces over \(E\), unique up to the composition from the left by a permissible automorphism of \(\wedge_E^j\mathrm{WD}\), satisfying the following property. For any given \(n\)-dimensional de Rham representation \(\rho_p\) of the absolute Galois group \(G_{\mathbb{Q}_p}\) over \(E\), with Hodge--Tate weights \(\underline{k}\), and such that \(D_{\mathrm{dR}}(\rho_p)\) is isomorphic to \(\mathrm{WD}\), the one-dimensional bottom step of the filtration of \(\wedge_{E}^j\mathrm{WD}\), arising from the filtration of \(D_{\mathrm{dR}}(\rho_p)\cong\mathrm{WD}\), corresponds via \(\mathcal{R}^j\) to a non-split extension
\[
\mathrm{Alg}\otimes_E \Pi^\infty \hbox{ -------- } \Pi^j(\underline{k},\mathrm{WD})
\]
which occurs in \(\Pi^{\mathrm{an}}(\rho_p)^?\). The extension in the codomain of \(\mathcal{R}^j\) is the extension in the category of admissible locally analytic representations.
The paper under review provides evidence for the conjecture, in particular, finds the candidates for the representation \(\Pi^j(\underline{k},\mathrm{WD})\) and the isomorphism \(\mathcal{R}^j\). In the case of \(\mathrm{GL}_2(\mathbb{Q}_p)\), the conjecture is proved, except for irreducible \(\mathrm{WD}\). In the case of \(\mathrm{GL}_2(L)\), where \(L\) is a non-trivial finite extension of \(\mathbb{Q}_p\), the conjecture is studied following the works of \textit{Y. Ding} [Forum Math. Sigma 4, Paper No. e13, 49 p. (2016; Zbl 1376.11082); Ann. Inst. Fourier 67, No. 4, 1457--1519 (2017; Zbl 1433.11131); Isr. J. Math. 231, No. 1, 47--122 (2019; Zbl 1442.22018)]. In the crystalline case of \(\mathrm{GL}_n(L)\), it is proved that the conjecture is compatible with the socle conjectures in [\textit{C. Breuil}, Math. Ann. 361, No. 3--4, 741--785 (2015; Zbl 1378.11060)]. And finally, the paper considers the semi-stable non-crystalline case of the group \(\mathrm{GL}_3(\mathbb{Q}_p)\).
In the case of \(\mathrm{GL}_3(\mathbb{Q}_p)\), let \(\rho_p\) be a three-dimensional semi-stable representation of the Galois group \(G_{\mathbb{Q}_p}\), such that the Hodge-Tate weights \(\underline{k}\) are not critical and the attached representation \(\mathrm{WD}\) of the Weil--Deligne group corresponds to the Steinberg representation \(\Pi^\infty=\mathrm{St}\) of \(\mathrm{GL}_3(\mathbb{Q}_p)\) under the local Langlands correspondence. Let \(\mathrm{Alg}\otimes_E\mathrm{St}\) be attached to \(\rho_p\) as above. In this setting, the author constructs the representations \(\Pi^j(\underline{k},\mathrm{WD})\), with \(j=1,2\), predicted by the conjecture. Suppose that \(\rho_p\) comes from an automorphic representation \(\pi\) of the unitary group \(G(\mathbb{A}_\mathbb{Q})\), where \(\mathbb{A}_\mathbb{Q}\) is the ring of adèles of \(\mathbb{Q}\), which is the compact unitary group at the archimedean place and split at the place \(p\), and there is a level \(U^p\) inside \(G(\mathbb{A}_\mathbb{Q}^{\infty,p})\), where \(\mathbb{A}_\mathbb{Q}^{\infty,p}\) is the subring of the ring of adèles with trivial component at the archimedean place and the place \(p\), such that the space \(U^p\)-invariants of \(\pi\) is non-trivial. Under these assumptions, it is proved that only one of the constructed locally analytic representations appears in the associated Hecke-isotypic subspace of the completed cohomology at level \(U^p\). The author also conjectures that this representation of \(\mathrm{GL}_3(\mathbb{Q}_p)\) does not depend on the Hodge filtration of the Deligne-Fontaine module, and determines it completely. Langlands correspondence; \(p\)-adic Langlands program; \(p\)-adic Galois representations; locally analytic \(p\)-adic representations of \(p\)-adic general linear groups Galois representations, Representations of Lie and linear algebraic groups over local fields, Langlands-Weil conjectures, nonabelian class field theory, Geometric Langlands program (algebro-geometric aspects) Locally analytic \(\mathrm{Ext}^1\) and local-global compatibility | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\Gamma\) be a finite subgroup of \(SL(2, {\mathbb C})\), and \(X\) be the corresponding Kleinian singularity. Gonzalez-Springberg and Verdier showed that there is natural bijection between the irreducible components of the exceptional fibre in the minimal resolution of \(X\), and the non-trivial irreducible representations of \(\Gamma\). \textit{Y. Ito} and \textit{I. Nakamura} [see e.g. Proc. Japan Acad., Ser. A 72, No. 7, 135-138 (1996; Zbl 0881.14002)] found a beautiful new interpretation of this bijection, by using an interpretation of the minimal resolution of \(X\) as a subset of the Hilbert scheme of codimension \(|\Gamma|\) ideals in \({\mathbb C}[x,y]\).
In this paper the author gives a new proof of the result of Ito and Nakamura. This proof avoids a case by case analysis, contrary to the proof of Ito and Nakamura. Kleinian singularity; McKay correspondence; minimal resolution; Hilbert scheme Dlab, V.: Representations of valued graphs. In: Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 73. Presses de l'Université de Montréal, Montreal, Que (1980) Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects) On the exceptional fibres 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 We present a local computation of deformations of the tangent bundle for a resolved orbifold singularity \(\mathbb{C}^d/G\). These correspond to \((0,2)\)-deformations of \((2,2)\)-theories. A McKay-like correspondence is found predicting the dimension of the space of first-order deformations from simple calculations involving the group. This is confirmed in two dimensions using the Kronheimer-Nakajima quiver construction. In higher dimensions such a computation is subject to nontrivial worldsheet instanton corrections and some examples are given where this happens. However, we conjecture that the special crepant resolution given by the \(G\)-Hilbert scheme is never subject to such corrections, and show this is true in an infinite number of cases. Amusingly, for three-dimensional examples where \(G\) is abelian, the moduli space is associated to a quiver given by the toric fan of the blow-up. It is shown that an orbifold of the form \(\mathbb{C}^3 / \mathbb{Z}_7\) has a nontrivial superpotential and thus an obstructed moduli space. Aspinwall, P. S.: A mckay-like correspondence for (0,2)-deformations String and superstring theories; other extended objects (e.g., branes) in quantum field theory, McKay correspondence, Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Deformations of singularities, Topology and geometry of orbifolds, Representations of quivers and partially ordered sets, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Blow-up in context of PDEs A McKay-like correspondence for \((0,2)\)-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 \(G\) be a finite subgroup of \(\mathrm{SL}(3,\mathbb{C})\) acting naturally on \(\mathbb{C}^3\), and let \(Y=G\text{-Hilb}( \mathbb{C}^3)\) be the moduli space of \(G\)-clusters introduced by Nakamura. It is proven in [\textit{T. Bridgeland, A. King} and \textit{M. Reid}, J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] that \(Y\) is irreducible and the natural Hilbert-Chow morphism \(f:Y\to \mathbb{C}^3/G\) is a crepant resolution. Furthermore, the derived category of coherent sheaves on \(Y\) is equivalent to the derived category of \(G\)-equivariant coherent sheaves on \(\mathbb{C}^3\) via a Fourier-Mukai transformation.
The question of whether other crepant resolutions of \(\mathbb{C}^3/G\) can be realized as moduli spaces was studied in [\textit{A. Craw} and \textit{A. Ishii}, Duke Math. J. 124, No. 2, 259--307 (2004; Zbl 1082.14009)]. Craw and Ishii conjectured that any projective crepant resolution of \(\mathbb{C}^3/G\) can be identified with a certain moduli space of representations of the McKay quiver with relations, called the moduli space of \(G\)-constellations. Craw and Ishii proved their conjecture in case \(G\) is abelian.
Let \(N\leq G\) be a normal subgroup. \(G/N\) acts on \(N\text{-Hilb} (\mathbb{C}^3)\) the crepant resolution of \(\mathbb{C}^3/N\). It follows from the main result of Bridgeland-King-Reid that \(G/N\text{-Hilb}( N\text{-Hilb}(\mathbb{C}^3))\) is a crepant resolution \(\mathbb{C}^3/G\).
The main result of the paper under review proves that the crepant resolution \(G/N\text{-Hilb}( N\text{-Hilb}(\mathbb{C}^3))\) is isomorphic to a moduli space of \(G\)-constellations, establishing Craw-Ishii conjecture for such crepant resolutions. The crepant resolutions \(Y=G\text{-Hilb}( \mathbb{C}^3)\) and \(G/N\text{-Hilb}( N\text{-Hilb}(\mathbb{C}^3))\) are usually nonisomorphic. In the case \(G\) is abelian, the paper under review presents a complete list of the cases where two crepant resolutions are isomorphic. For non-Abelian subgroups, the paper under review shows that these two crepant resolutions are not isomorphic when \(G\) is a finite small subgroup of \(\mathrm{GL}(2,\mathbb{C})\subset \mathrm{SL}(3,\mathbb{C})\). \(G\)-clusters; \(G\)-constellations; crepant resolution; Hilbert scheme A. Ishii, Y. Ito and Á. Nolla de Celis, On \(G/N\)-Hilb of \(N\)-Hilb, Kyoto J. Math. 53 (2013), no. 1, 91-130. MR3049308 McKay correspondence, Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets On \(G/N\)-Hilb of \(N\)-Hilb | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Around 1980 J. McKay discovered a remarkable bijection between isomorphism classes of non trivial irreducible representations of a finite subgroup \(G\) of \(\text{SL} (2,{\mathbb C})\) and irreducible components of the exceptional divisor of the minimal resolution of the (2-dimensional) quotient singularity \(X_G = {\mathbb C}^2/G\). The original presentation was rather formal (it used Dynkin diagrams). Later considerable work was done to describe the correspondence more geometrically, as well as trying to generalize it to other groups and higher dimension. To achieve this goal, different techniques were used: reflexive sheaves, \(G\)-Hilbert schemes of points of the complex plane (where \(G\) is a suitable finite subgroup of \(\text{SL}(2,{\mathbb C})\)), \(K\)-theory, derived categories, etc. In the two dimensional case, to obtain a generalized ``McKay correspondence'' it seems necessary to consider small subgroups of \(\text{GL}(2,{\mathbb C})\) (i.e., those acting freely on \({\mathbb C}^2\) away from the origin) and certain representations of \(G\), called special.
In this paper the author, a leading expert in this area, gives a brief survey of the theory, primarily in the two-dimensional case. Certain items are discussed in some detail, e.g., special representations and their associated special reflexive sheaves, \(G\)-Hilbert schemes and their applications. The paper is mainly expository, but it contains a new, more combinatorial description of a generalized McKay correspondence in the case where \(G\) is a finite cyclic subgroup of \(\text{GL}(2,{\mathbb C})\), as well as some examples. The article includes some comments about the three-dimensional case. quotient singularity; minimal resolution; irreducible representation Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients) An introduction to the special 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 We study the algebraic symplectic geometry of multiplicative quiver varieties, which are moduli spaces of representations of certain quiver algebras, introduced by \textit{W. Crawley-Boevey} and \textit{P. Shaw} [Adv. Math. 201, No. 1, 180--208 (2006; Zbl 1095.15014)], called \textit{multiplicative preprojective algebras}. They are multiplicative analogues of Nakajima quiver varieties. They include character varieties of (open) Riemann surfaces fixing conjugacy class closures of the monodromies around punctures, when the quiver is ``crab-shaped.'' We prove that under suitable hypotheses on the dimension vector of the representations, or the conjugacy classes of monodromies in the character variety case, the normalizations of such moduli spaces are symplectic singularities and the existence of a symplectic resolution depends on a combinatorial condition on the quiver and the dimension vector. These results are analogous to those obtained by Bellamy and the first author in the ordinary quiver variety case and for character varieties of closed Riemann surfaces. At the end of the paper, we outline some conjectural generalizations to moduli spaces of objects in 2-Calabi-Yau categories. Momentum maps; symplectic reduction, Representations of quivers and partially ordered sets, Character varieties, Singularities in algebraic geometry, Symplectic structures of moduli spaces, Global theory and resolution of singularities (algebro-geometric aspects) Symplectic resolutions for multiplicative quiver varieties and character varieties for punctured surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let K be any field with the property that every singular K-variety admits a resolution of singularities. The authors present an easy, and very natural, proof of the Riemann-Roch theorem for any, possibly non- singular, algebraic variety X (locally of finite type and separated), defined on K.
The precise statement is the following: There is a homomorphism \(\tau_ X\) from the Grothendieck group \(K_ 0(X)\) to the rational Chow ring \(A_*(X)\otimes {\mathbb{Q}}\), which is covariant for proper morphisms, and coincide with \(ch\cap Todd(T_ X)\) if X is nonsingular. - The proof is by induction on the dimension of X, using resolutions of singularities, the Chow envelopes of Fulton and Gillet, and standard exact sequences in K-theory.
Given a proper morphism f: \(Y'\to Y\), of regular quasiprojective varieties defined on a field of characteristic 0, and closed immersions \(X\to Y\) and \(X'=f^{-1}(X)\to Y'\) such that f induces an isomorphism \(Y'-X'\to Y-X\), assumed to be open and dense subschemes of \(Y'\) and Y, respectively, the authors also prove the existence of an exact sequence \(0\to K_ i(X)\to K_ i(X')\oplus K_ i(Y)\to K_ i(Y')\to 0.\) This sequence reduces the problem of calculating Quillen's K-groups for general quasiprojective varieties to the same problem for resolutions with at most normal crossing divisors. This is then used to prove a Riemann-Roch theorem ``sans dénominateurs'' and to compute Chern classes in the case of normal crossing divisors. resolution of singularities; Riemann-Roch theorem; rational Chow ring; K- theory Riemann-Roch theorems, Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry Théorème de Riemann-Roch par désingularisation. (Theorem of Riemann-Roch for 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 \(X\) be a smooth connected complex surface.
Haiman defined the isospectral Hilbert scheme of \(n\) points on \(X\) to be the blow up \(B^n \to X^n\) of the product variety
\(X^n\) along the big diagonal \(\Delta_n\) (in [\textit{M. Haiman}, Math. Sci. Res. Inst. Publ. 38, 207--254 (1999; Zbl 0952.05074); J. Am. Math. Soc. 14, No. 4, 941--1006 (2001; Zbl 1009.14001)] and
proved that \(B^n\) is normal, Cohen-Macaulay and Gorenstein.
Here the author further investigates singularities of \(B^n\), that is, of the pair \((B^n,\emptyset)\), which has log-canonical singularities if and only if \((X^n, {\mathcal I}_{\Delta_n})\) does.
Using Haiman's description of generators of the ideal \({\mathcal I}_{\Delta_n}\) for the local model \(\mathbb C^2\), the
author proves the upper bound on the log-canonical threshold
\[
\text{lct} (X^n,{\mathcal I}_{\Delta_n}) \leq \frac{2n-2}{d_n}
\]
where \(d_n\) is the minimal degree of a generator of \({\mathcal I}_{\Delta_n}\).
This bound leads to the main theorem, which states that the singularities of \(B^n\) are canonical if \(n \leq 5\)
and log-canonical for \(n \leq 7\), but for \(n \geq 9\) they are not log-canonical.
The author conjectures that the log-canonical threshold bound is sharp, which would imply that \(B^n\) has
canonical singularities if and only if \(n \leq 7\) and log-canonical singularities if and only if \(n \leq 8\).
The author also provides two log-resolutions of \(B^3\), one crepant and the other \(S_3\)-equivariant. isospectral Hilbert scheme of points; canonical singularities; log-canonical thresholds; crepant resolutions Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Singularities of the isospectral Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a given finite small binary dihedral group \(G\subset\mathrm{GL}(2,\mathbb{C})\) we provide an explicit description of the minimal resolution \(Y\) of the singularity \(\mathbb{C}^{2}/G\). The minimal resolution \(Y\) is known to be either the moduli space of \(G\)-clusters \(G\)-Hilb\((\mathbb{C}^{2})\), or the equivalent \(\mathcal{M}_{\theta}(Q,R)\), the moduli space of \(\theta\)-stable quiver representations of the McKay quiver. We use both moduli approaches to give an explicit open cover of \(Y\), by assigning to every distinguished \(G\)-graph \(\Gamma\) an open set \(U_{\Gamma}\subset\mathcal{M}_{\theta}(Q,R)\), and calculating the explicit equation of \(U_{\Gamma}\) using the McKay quiver with relations \((Q,R)\). mckay correspondence; \(G\)-Hilbert scheme; quiver representations Á. Nolla de Celis, Dihedral \({G}\)-Hilb via representations of the McKay quiver , Proc. Japan Acad. Ser. A 88 (2012), 78-83. McKay correspondence, Algebraic moduli problems, moduli of vector bundles, Parametrization (Chow and Hilbert schemes), Representations of quivers and partially ordered sets, Global theory and resolution of singularities (algebro-geometric aspects) Dihedral \(G\)-Hilb via representations of the McKay quiver | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities ``If \(f : X^n \to Y^p\) is a morphism of smooth complex analytic varieties with \(n < p\), then the multiple points of order \(k\) of \(f\) in the target are those \(y \in Y^p\) with \(k\) preimages, each preimage counted with multiplicity. ... In this paper we give a new construction, using punctual Hilbert schemes, which we offer as an alternative to multi-jets in the study of multiple point singularities. As an illustration of its usefulness, we use it to find a resolution of the closure of the triple point set of any ``good'' map \(f\), and of the multiple point set of a ``good'' \(f\) of any order, provided the map \(f\) has kernel rank at most 2. ... This construction also provides a useful starting point for finding resolutions of multiple point sets for general \(f\), by reducing the problem of resolving the singularities of \(f\) to the problem of resolving the singularities of the corresponding Hilbert schemes''. punctual Hilbert schemes; multiple point singularities; triple point Gaffney, T.; Counting Double Point Singularities, preprint. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Punctual Hilbert schemes and resolutions of multiple point 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.
For Part I see [Zbl 1275.14001]. Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Singularities in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Proceedings, conferences, collections, etc. pertaining to associative rings and algebras, Proceedings, conferences, collections, etc. pertaining to nonassociative rings and algebras, Proceedings, conferences, collections, etc. pertaining to category theory, Representations of quivers and partially ordered sets, Quantum groups (quantized enveloping algebras) and related deformations, Derived categories, triangulated categories, Proceedings of conferences of miscellaneous specific interest Geometric methods in representation theory. II. Selected papers based on the presentations at the summer school, Grenoble, France, June 16 -- July 4, 2008 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 subgroup \(\Gamma\subset \mathrm{SL}(2,\mathbb{C})\) and \(n\geqslant 1\), we construct the (reduced scheme underlying the) Hilbert scheme of \(n\) points on the Kleinian singularity \(\mathbb{C}^2/\Gamma\) as a Nakajima quiver variety for the framed McKay quiver of \(\Gamma\), taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed
McKay quiver by removing an arrow and then `cornering', and we show that fine moduli spaces of cyclic modules over these new algebras are isomorphic to quiver varieties for the framed McKay quiver and certain non-generic choices of the stability parameter. Hilbert scheme of points; quiver variety; Kleinian singularity; preprojective algebra; cornered algebra Representations of quivers and partially ordered sets, Parametrization (Chow and Hilbert schemes), McKay correspondence, Singularities in algebraic geometry Punctual Hilbert schemes for Kleinian singularities as quiver varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper studies in detail the \(F\)-blowups of certain normal surface singularities. The \(F\)-blowup of a variety was introduced by \textit{T. Yasuda} [Am. J. Math. 134, No. 2, 349--378 (2012; Zbl 1251.14002)]. Its interaction to \(F\)-pure and \(F\)-regular singularities has not been fully expored. This article deals with understanding the properties of \(F\)-blowups of non \(F\)-regular rational normal double points and simple elliptic singularities, in relation to normality, smoothness, and stabilization of the \(F\)-blowup sequence. The techniques used combine classical results on normal surface singularities with computations performed with the help of Macaulay2 using two computational tools implemented here. Given a module, the first computes an ideal such that the blowups at the ideal and module coincide, based upon \textit{O. Villamayor U.}'s work [J. Algebra 295, No. 1, 119--140 (2006; Zbl 1087.14011)]. The second tool computes the Frobenius pushforward \(F_*M\) of a given module \(M\). Among other things, in the case of the rational normal double points, the authors exhibit two non-\(F\)-regular such surfaces for which the \(e\)th \(F\)-blowup is the minimal resolution, for \(e \geq 2\). For simple elliptic singularities, the authors determine the structure of \(F\)-blowups up to normalization. Some of the results build upon previous work of \textit{N. Hara} and \textit{T. Sawada} [RIMS Kôkyûroku Bessatsu B24, 121--141 (2011; Zbl 1228.13009)]. \(F\)-blowups; \(F\)-pure surface; \(F\)-regular surface; rational double points; simple elliptic singularities \beginbarticle \bauthor\binitsN. \bsnmHara, \bauthor\binitsT. \bsnmSawada and \bauthor\binitsT. \bsnmYasuda, \batitle\(F\)-blowups of normal surface singularities, \bjtitleAlgebra Number Theory \bvolume7 (\byear2013), page 733-\blpage763. \endbarticle \OrigBibText N. Hara, T. Sawada and T. Yasuda, \(F\)-blowups of normal surface singularities, Algebra Number Theory 7 (2013), 733-763. \endOrigBibText \bptokstructpyb \endbibitem Singularities in algebraic geometry, Positive characteristic ground fields in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) \(F\)-blowups 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 Let \(X\) be a smooth variety over an algebraically closed field \(k\) of characteristic zero, \(P_ 0\in X\) a closed point and \(E_ 1\) the exceptional divisor of the blowing-up of \(X\) with center \(P_ 0\). A closed point \(P_ 1\in E_ 1\) is said to be an infinitely near point (shortly i.n.p.) of \(P_ 0\). A sequence of i.n.p. is defined obviously by recurrence. This paper studies some invariants of the curves \(C\) passing through these i.n.p. as e.g. the multiplicity sequence \(e(C_ i)\) of \(C\), \(C_ i\) being the \(i\)-th quadratic transform of \(C\), the Arf dimensions of \(C_ i\), the semigroups of the saturated rings (cf. Zariski or Campillo saturation) of \(A_ i\), where \(C_ i=\text{Spec}(A_ i)\) and the semigroups of saturated rings of \(A_ i\) relative to some exceptional divisors of the chain of blowing up's. blowing-up; infinitely near point; multiplicity sequence; Arf dimension; Campillo saturation; Zariski saturation; equisingularity Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Infinitesimal methods in algebraic geometry On the geometry of the sequence of infinitely near points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The book has three parts. The first is an elementary introduction to theory of representations of quivers. In this part, many aspects of representation theory of Dynkin quivers are described: Euler forms, root lattices, projective and injective modules, geometry of orbits of certain group actions on the representation space, Gabriel theorem, Hall algebras and preprojective algebras.
In the second part, representation theory of infinite type quivers is discussed. In particular, the author describes preprojective and preinjective representations, Auslander-Reiten quivers, tame and wild quivers, tame-wild dichotomy, Kac's theorem.
The third part is an excursion in quiver varieties theory. This part is more advanced. It contains for example: Hamiltonian reduction and geometric invariant theory, Hilbert schemes, Kleinian singularities, geometric McKay correspondence, geometric realization of Kac-Moody algebras.
The book can be a concise guide to representation theory of quiver representations for beginner and advanced researchers. quiver representations; finite representation type; infinite representation type; quiver varieties; Hall algebras Research exposition (monographs, survey articles) pertaining to associative rings and algebras, Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Parametrization (Chow and Hilbert schemes) Quiver representations and quiver varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We prove that the wreath product orbifolds studied earlier by the first author [Duke Math. J. 103, 1--23 (2000; Zbl 0947.19004); Hilbert schemes, wreath products, and the McKay correspondence, preprint, \texttt{http://arxiv.org/abs/math.AG/9912104}] provide a large class of higher dimensional examples of orbifolds whose orbifold Hodge numbers coincide with the ordinary ones of suitable resolutions of singularities. We also make explicit conjectures on elliptic genera for the wreath product orbifolds. Hilbert schemes; elliptic genera W. Wang and J. Zhou, ''Orbifold Hodge Numbers of Wreath Product Orbifolds,'' J. Geom. Phys. 38, 152--169 (2001). Modifications; resolution of singularities (complex-analytic aspects), Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Transcendental methods of algebraic geometry (complex-analytic aspects), Elliptic genera Orbifold Hodge numbers of the wreath product orbifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Through a geometric approach, we explain the origin of the crepant resolution conjecture of Y. Ruan. More precisely, we calculate the Chen-Ruan cohomology and the quantum corrections of Ruan for the cohomology of Hilbert schemes in the particular case of the two-fold symmetric product of \(\mathbb C\mathbb{P}^2\), which corresponds to the invariant part by the action of the symmetric group \(\mathfrak S_2\) on the blow-up of \(\mathbb C\mathbb{P}^2 \times \mathbb C\mathbb{P}^2\) along the diagonal. Chen-Ruan cohomology; Hilbert schemes; crepant resolution conjecture; symmetric product Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) On Hilbert schemes and Chen-Ruan cohomology | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 book is based on a series of lectures on equisingularity theory given by the author about results of Zariski, Greuel, Lê, K. Saito, Teissier and many others related to Zariski's multiplicity conjecture.
Let \(f_i:(\mathbb{C}^n,0)\to (\mathbb{C}, 0)\) be germs of analytic functions, \(i=1,2\). \(f_1\) and \(f_2\) are called topologically \(V\)-equivalent if there exist open neighbourhoods \(U_1, U_2\) of \(0\in \mathbb{C}^n\) and a homeomorphism \(\varphi:(U_2, 0)\to (U_1, 0)\) such that \(f_i\) is defined on \(U_i\) and \(\varphi(V(f_2))\cap U_2=V(f_1)\cap U_1\).
Zariski's multiplicity conjecture states that \(f_1\) and \(f_2\) being topologically \(V\)-equivalent implies that the multiplicities \(\text{mult}_0(f_1)\) and \(\text{mult}_0(f_2)\) are equal. A family \(\{f_t\}\) of germs of functions is weakly topologically \(V\)-equisingular if for all sufficiently small \(t\), the function \(f_t\) is topologically \(V\)-equivalent to \(f_0\), \(t\in D\) an open disc in \(\mathbb{C}\). The family is called \(\mu\)-constant if the Milnor number of \(f_t\) at \(0\) is independent on \(t\).
If \(n\neq 3\) the family is \(\mu\)-constant if and only if it is weakly topologically \(V\)-equisingular. The conjecture that a \(\mu\)-constant family is equimultiple is still open, only proved in special cases.
The first chapter presents Ephraim's homology approach to Zariskis conjecture. The second chapter introduces and studies the Lê number replacing the Milnor number for non-isolated singularities. Deformations with constant Milnor number are studied in chapter 3. It is proved that in the case \(n\neq 3\) in a \(\mu\)-constant family \(\{f_t\}\) the diffeomorphism type of the Milnor fibration of \(f_t\) at \(0\) is independent on \(t\) (for small \(t\)) and the family is weakly topologically \(V\)-equisingular. Criteria for a family being \(\mu\)-constant are discussed. As a consequence Zariski's conjecture is proved for quasihomogeneous polynomials.
In chapter 4 deformations with constant Lê number are studied. In chapter 5 equisingular deformations of aligned singularities (introduced by Massey) are investigated. Zariski's multiplicity conjecture; topological V-equivalent; topologically equisingular; Le number; aligned singularities Eyral C., IMPAN Lecture Notes 3, in: Topics in Equisingularity Theory (2016) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities in algebraic geometry, Deformations of singularities, Global theory and resolution of singularities (algebro-geometric aspects), Equisingularity (topological and analytic), Milnor fibration; relations with knot theory Topics in equisingularity theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We provide here an infinite family of finite subgroups \(\{G_n\subset\text{SL}(\mathbb{C}\}_{n\geq 2}\) for which the \(G\)-Hilbert scheme \(G_n\)-Hilb \(\mathbb{A}^n\) is a crepant resolution of \(\mathbb{A}^n/G_n\), via the Hilbert-Chow morphism. The proof is based on an explicit description of the toric structure of \(G_n\)-Hilb \(\mathbb{A}^n\), \(n\geq 2\), in terms of Nakamura's \(G_n\)-graphs. DOI: 10.1016/j.crma.2006.11.033 Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Smooth toric \(G\)-Hilbert schemes via \(G\)-graphs | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 a very old and interesting open problem to characterize those collections of embedded topological types of local plane curve singularities which may appear as singularities of a projective plane curve \(C\) of degree \(d\). The goal of the present article is to give a complete (topological) classification of those cases when \(C\) is rational and it has a unique singularity which is locally irreducible (i.e., \(C\) is unicuspidal) with one Puiseux pair. cuspidal rational plane curves; logarithmic Kodaira dimension Fernández de Bobadilla, Javier; Luengo, Ignacio; Melle Hernández, Alejandro; Némethi, Andras, Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair.Real and complex singularities, Trends Math., 31-45, (2007), Birkhäuser, Basel Singularities of curves, local rings, Singularities in algebraic geometry, Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(G \subseteq GL_ 3(\mathbb{C})\) be a finite subgroup. The singularities of the quotient \(\mathbb{C}^ 3/G\) are studied. -- If \(G\) is abelian then \(\mathbb{C}^ 3/G\) is a simplicial toric variety. In general, a variety \(X_ G \to \mathbb{C}^ 3/G\), proper, birational, and isomorphic outside of the singular locus, is constructed having only toric singularities. The singularities of \(X_ G\) are studied in terms of isotropy subgroups of \(G\).
Finally, the classification of all \(\mathbb{C}^ 3/G\) with \(G\) abelian and having canonical singularities is given. singularities of quotient; simplicial toric variety; canonical singularities Pouyanne, Nicolas: Une résolution en singularités toriques simpliciales des singularités-quotient de dimension trois. Ann. fac. Sci. Toulouse (6) 1, 363-398 (1992) \(3\)-folds, Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) A resolution of three dimensional quotient-singularities in simplicial toric singularities. Three dimensional quotients | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0653.00009.]
During his study of the resolution of curve singularities, Zariski introduced the notion of the saturation of a one dimensional complete local domain of equicharacteristic zero. Zariski's saturation is not appropriate in positive characteristic. By using Hamburger-Noether expansion of power series, instead of Puiseux expansion, the author [in Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 1, 211-220 (1983; Zbl 0553.14013)] has introduced a notion of saturation which agrees with Zariski saturation in characteristic zero, but which remains meaningful in positive characteristic. The present paper is a well-written survey article about the author's saturation; nearly all of the results in it appear elsewhere. resolution of curve singularities; positive characteristic; saturation Campillo, A.: Arithmetical aspects of saturation of singularities. Pol. Acad. Publ. 20, 121-137 (1988) Singularities of curves, local rings, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Local rings and semilocal rings Arithmetical aspects of saturation 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 author considers algebroid varieties: \(V(I):=Spec(k[[x_ 1,...,x_ n]]/I)\) where \(I\subset k[[x_ 1,...,x_ n]]\) is a radical ideal. He studies transversality of regular varieties with V(I) in terms of the tangent cone \(Spec(k[x_ 1,...,x_ n]/in(I))]\) where in(I) is generated by the initial parts of elements in I. Transversality is an important point in resolution of singularities. algebroid varieties; transversality; resolution of singularities Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Transversality of varieties. A characterization | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is an obituary remembering the work and life of Shreeram Abhyankar. He was a leading algebraist and geometer who for almost sixty years made important contributions to Mathematics. He passed away on November 2, 2013. He was active in research until his death.
In this note there is a non technical but accurate description of Abhyankar's work in different fields, such as resolution of singularities, valuation theory, ramified coverings, some difficult problems of affine geometry, etc. Some of his most noteworthy work was in resolution of singularities over fields of positive characteristic, where some of the best results available at present are still essentially due to him. The author makes interesting remarks on the preferred methods of Abhyankar (explicit techniques, such as fine calculations with polynomials and series) and on aspects of his life and personality. A brief but useful bibliography including some of his more important works and papers by other authors related to them is included. resolution of singularities; valuation; ramification; affine geometry History of algebraic geometry, Biographies, obituaries, personalia, bibliographies, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Coverings in algebraic geometry, Jacobian problem The mathematical life of Shreeram Abhyankar | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(Y\) be a non-singular complex algebraic variety of dimension \(d\) and let \(f\) be a non-constant regular function on \(Y\). Some kinds of global and local zeta functions associated with \(f\) have been defined and studied by Y. Denef and F. Loeser. These are interesting singularity invariants of the hypersurface \(f^{-1}(\{0\})\) and its germ at some point \(0\in f^{-1} (\{0\})\). We denote by \(Z_{\text{top}} (f,s)\) and \(Z_{\text{mot}}(f,s)\) the topological zeta function and the motivic zeta function, respectively. The main problem of this paper is the poles of these zeta functions. The authors gives a necessary and sufficient condition that \(Z_{\text{top}}(f,s)\) has a pole at \(s=s_0\) for a non-singular \(Y\) with \(d=2\). It is also given a necessary and sufficient condition that \(Z_{\text{top}} (f,s)\) has a pole at \(s=s_0\) when an embedded resolution \(h:X\to Y\) of \(f^{-1} (\{0\})\) in \((Y,0)\) is given. singularity; zeta functions; poles Rodrigues, B., Veys, W.: Poles of zeta functions on normal surfaces. Proc. London Math. Soc. 87, 164--196 (2003) Singularities in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants Poles of zeta functions on normal surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\alpha\) and \(\beta\) be primitive \(m\)-th root and \(n\)-th root of unity, respectively, let \(K\) be an algebraically close field of characteristic zero and let \(R=K[X,Y]\) be a polynomial ring in two variables over \(K\) with \(m=(X,Y)\) its maximal ideal. The author gives a characterization for the existence of a rational rank one non discrete valuation dominating \(R_m\) with a generating sequence of eigenfunctions for a subgroup, \(H\), of \(\langle\alpha\rangle\times\langle\beta\rangle\) using the greatest common divisor of \(m\) and \(n\).
He also gives a method for computing the semigroup of values of elements of \(K[X,Y]^H\). Finally, he shows that this kind of valuation restricted to the quotient field of a local domain does not split in \(K[X,Y]_m\).
The paper es quite technical so a good backgroud is required to fully understand it. polynomial ring; valuations Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Semigroup rings, multiplicative semigroups of rings Generating sequences and semigroups of valuations on 2-dimensional normal local rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 about a formalization of known procedures to resolve singularities of algebraic varieties defined over fields of characteristic zero. This formalization is presented as a game, called \textit{Stratify}. Probably the name is due to the fact that, after all, a canonical resolution of singularities means to stratify, in a suitable natural way, the singular locus \(S\) of the variety (and those of its transforms) as a union of locally closed regular subvarieties, at each step of the desingularization process we blow up the ``worst'' stratum (which is closed).
At each stage of the game we have a finite weighted graph, which is successively modified by two players \(A\) and \(B\), according to certain rules. It is a rather curious game, because only player \(A\) can win (by reaching a \textit{final stage.} Player \(B\) can only prevent \(A\) from winning, perhaps forever. The game mimics, in a formal or combinatorial way, the different stages in an attempt to resolve singularities of varieties (in characteristic zero, where we may perform induction by using hypersurfaces of maximal contact), specially by taking blowups with permissible centers.
In the paper, after a review of known results on desingularization, the authors explain the game. The list of rules is rather long and complicated. Then the necessary algebro-geometric concepts are briefly reviewed. Next they explain how a resolution process can be translated into the rules of \textit{Stratify}. Actually, the process considered is not one trying to directly resolve singular algebraic varieties, but rather a resolution of pairs \((I,b)\), where \(I\) is a sheaf of nonzero ideals on a regular ambient variety and \(b\) is a natural number (or rather of some closely related objects, the \textit{singularity data}). It is known (and sketched in the paper) that solving this problem implies resolution of varieties. The authors prove that this resolution problem is equivalent to having a winning strategy for ``Stratify'', and finally they show that such a winning strategy is available. singularity; resolution; graph; transversality; singular datum; Rees algebra Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Polynomials in real and complex fields: location of zeros (algebraic theorems), Directed graphs (digraphs), tournaments A game for the 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 I be an ideal of \({\mathbb{C}}\{z_ 1,...,z_ n\}\) which defines the origin in \({\mathbb{C}}^ n\). Given a suitable choice of generators of \(I=(f_ 1,...,f_{n+p})\), we give an explicit method to determine the cycle of \({\mathbb{P}}^{n+p-1}\) associated with the exceptional fiber of the blowing-up of I in \({\mathbb{C}}^ n\). We also study the blowing-up of an equimultiple family of punctual ideals parametrized by a germ of a reduced complex analytic space. exceptional fiber; blowing-up of an equimultiple family of punctual ideals; multiplicities Henaut A., Ann. Inst. Fourier Grenoble 37 pp 143-- (1987) Global theory and resolution of singularities (algebro-geometric aspects), Algebraic cycles, Local complex singularities, Singularities in algebraic geometry Cycle exceptionnel de l'éclatement d'un idéal définissant l'origine de \({\mathbb C}^ n\) et applications. (Exceptional cycle of the blowing-up of an ideal defining the origin of \({\mathbb C}^ n\) and applications) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The articles of this volume will be reviewed individually.
For Part II see [Zbl 1275.14002]. Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Geometric invariant theory, Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Proceedings of conferences of miscellaneous specific interest Geometric methods in representation theory. I. Selected papers based on the presentations at the summer school, Grenoble, France, June 16 -- July 4, 2008 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 was observed by \textit{V. I. Arnol'd} [Russ. Math. Surv. 30, No.5, 1-75 (1975); translation from Usp. Mat. Nauk 30, No.5(185), 3-65 (1975; Zbl 0338.58004)] that for each exceptional unimodular hypersurface singularity (X,0) there is a unique exceptional unimodular hypersurface singularity (Y,0) with the following property: The Dolgachev numbers (these are data given by the resolution graph) of (X,0) coincide with the Gabrielov numbers (these are invariants of the system of vanishing cycles in the homology of the Milnor fibre) of (Y,0), and conversely. The correspondence (X,0)\(\leftrightarrow (Y,0)\) is called the ''strange duality'' among the exceptional unimodular hypersurface singularities. The authors define an extension of this duality to the class of all Kodaira singularites (i.e. singularities for which the exceptional set in the minimal resolution \(\tilde X\to X\) of the singularity is isomorphic to a minimal exceptional fibre in an elliptic pencil - but with different neighbourhood in \(\tilde X)\). They discuss the relation between resolution graphs, Dynkin diagrams and Milnor lattices and other invariants of dual Kodaira singularities. Most of these invariants are also computed explicitly for all Kodaira singularities. deformation; Dolgachev numbers; Gabrielov numbers; strange duality; exceptional unimodular hypersurface singularities; Kodaira singularites; resolution graphs; Dynkin diagrams; Milnor lattices Ebeling W., Wall C.T.C.: Kodaira singularities and an extension of Arnold's strange duality. Compositio Math. 56, 3--77 (1985) Singularities of surfaces or higher-dimensional varieties, Deformations of singularities, Complex singularities, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Local complex singularities Kodaira singularities and an extension of Arnold's strange duality | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper the authors prove a motivic analogue of the rationality of the Poincaré series associated to the \(p\)-adic points on algebraic varieties. They consider the Grothendieck ring on algebraic varieties over a field of characteristic 0: The class of a variety \(S\) is the sum of the classes of \(S'\) and \(S''\) if \(S'\) is closed in \(S\) and \(S''\) is isomorphic to \(S-S'\). The multiplication is the product of varieties and the ring they consider is a certain subring of the localization of the above Grothendieck ring where multiplication by the affine line is made invertible. The Poincaré series is viewed as an element of the formal power series ring in one variable over the above ring. The main techniques used in the proof are quantifier elimination for semi-algebraic sets of power series in characteristic 0 and M. Kontsevich's idea of motivic integration. motivic integration; rationality of the Poincaré series; \(p\)-adic points; quantifier elimination; semi-algebraic sets J. Denef and F. Loeser, \textit{Germs of arcs in singular algebraic varieties and motivic integration}, \textit{Invent. Math.}\textbf{135} (1999) 201 [math/9803039]. Arcs and motivic integration, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Semialgebraic sets and related spaces, Quantifier elimination, model completeness, and related topics, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Motivic cohomology; motivic homotopy theory Germs of arcs on singular algebraic varieties and motivic integration | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0607.00004.]
The author describes a minimal embedded resolution of the affine variety given by the vanishing of the only nontrivial invariant polynomial for the fundamental linear representation of dimension 27 (resp. 56) of a simple Lie group G of type \(E_ 6\) (resp. \(E_ 7\)). We describe the results only in the case of \(E_ 6\). In this case the invariant polynomial is a famous Jordan-Cartan-Dickson cubic with each of its 45 monomials corresponding to tritangent planes of a nonsingular cubic surface. The variety is resolved by one blowing up of \({\mathbb{A}}^{27}\) at its vertex followed by the blowing up of the exceptional divisor \({\mathbb{P}}^{26}\) at the singular locus of the projectivized cubic. The latter is a nonsingular 16-dimensional variety isomorphic to G/P, where P is a maximal parabolic subgroup of G. The total inverse transform of the cubic is a normal crossing divisor with irreducible components of multiplicities 1, 3 and 2 corresponding to the proper inverse transform, the first and the second exceptional divisor respecively. minimal embedded resolution; \(E_ 6\); \(E_ 7\); Jordan-Cartan-Dickson cubic; total inverse transform Global theory and resolution of singularities (algebro-geometric aspects), Other algebraic groups (geometric aspects), Representation theory for linear algebraic groups, Singularities in algebraic geometry, Semisimple Lie groups and their representations The singularities of some invariant 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 See the review of the first edition (1975; Zbl 0317.14004). local moduli problem; moduli space of the equisingularity class; deformation theory Zariski O.: Le problème des modules pour les courbes planes. Hermann, Paris (1986) Global theory and resolution of singularities (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic), Singularities in algebraic geometry, Power series rings Le problème des modules pour les branches planes. Cours donné au Centre de Mathématiques de l'École Polytechnique. Nouvelle éd. revue par l'auteur. Rédigé par François Kmety et Michel Merle. Avec un appendice de Bernard Teissier | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a variety defined over a field \(k\). One of the fundamental problems of algebraic geometry is the existence of a proper birational morphism \(f : X^{\prime} \rightarrow X\) such that \(X\) is nonsingular over \(k\). Such an \(f\) is called a resolution of singularities of \(X\).
If \(X\) has characteristic zero then it is well known that a resolution of singularities exists. If on the other hand \(k\) has characteristic \(p>0\), the following is known at the moment of this writing. Resolution of singularities exist in dimensions 1 and 2 and in dimension 3 if \(p>5\).
A weaker form of resolution of singularities was proved by \textit{J. de Jong} who has shown that for any variety \(X\) defined over a field \(k\), there exists an alteration \(f : X^{\prime} \rightarrow X\), such that \(\dim X =\dim X^{\prime}\), where \(f\) is proper and dominant, and \(X^{\prime}\) is nonsingular [Publ. Math., Inst. Hautes Étud. Sci. 83, 51--93 (1996; Zbl 0916.14005)].
In this paper, the authors give evidence by providing low dimensional examples, that if \(X\) is an \(n\)-dimensional variety defined over a field \(k\), then there exists a proper dominant morphism \(f : X^{\prime} \rightarrow X\), such that \(\dim X^{\prime}=n\), \(X\) is nonsingular and there exists a tower of smooth fibrations \[ X^{\prime}\rightarrow X^{\prime}_1 \rightarrow \cdots \rightarrow X^{\prime}_n, \] such that \(\dim X^{\prime}_i =n-i\). resolution of singularities; alterations Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Dominant classes of projective varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0614.00007.]
From the introduction: ``Let Z be a regular noetherian scheme, let X be a closed two dimensional subscheme of Z and let P be a closed point of X. the aim of this paper is to associate a polygon \(\Delta\) (P) to P, which turns out to be an intrinsic invariant of the singularity of X at P... The invariant \(\Delta\) (P) is expected to be useful for formulating a ``fine'' version of the resolution game...''
The original ``resolution game'' was unfortunately found to be inadequate by \textit{M. Spivakoski} [cf. Arithmetic and Geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 419-432 (1983; Zbl 0531.14009)]. resolution game Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Characteristic polygon 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 Let \(A\) and \(B\) be finite dimensional self-injective algebras over an algebraically closed field \(k\). Then, one defines an algebraic variety \(\text{mod}(A,d)\) of \(d\)-dimensional \(A\)-modules as the variety of possible \(A\)-module structures on a fixed \(d\)-dimensional vector space. The base change group \(\text{Gl}_d(k)\) acts on \(\text{mod}(A,d)\) and a module \(M\) is said to be a degeneration of a module \(N\) if \(M\) belongs to the Zariski closure \(\overline O_N=\overline{\text{Gl}_d(k)\cdot N}\) of the \(\text{Gl}_d(k)\)-orbit of \(N\). Now, any two pointed algebraic varieties \((X_1,x_1)\) and \((X_2,x_2)\) are said to be smoothly equivalent if there is a third \((Z,z_0)\) and smooth morphisms \(\xi_i\colon(Z,z_0)\to(X_i,x_i)\) for \(i=1\) and \(i=2\). The equivalence class of \((X,x)\) is denoted \(\text{Sing}(X,x)\).
The main result of the paper under review is the following statement. Suppose there is an \(A\)-\(B\)-bimodule \(M\) and a \(B\)-\(A\)-bimodule \(N\) so that \(M\otimes_BN\) is isomorphic to the direct sum of \(A\) and a projective \(A\)-\(A\)-bimodule, and likewise for \(N\otimes_AM\). If the \(A\)-module \(V\) degenerates to \(W\), then \(\text{Sing}(\overline O_V,W)=\text{Sing}(\overline O_{M\otimes_AV},M\otimes_AW)\). As an application the authors show that the closures of the orbits of the module varieties of a Brauer tree algebra are Cohen-Macaulay, normal, and, if the characteristic of \(k\) is \(0\), have rational singularities.
The proof of the main theorem is a reduction to a statement on Grothendieck groups, and then the authors apply an earlier result of the second author. The existence of such bimodules \(M\) and \(N\) is implied by an equivalence between the derived categories of the two algebras. degenerations of modules; singularities; modules varieties Skowroński, A.; Zwara, G.: Derived equivalences of selfinjective algebras preserve singularities. Manuscripta math. 112, 221-230 (2003) Representations of quivers and partially ordered sets, Singularities in algebraic geometry, Module categories in associative algebras, Group actions on varieties or schemes (quotients) Derived equivalences of selfinjective algebras preserve 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 monodromy conjecture of Denef and Loeser predicts that the poles of the topological zeta function and related zeta functions associated to a polynomial \(f\) induce monodromy eigenvalues of \(f\). However, not every monodromy eigenvalue can be recovered from a pole. This motivates to generalise the conjecture in order to obtain more poles: one also considers zeta functions associated to a polynomial and a differential form. The authors attach to \(f\) a suitable family of differential forms, such that each pole of the topological zeta function of \(f\) and a form from the family induce a monodromy eigenvalue, and moreover, such that all monodromy eigenvalues are obtained this way. This answers positively a question (regarding the existence of a family of such forms) of \textit{W. Veys} [Adv. Math. 213, No. 1, 341--357 (2007; Zbl 1129.14005)]. monodromy conjecture; monodromy eigenvalues; topological zeta function Cauwbergs, Thomas; Veys, Willem, Monodromy eigenvalues and poles of zeta functions, Bull. Lond. Math. Soc., 49, 2, 342-350, (2017) Singularities in algebraic geometry, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Local complex singularities Monodromy eigenvalues and poles of 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 P be a point of an algebroid hypersurface V. The similarity between Zariski's notion of V being ''equisingular'' at P and Abhyankar's notion of P being a ''good point'' of V is illustrated by proving a criterion for P to be a good point, which is analogous to Zariski's definition of equisingularity of V at P. equisingular; good point Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry A criterion for a point of a hypersurface to be good | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author generalizes results of \textit{O. A. Platonova} [Russ. Math. Surv. 36, No.1, 248-249 (1981); translation from Usp. Mat. Nauk 36, No.1(217), 221-222 (1981; Zbl 0458.14014)] about the number of inflection points on a planar algebraic curve to surfaces in \(P^ 3\). The five types of inflection points on a general surface due to Platonova are: Type (P\({}_ 1):\) curve of parabolic points; Type (P\({}_ 2):\) finite set of points of tangency of \(P_ 1\) to an asymptotic direction; Type (H\({}_ 2):\) curve of inflections of asymptotic lines; Type (H\({}_ 3):\) finite set of selfintersections of curves of type \(H_ 2\); Type (H\({}_ 4):\) finite set of points where curve \(H_ 2\) is tangent to an asymptotic direction.
The author shows that, if d is the degree of the surface, then: deg \(P_ 1=4d(d-2)\); deg \(H_ 2=d(11d-24)\); {\#}P\({}_ 2=2d(d-2)(11d-24)\); {\#}H\({}_ 3=5d(7d^ 2-28d+30)\); {\#}H\({}_ 4=5d(d-4)(7d-12)\). The connection between the singularities of the embedding \(X\subseteq P^ 3\) and the Thom-Bordman singularities of type \(S_{\ell^ k}\) established in {\S} 2 of the article under review is the key point of the calculation. inflection points on a general surface; Thom-Bordman singularities Kulikov, V.S. 1983. The calculation of the singularities of the embedding of a generic algebraic surface in the projective space P 3. Functional Analysis and Applications, 17:176-186. Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Calculation of singularities of an imbedding of a generic algebraic surface in projective space \(P^ 3\) | 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 There are well-known algorithms to resolve algebraic curve singularities, based on Puiseux series and on birational transform. In the 1990s an algorithm based on the monomial transform appeared [see \textit{M. Oka} in: Algebraic geometry and singularities, Proc. 3rd Int. Conf. Algebraic Geometry, La Rabida 1991, Prog. Math. 134, 95--118 (1996; Zbl 0857.14014)]. In this paper the author presents a new algorithm to resolve local singularities of algebraic plane curves based on the monomial transform and independent of additional coordinate changes. Moreover, by introducing two singularity invariants, he can give a sharp estimate of the number of steps in the algorithm. Singularities of curves, local rings, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Plane and space curves, Computational aspects of algebraic curves A fast algorithm for curve singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A normal surface singularity \((X, Q)\) is said to be sandwiched if it dominates birationally a non-singular surface. They arise when a complete \({\mathbf m}\)-primary ideal in a local regular \(\mathbb{C}\)-algebra \(R\) of dimension two is blown up. A sandwiched singularity is said to be primitive if it can be obtained by blowing up a simple ideal, that is, a complete irreducible ideal of \(R\). It is known that any sandwiched singularity is the birational join of some primitive singularities [\textit{M. Spivakovsky}, Ann. Math. (2) 131, 411--491 (1990; Zbl 0719.14005)]. In this note, we prove that the Nash conjecture for sandwiched singularities and for primitive singularities are equivalent. arc; infinitely near point; birational join Fernandez-Sanchez, J.: Equivalence of the Nash conjecture for primitive and sandwiched singularities. Proc. amer. Math. soc. 133, 677-679 (2005) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Equivalence of the Nash conjecture for primitive and sandwiched 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 Singularities of plane curves occupy a central position in singularity theory: numerous definitions of equisingularity coincide and the classification up to this is well known, and has been the starting point of many investigations. This manuscript, which originates from 1973, gives an account of the theory from the viewpoint of maximal contact; in particular, equisingularity class is characterised via `simpler' curves having maximal contact with the given one. This is given for reducible, as well as for irreducible curves. Moreover an algorithm to determine the class (via Newton polygons and blowing-up) is described, but is not fully implemented. The theory is developed in detail, but not independently of other versions of classification. This viewpoint, unlike some of the alternatives, works well in finite characteristic. Singularities of plane curves; equisingularity; maximal contact; Newton polygons; blowing-up; finite characteristic M. Lejeune , Sur l'équivalence des singularités des courbes algébroïdes planes . Coefficients de Newton. Thèse. Singularities in algebraic geometry, Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Finite ground fields in algebraic geometry, Analytic subsets of affine space On the equivalence of singularities of plane algebroid curves. Newton coefficients | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((Y,0)\) be a germ of a complex normal surface singularity, \((X,E) \to (Y,0)\) a minimal resolution, \(E_1, \ldots,E_n\) the irreducible components of the (necessarily connected) exceptional divisor \(E\). Then
\[
{\mathcal E} = \{ D=m_1E_1 + \cdots + m_n E_n : m_i \in \mathbb{Z} , \, D.E_i \leq 0 , \, i=1, \ldots, n \}
\]
is an additive semigroup, called the \textit{Lipman semigroup} of the singularity. It has a number of interesting properties, in particular it has a unique minimal generating set \({\mathcal H}_{\mathcal E} \), called the Hilbert basis of the semigroup. The toric variety associated to \(\mathcal E \) is \(V_{\mathcal E}:={\text{Spec}}\, ({\mathbb{C}}[\mathcal E]\)). Both the semigroup \(\mathcal E\) and the variety \(V_{\mathcal E}\) are important invariants of the singularity.
A natural question, addressed before by several authors, is to give an algorithm to construct the Hilbert basis \({\mathcal H}_{\mathcal E}\). In this paper the author works indirectly, by introducing an auxiliary semigroup \(\mathcal S = {\mathcal S}_{\mathcal E}\), which has a Hilbert basis \({\mathcal H}_{\mathcal S}\) that can be computed. It is proved that this basis immediately gives a Hilbert basis of \(\mathcal E\) and a parametrization of the toric variety \(V_{\mathcal E}\). The semigroup \(\mathcal S\) is defined as follows. Let \(M=M(\mathcal E)\) be the intersection matrix, i.e., its entry \(M_{ij}\) is the integer \(E_i.E_j\) and \(A=[A|I_n]\) (an \(n \times 2n\) matrix). Then,
\[
\mathcal S= \{(v_1, \ldots, v_n)\in {\mathbb{N}}^{2n}:A[v_1, \ldots,v_n]^T =0 \}\, .
\]
For semigroups of this kind (kernels of integral matrices) there are efficient combinatorial algorithms to find a Hilbert basis, e.g., see Chapter 6 of [\textit{M. Kreuzer, L. Robbiano}, Computational commutative algebra. II. Berlin: Springer. ( 2005; Zbl 1090.13021)].
The paper concludes with an example involving a singularity of type \(A_2\). Normal surface singularity; Lipman semigroup; Hilbert basis of a semigroup; toric variety; intersection matrix. Global theory and resolution of singularities (algebro-geometric aspects), Compact complex \(3\)-folds, Singularities in algebraic geometry, Local complex singularities, Toric varieties, Newton polyhedra, Okounkov bodies, Real algebraic and real-analytic geometry Hilbert basis of the Lipman semigroup | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities See the preview in Zbl 0539.14008. decomposition of birational morphism; smooth curves on surfaces; with rational double points; blow-up of a smooth point; Gorenstein; threefold singularities D. Morrison, ''The biratioanl geometry of surfaces with ratiional double points,''Math. Ann.,271, 415--438 (1985). Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, Singularities of surfaces or higher-dimensional varieties, \(3\)-folds, Singularities in algebraic geometry The birational geometry of surfaces with rational double points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities There is a notion, due to Nakumura, of \(G\)-Hilbert scheme \(\mathrm{Hilb}^G{\mathbb C}^n\) for any finite, abelian subgroup \(G\) of \(\mathrm{GL}(n,{\mathbb C})\). The \(G\)-Hilbert scheme can be described in terms of \(G\)-sets. In the article under review the author describes the \(G\)-Hilbert scheme when \(G\) is a finite cyclic group generated by a \((3\times 3)\) matrix of a particular form. After giving an classification of all possible \(G\)-sets, the author also obtains a formula for the number of different \(G\)-sets that appear for these groups. \(G\)-sets; \(G\)-Hilbert scheme O. Kȩdzierski, The G-Hilbert scheme for \(\frac{1}{r}\)(1,a,r-a), Glasg. Math. J. 53 (2010), 115 -129. McKay correspondence, Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry The \(G\)-Hilbert scheme for \(\frac 1{r} (1,a, r-a)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 shows in a very synthetic manner some results concerning algorithmic resolution in characteristic zero (in the sense of \textit{S. Encinas} and \textit{O. Villamayor} [Prog. Math. 181, 147--227 (2000; Zbl 0969.14007)], and \textit{A. Bravo, S. Encinas} and \textit{O. Villamayor} [Rev. Mat. Iberoam. 21, No. 2, 349--458 (2005; Zbl 1086.14012)]) and their relationship with the weak equivalence introduced by Hironaka in the latest 70's, see for example [\textit{H. Hironaka}, ``Idealistic exponents of singularity'', Algebraic geometry, The Johns Hopkins centen. Lect., Symp. Baltimore/Maryland 1976, 52--125 (1977; Zbl 0496.14011)]. It simplifies some arguments and gives a very simple way to prove the ``naturality'' of the main step of the proof: the inductive step. singularities; marked ideals; blowing-up; permissible centers; resolution; equivalence O. Fujino, \textit{Semipositivity theorems for moduli problems}, preprint, arXiv:1210.5784v2 [math.AG] (2012). Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Singularities in algebraic geometry, Local theory in algebraic geometry, Birational geometry On the use of naturality in algorithmic resolution | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0527.00002.]
The paper deals with some properties of the 3-dimensional singularities with equations arising from Arnold's list, considered over an algebraically closed field of characteristic \(\neq 2,3\). These singularities are absolutely isolated, i.e. they have resolutions obtained by successively blowing up points, the ''canonical'' resolutions. These were considered first by P. J. Giblin in the topological context. - The article under review gives the algebraic description of the canonical resolution together with the intersections. Fundamental cycles (i.e. minimal negative embeddings of the exceptional loci) are computed. A method of calculating some cohomology groups on the nonreduced exceptional divisors is developed and applied to the normal bundles and the structural sheaves, thus getting vanishing theorems, applied to study the local moduli of the exceptional loci (embedded into the canonical resolution). For the \(A_ n\)-resolutions, the moduli space turns out to be smooth, and the dimension of its tangent space is the number of irreducible components isomorphic to the ruled surface \(F_ 2\). deformation; exceptional divisor; \(D_ n\); \(E_ n\); 3-dimensional singularities; vanishing theorems; local moduli of the exceptional loci; canonical resolution; \(A_ n\) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Some properties of the canonical resolutions of the 3-dimensional singularities \(A_ n\), \(D_ n\), \(E_ n\) over a field of characteristic \(\neq 2\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors study ideals in the power series ring \(R= {\mathbb C}[[x,y]]\) by introducing and using certain auxiliary objects of a combinatorial nature. They are, in a sense, generalizations, or refinements, of the classical Newton polygon.
First they introduce and study \textit{Newton maps}, which are certain ring endomorphisms of \(R\) of a simple form, and next the \textit{Newton algorithm}, a sequence of Newton maps which transforms an arbitrary ideal \(I\) of \(R\) into a principal ideal, generated by a ``monomial like'' element. Then they attach to an ideal \(I\) of \(R\) a certain graph (with some extra structure), called the \textit{Newton tree} of \(I\), which ``codifies'' much of the information available from the implementation of the Newton algorithm of \(I\), and also the \textit{Newton process} of \(I\). The latter is a more complicated object, it consists of sets of pairs \((\Sigma, Z)\) where \(\Sigma\) is a finite sequence of Newton maps and \(Z\) is either a positive integer of a certain series in \(x{\mathbb C}[[x]]\). They extensively study these concepts.
They prove, among others, the following facts. The Newton tree can be recovered from the Newton process, but not the other way around. Two ideals of \(R\) have the same Newton process if and only if they have the same integral closure. In the course of the proof of this result the authors use Rees valuations, which correspond to certain vertices in the Newton tree, called the \textit{dicritical} ones. From the Newton process of an integrally closed ideal \(I\) of \(R\) they obtain the Zariski factorization of \(I\) as product of ideals of \(R\), each factor being either a principal prime or a simple, integrally closed ideals of \(R\).
When the ideal \(I\) has finite codimension, the authors explain how to compute from its Newton tree the multiplicity (or order) of \(I\) (the highest power of the maximal ideal \((x,y)\) containing \(I\)), the Hilbert-Samuel multiplicity \(e(I)\) as well as its Łojasiewicz exponent of \(I\). They also show a new formula for \(e(I)\) in terms of areas defined by the successive Newton polygons found in the Newton algorithm.
Some of these results were known for more restricted classes of ideals of finite codimension of, the proof involved the use of the Newton polygon.
The authors announce a continuation of this paper, where some of the theory will be treated more geometrically.
The paper is well written and contains numerous examples. Newton polygon; Newton map; Newton algorithm; Newton tree; Newton process; Rees valuation; multiplicity; integral closure; dicritical. Cassou-Noguès, P., Veys, W.: Newton trees for ideals in two variables and applications. Proc. Lond. Math. Soc. (3) 108(4), 869-910 (2014) Structure, classification theorems for modules and ideals in commutative rings, Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects), Integral closure of commutative rings and ideals, Global theory and resolution of singularities (algebro-geometric aspects), Formal power series rings Newton trees for ideals in two variables and applications | 0 |
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