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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 The author presents (in expository form) results and recent questions on the relation between simple singularities and the corresponding finite groups. The questions treated here arose from conjectures of Grothendieck, solved by Brieskorn and published first time with complete proofs and additional contributions by the author in his book ``Simple singularities and simple algebraic groups'', Lect. Notes Math. 815 (1980; Zbl 0441.14002). Let \(X=\mathbb{C}^ 2/F\), \(F\subset SU(2,\mathbb{C})\) a finite subgroup, then \(X\) is a surface with an isolated singularity, and the dual graph of its minimal resolution is a Dynkin diagram of type \(A_ n\), \(D_ n\), \(E_ 6\), \(E_ 7\), \(E_ 8\) respectively, corresponding to the cyclic group of order \(n\), the binary dihedral group of order \(4n\), the binary tetrahedral, octahedral, icosahedral group, respectively. The 2-dimensional homology of the minimal resolution can be obtained from the corresponding root system \(A_ n\), \(D_ n\), \(E_ n\), its intersection form is given by the Cartan matrix \(C\). Conversely, if \(C\) is given corresponding to some Dynkin diagram, the group \(F\) can be expressed in terms of generators and relations by the elements of \(C\).
On the other hand, let \(G\) be a simply connected simple Lie group of type \(A_ n\), \(D_ n\), \(E_ n\), \(T\subset G\) a maximal torus, \(W=N_ G(T)/T\) the corresponding Weyl group, \({\mathcal X}:G\to T/W\) the quotient by the adjoint action of \(G\) on itself. Then the theorem of Brieskorn asserts: If \(S\) is a section transversal to the subregular unipotent orbit, then \(\mathcal X\) restricted to \(S\) is a versal deformation of the singularity of the same type. [Note that a different construction was recently found by \textit{F. Knop}, Invent. Math. 90, 579-604 (1987; Zbl 0648.14002); it gives some surprising deformations of the singularities in characteristic 2 and 3.]
There arises the question of a more direct connection between the Lie group we started with and the finite group \(F\). A remark of \textit{J. McKay} [cf. Proc. Am. Math. Soc. 81, 153-154 (1981; Zbl 0477.20006)] relates \(F\) with the Dynkin diagram by means of representation theory: If \(N\) is the 2-dimensional representation \(F\subset SU(2,\mathbb{C})\), \(R_ 0,\ldots,R_ r\) (representants of) the irreducible representations of \(F\) and \(N\otimes R_ i=\oplus_ ja_{ij}R_ j\), then \(2E_{r+1}- (a_{ij})\) is the Cartan matrix of the extended Dynkin diagram associated to the group \(F\) [later efforts to understand McKay's correspondence are related with the socalled Auslander-Reiten theory, cf. \textit{M. Auslander} and \textit{I. Reiten}, Trans. Am. Math. Soc. 302, 87-97 (1987; Zbl 0617.13018); results of Artin, Verdier, Esnault, Knörrer, Buchweitz, Greuel, Schreyer and others concern the maximal Cohen-Macaulay modules over simple singularities, cf. e.g. \textit{H. Knörrer} in Representations of algebras, Proc. Symp., Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser. 116, 147-164 (1986; Zbl 0613.14004)]. Let \({\mathcal X}_ i\) denote the character of the representation \(R_ i\), \(d_ i={\mathcal X}_ i(1)\), then \(d=(d_ 0,\ldots,d_ r)\) generates the kernel of \(2E_{r+1}-(a_{ij})\), \(d_ i\) are the coefficients of the maximal root in the corresponding root system and give the fundamental cycle of the minimal resolution of \(\mathbb{C}^ 2/F\). Further, \(\sum d_ i\) is the Coxeter number of the root system. Until now, the Dynkin diagrams of type \(B\), \(C\), \(F\) and \(G\) are missing; this is explained in the following way: They can be obtained by factorizing some of the homogeneous diagrams by a group \(\Gamma\) of symmetries. A simple singularity of type \(B_ n\), \(C_ n\), \(F_ 4\), \(G_ 2\), respectively is defined to be a couple \((X,\Gamma)\), where \(X\) is one of \(A_{2n-1}\), \(D_{n+1}\), \(E_ 6\), \(D_ 4\), respectively, such that \(\Gamma\) acts as a group of automorphisms. The author's results generalize the Brieskorn theorem to singularities of type \(B\), \(C\), \(F\), \(G\): If \(G\) is a simple Lie group of one of the above types, \(T\subset G\) a maximal torus, \(W\) the corresponding Weyl group, \({\mathcal X}:G\to T/W\) the adjoint quotient, a versal deformation of the singularity of the same type is obtained as follows: Take a transversal section \(S\) to the orbit of a unipotent subregular element \(x\in S\) such that \(S\) is stabilized by a reductive subset of \(Z_ G(x)\). Then \(\mathcal X\) restricted to \(S\) induces an equivariant versal deformation. \(S\cap\text{Uni}(G)\) is a singularity of type \(A_{2n-1}\), \(D_{n+1}\), \(E_ 6\), \(D_ 4\), respectively, and the action of \(\Gamma\) is given by a subgroup of \(Z_ G(x)\). A refined result is obtained considering the mentioned groups together with their automorphisms. -- Again, the preceding diagrams \(B\), \(C\), \(F\), \(G\) are considered from the viewpoint of representation theory of finite groups, this time in the relative case.
The author formulates as a dominating question the explanation of the relation between the finite group \(F\) and the corresponding Lie group, to show why the simple singularity \(\mathbb{C}^ 2/F\) appears on the unipotent variety of \(G\). finite subgroup of \(Sl(2,\mathbb{C})\); simple singularities; Dynkin diagram; homology of minimal resolution; root system; Cartan matrix; simple Lie group; McKay correspondence; Coxeter number Singularities in algebraic geometry, Representations of finite symmetric groups, Singularities of surfaces or higher-dimensional varieties, Local complex singularities, General properties and structure of complex Lie groups On finite groups associated to 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 The paper studies an ODE arising geometrically in the study of the equivariant quantum cohomology of the Hilbert scheme of points on \(\mathbb C^2\) and the Gromov-Witten/Donaldson-Thomas theory of \(\mathbb P^1\times \mathbb C^2\).
Specifically, let \(\mathcal F\) denote the Fock space with creation operators \(\alpha_{-k}\) acting on the vacum vector \(|\emptyset\rangle\), and annihilation operators killing the vacuum. The commutation relations are
\[
[\alpha_k, \alpha_l]=k\delta_{k+l}.
\]
The differential equation studied in the paper takes the form
\[
q\frac{d}{dq}\Psi=\mathsf M_{D}\Psi,\,\,\,\Psi\in \mathcal F
\]
for the differential operator
\[
\mathsf M_{D}=\frac{t_1+t_2}{2}\sum_{k>0} \left(k\frac{(-q)^k+1}{(-q)^k-1}-\frac{(-q)+1}{(-q)-1}\right)\alpha_{-k}\alpha_k +\frac{1}{2}\sum_{k, l>0} (t_1t_2\alpha_{k+l}\alpha_{-k}\alpha_{-l}-\alpha_{-k-l}\alpha_k\alpha_l).
\]
This equation has regular singularities at \(q=0\), \(q=\infty\) and certain roots of unity.
Geometrically, the Fock space can be identified via the Nakajima basis with the equivariant cohomology of the Hilbert scheme of points on the plane,
\[
\mathcal F\otimes \mathbb C[t_1, t_2]\cong \bigoplus_n H^{\star}_{T}((\mathbb C^2)^{[n]}).
\]
A natural divisor \(D\) can be constructed from the first Chern class of the tautological quotient. Classical multiplication by \(D\) is related to the Hamiltonian of the Calogero-Sutherland system. The operator \(\mathsf M_D\) corresponds to the small quantum multiplication by \(D\) in the equivariant quantum cohomology of the Hilbert scheme.
For \(q=0\), the operator \(\mathsf M_D(0)\) has as eigenvalues the Jack symmetric functions \(\mathsf J^{\lambda}\) with eigenvalues
\[
c(\lambda, t_1, t_2)=\sum_{(i,j)\in \lambda}((j-1)t_1+(i-1)t_2),
\]
where \(\lambda\) is a partition. A solution of the ODE of the form
\[
\Psi=\mathsf Y^{\lambda}(q)q^{-c(\lambda, t_1, t_2)},\,\, \mathsf Y^{\lambda}(0)=\mathsf J^{\lambda}
\]
with \(\mathsf Y^{\lambda}(q)\in \mathbb C[[q]]\) can be constructed for \(|q|<1\). The value of \(\mathsf Y^{\lambda}(q)\) for \(q=-1\) is found and related to the Macdonald polynomials. Okounkov, A.; Pandharipande, R., The quantum differential equation of the Hilbert scheme of points in the plane, Transform. Groups, 15, 965, (2010) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Relationships between surfaces, higher-dimensional varieties, and physics The quantum differential equation of the Hilbert scheme of points in the plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a scheme \(X\) over \(k\), a generalized jet scheme parametrizes maps \(\operatorname{Spec} A \to X\), where \(A\) is a finite-dimensional, local algebra over \(k\). We give an overview of known results concerning the dimensions of these schemes for \(A=k[t]/(t^m)\), when they are related to invariants of singularities in birational geometry. We end with a discussion of more general jet schemes. jet scheme; log canonical threshold; minimal log discrepancy Arcs and motivic integration, Singularities in algebraic geometry The dimension of jet schemes of singular varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper gives irreducibility results on Hilbert schemes of space curves. Let \(H_{d,g}\) denote the Hilbert scheme parametrizing smooth connected curves \(C \subset \mathbb P^3\) of degree \(d\) and genus \(g\). Confirming part of a claim of Severi, \textit{L. Ein} showed that \(H_{d,g}\) is irreducible for \(d \geq g+3\) [Ann. Scient. Ec. Norm. Sup. 19, 469--478 (1986; Zbl 0606.14003)]. The authors proved irreducibility of \(H_{d,g}\) for \(d=g+2, g \geq 5\) and \(d=g+1, g \geq 11\) [J. Algebra 145, 240--248 (1992; Zbl 0783.14002)] and \textit{H. Iliev} proved that \(H_{g,g}\) is irreducible for \(g \geq 13\) [Proc. Amer. Math. Soc. 134, 2823--2832 (2006; Zbl 1097.14022)].
Here the authors complete Iliev's result, showing that every non-empty \(H_{g,g}\) is irreducible. Noting that \(H_{g,g}\) is empty for \(1 \leq g \leq 7\) and irreducible for \(g=8\) and \(9\) by \textit{K. Dasaratha} [``The reducibility and dimension of Hilbert schemes of complex projective curves'', undergraduate thesis, Harvard University, Department of Mathematics, available at \url{http://www.math.harvard.edu/theses/senior/dasaratha/dasaratha}], the authors need only consider \(g \geq 10\). Here they build on the proof of Iliev [loc. cit.], making a careful study of an irreducible component \(\mathcal G\) of the space \(\mathcal G_g^3\) of pairs \((C,D)\) with \(C\) a smooth connected curve and \(D\) a linear series of degree \(g\) and dimension \(3\) whose general element is very ample. The key result here is that \(D\) is in fact a complete linear system of dimension \(4g-15\) and that the general member of the component of the residual series is base point free, complete, and birationally very ample. Hilbert schemes; space curves \textsc{C. Keem and Y.-H. Kim}, Irreducibility of the Hilbert scheme of smooth curves in \({\mathbb{P}}^3\), Arch. Math. \textbf{108} (2017), 593-600. Plane and space curves, Parametrization (Chow and Hilbert schemes) Irreducibility of the Hilbert scheme of smooth curves in \(\mathbb {P}^3\) of degree \(g\) and genus \(g\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Many authors have investigated the claim of \textit{F. Severi} [Vorlesungen über algebraische Geometrie. (Übersetzung von E. Löffler.). Leipzig-Berlin: B. G. Teubner (1921; JFM 48.0687.01)] that the Hilbert scheme \({\mathcal H}_{d,g,r}\) of smooth connected non-degenerate curves of degree \(d\) and genus \(g\) in \(\mathbb P^r\) is irreducible in the Brill-Noether range \(0 \leq \rho (d,g,r) = g - (r+1)(g-d+r)\), in which case there is a distinguished component of \({\mathcal H}_{d,g,r}\) dominating the moduli space \(\mathcal M_g\) of genus \(g\) curves. While Severi's claim is false, in each counterexample the locus \({\mathcal H}^L_{d,g,r}\) of linearly normal curves is irreducible and it is possible that this is what Severi intended in the first place (see [\textit{C. Ciliberto} and \textit{E. Sernesi}, in: Proceedings of the first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987. Teaneck, NJ: World Scientific Publishing Co.. 428--499 (1989; Zbl 0800.14002)] and Lopez in his Mathematical Review of [\textit{C. Keem}, Proc. Am. Math. Soc. 122, No. 2, 349--354 (1994; Zbl 0860.14003)]).
In the paper under review the authors study the case \(r=4\) and \(d=g+1\) outside the Brill-Noether range, meaning that \(g \leq 14\). In their previous work [\textit{C. Keem} and \textit{Y.-H. Kim}, Arch. Math. 113, No. 4, 373--384 (2019; Zbl 1423.14028)] they showed that \({\mathcal H}_{g+1,g,4}\) is empty for \(g \leq 8\), \({\mathcal H}_{10,9,4}\) is reducible, \({\mathcal H}_{11,10,4} = {\mathcal H}^L_{11,10,4}\) is irreducible, and \({\mathcal H}_{13,12,4}\) is reducible. Therefore they focus here on the cases \(g = 11, 13, 14\), where they show through a case by case analysis that \({\mathcal H}_{g+1,g,4} = {\mathcal H}^L_{g+1,g,4}\) is irreducible and generically reduced. Hilbert scheme; algebraic curves; linear series; Brill-Noether range Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) On the Hilbert scheme of smooth curves in \(\mathbb{P}^4\) of degree \(d=g+1\) and genus \(g\) with negative Brill-Noether number | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors introduce symmetric obstruction theories, and compute the weighted Euler characteristic for schemes \(X\) that admit a \(\mathrm{G}_m\)-action and a compatible symmetric obstruction theory. They show that the weighted Euler characteristic is given by the formula
\[
\tilde{\chi} (X) = \sum (-1)^{\mathrm{dim}T_{X\mid P}},
\]
where the sum runs over the fixed points for the \(\mathrm{G}_m\)-action, where the fixed points are assumed finite and isolated. If the scheme \(X\) furthermore is projective, their formula above expresses the virtual count; the degree of the associated virtual fundamental class of \(X\).
A scheme which is locally the critical locus of a regular function on a smooth manifold, have a canonical symmetric obstruction theory. An example of such a scheme is the Hilbert scheme \(\mathrm{Hilb}^nY\) of \(n\)-points on a smooth three-fold \(Y\).
In the article it is shown that the weighted Euler characteristic of \(\mathrm{Hilb}^nY\), where \(Y\) is a smooth three-fold, is up to a sign, the ordinary Euler characteristic of \(\mathrm{Hilb}^nY\). For projective \(Y\) they then obtain the formula for the virtual count of \(\mathrm{Hilb}^nY\), as conjectured in [\textit{D. Maulik, N. Nekrasov, A. Okounkov} and \textit{R. Pandharipande}, Compos. Math. 142, No. 5, 1263--1285 (2006; Zbl 1108.14046)] symmetric obstruction theories; Hilbert schemes; Calabi-Yau threefolds; \(C^*\) actions; Donaldson-Thomas invariants; MNOP conjecture K. Behrend and B. Fantechi, 'Symmetric obstruction theories and Hilbert schemes of points on threefolds', \textit{Algebra Number Theory}2 (2008) 313-345. Parametrization (Chow and Hilbert schemes), Calabi-Yau manifolds (algebro-geometric aspects), \(3\)-folds Symmetric obstruction theories and Hilbert schemes of points on threefolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities From the fact that a foliation by curves of degree greater than one, with isolated singularities, in the complex projective plane \(\mathbb P^2\) is uniquely determined by its subscheme of singular points (the singular subscheme of the foliation), we pose the problem of existence of proper closed subschemes \(Z\) of the singular subscheme which still determine the foliation in a unique way. We prove the existence of such subschemes \(Z\) for foliations with reduced singular subscheme. Unlike the degree \(\deg Z\) of such subschemes is not sharp for the posed problem, we show that it is so in the sense that \(Z\) defines the so-called polar net of the foliation. Campillo, A., Olivares, J.: Special subschemes of the scheme of singularities of a plane foliation. Comptes Rendus de l'Académie de Sciences - Série I - Mathématiques 344(9), 581-585 (2007) Singularities of holomorphic vector fields and foliations, Cycles and subschemes Special subschemes of the scheme of singularities of a plane foliation | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be an Azumaya algebra over a smooth projective variety \(X\) or more generally, a torsion free coherent sheaf of algebras over \(X\) whose generic fiber is a central simple algebra. The authors show that generically simple torsion free \(A\)-module sheaves have a projective coarse moduli scheme; it is smooth and even symplectic if \(X\) is an abelian or \(K3\) surface and \(A\) is Azumaya. The relation to the classical theory of Brandt groupoid is explained. moduli space; torsion free sheaf; Azumaya module; Brandt groupoid Hoffmann, N.; Stuhler, U.: Moduli schemes of generically simple Azumaya modules. Doc. math. 10, 369-389 (2005) Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) Moduli schemes of generically simple Azumaya modules | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert scheme of points Hilb\(_n(\mathbb{C}^2)\) is the crepant resolution of the orbifold quotient \((\mathbb{C}^2)^n/S_n\) containing the configurations of \(n\) distinct points in \(\mathbb{C}^2\). Its geometry and algebraic invariants have been a subject of intense study in the last decade due to connections with the string theory. In particular, the ring structure of its torus equivariant quantum cohomology has been established (the natural action of the complex torus on \((\mathbb{C}^2)^n/S_n\) extends to the Hilbert scheme).
In the paper under review, the authors give an explicit formula for \(M_D\), the operator of multiplication by the first Chern class \(D\) of the Hilbert scheme in small quantum cohomology. The matrix elements of \(M_D\) give counts of rational curves meeting three given subvarieties of the scheme, and an associated differential equation is an integrable non-stationary deformation of the Calogero-Sutherland equation for quantum particles on a torus. The formula implies that \(D\) generates the small quantum cohomology ring, and completes establishing the four-way correspondence between the quantum cohomology of Hilb\(_n(\mathbb{C}^2)\), the equivariant orbifold cohomology of \((\mathbb{C}^2)^n/S_n\), and the Gromov-Witten and Donaldson-Thomas invariants of \(\mathbb{P}^1\times\mathbb{C}^2\). Hilbert scheme of points; small quantum cohomology; Calogero-Sutherland operator A. Okounkov and R. Pandharipande, \textit{Quantum cohomology of the Hilbert scheme of points in the plane}, math/0411210. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Quantum cohomology of the Hilbert scheme of points in the plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S \rightarrow C\) be a smooth projective surface with numerically trivial canonical bundle fibered onto a curve. We prove the multiplicativity of the perverse filtration with respect to the cup product on \(H^\ast(S^{[n]}, \mathbb{Q})\) for the natural morphism \(S^{[n]} \rightarrow C^{(n)}\). We also prove the multiplicativity for five families of Hitchin systems obtained in a similar way and compute the perverse numbers of the Hitchin moduli spaces. We show that for small values of \(n\) the perverse numbers match the predictions of the numerical version of the de Cataldo-Hausel-Migliorini \(P = W\) conjecture and of the conjecture by \textit{T. Hausel} et al. [Duke Math. J. 160, No. 2, 323--400 (2011; Zbl 1246.14063); Adv. Math. 234, 85--128 (2013; Zbl 1273.14101)]. Hitchin fibration; Hilbert scheme; perverse filtration Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Parametrization (Chow and Hilbert schemes) Multiplicativity of perverse filtration for Hilbert schemes of fibered surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study the local Gromov-Witten invariants of \(\mathcal{O} (k)\oplus \mathcal{O}( - k - 2) \rightarrow \mathbb P^{1}\) by localization techniques and the Mariño-Vafa formula, using suitable circle actions. They are identified with the equivariant Riemann-Roch indices of some power of the determinant of the tautological sheaves on the Hilbert schemes of points on the affine plane. We also compute the corresponding Gopakumar-Vafa invariants and make some conjectures about them. local Gromov-Witten invariants; localization; Riemann-Roch indices; Gopakumar-Vafa invariants Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Local Gromov-Witten invariants and tautological sheaves on 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 Consider a smooth family of irreducible surfaces \(F/Y\) and an integer \(n \geq 0\). Let us consider a \textit{sequence of arbitrarily near \(T\)-points of} \(F/Y\): it is a \((n+1)\)-tuple \((t_0, \dots, t_n)\) where \(t_0\) is a \(T\) point of \(F_T^{(0)}:=F \times_{Y} T\) and where \(t_i\) (\(i \geq 1\)) is a \(T\)-point of the blowup \(F_T^{(i)}\) of \(F_T^{(i-1)}\) at \(t_{i-1}\). These sequences form a functor in \(T\), representable by a smooth \(Y\)-scheme \(F^{(n)}\). The sequence \((t_0, \dots, t_n)\) is \textit{strict} if for each \(i,j\) with \(1 \leq j \leq i\), the image \(T^{(i)}\) of \(t_i\) is either disjoint from or contained in the strict transform of the exceptional divisor \(E_T^{(j)}\) of \(F_T^{(j)}\). To each strict sequence an unweighted Enriques diagram is associated and it is shown that the various sequences with a fixed diagram form a functor which is represented by a smooth \(Y\)-scheme.
Let us continue this review with the second paragraph of the abstract:
``We equip this \(Y\)-scheme with a free action of the automorphism group of the diagram. We equip the diagram with weights, take the subgroup of those automorphisms preserving the weights, and form the corresponding quotient scheme. Our main theorem constructs a canonical universally injective map from this quotient scheme to the Hilbert scheme of \(F/Y\); further, this map is an embedding in characteristic \(0\). However, in every positive characteristic, we give an example, in Appendix B, where the map is purely inseparable.'' arbitrarily near points; Enriques diagrams; Hilbert schemes Kleiman, S., Piene, R., Tyomkin, I.: Enriques diagrams, arbitrarily near points, and Hilbert schemes. Rendiconti Lincei-Matematica e Applicazioni. 22(4), 411--451 (2011) Enumerative problems (combinatorial problems) in algebraic geometry, Divisors, linear systems, invertible sheaves, Jacobians, Prym varieties, Algebraic theory of abelian varieties Enriques diagrams, arbitrarily near points, and 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 Vector bundles on noncommutative spaces play an important role in representation theory. The authors claim to make the first systematic application of moduli-theoretic techniques from algebraic geometry to a problem in this area, the study of coherent sheaves on noncommutative projective planes. In accordance with Grothendieck's philosophy, a noncommutative projective plane is an abelian category with the fundamental properties of the category \(\text{coh}(\mathbb{P}^2)\) of coherent sheaves on the projective plane. For a connected graded noetherian algebra \(S\), one regards \(\text{qgr}-S\), the category of finitely generated graded \(S\)-modules modulo those of finite length, as the category of coherent sheaves on the noncommutative projective variety \(\text{Proj}(S).\) The noncommutative analogues of \(\text{coh}(\mathbb{P}^2)\) are exactly the categories of the form \(\text{qgr}-S\) where \(S\) is an Artin Schelter (AR) regular algebra with the Hilbert series \((1-t)^{-3}\) of a polynomial ring in three variables \((S\in\underline{AS}_3.)\) Earlier work relates a noncommutative plane \(\mathbb{P}^2_{\hbar}\) to the first Weyl algebra \(A_1\). Then one can regard \(\mathbb{P}^2_{\hbar}\) as a deformation of \(\mathbb{P}^2\), \(A_1\) as a deformation of the ring of functions on \(\mathbb{A}^2\subset\mathbb{P}^2\), and the line bundles on \(\mathbb{P}^2_{\hbar}\) with \(c_2=n\) correspond naturally to points of a deformation of \((\mathbb{A}^2)^{[n]},\) the Hilbert scheme of \(n\) points in the plane.
The authors generalize this by working out the following results: The plane \(\mathbb{P}^2_{\hbar}\) is one of several families of noncommutative planes. The authors construct moduli spaces that classify vector bundles and torsion-free coherent sheaves on all such planes. They prove that previous results are special cases of this. The authors prove that these moduli spaces behave well in families. For example, when the noncommutative plane is a deformation of \(\mathbb{P}^2,\) this provides a deformation of the Hilbert schemes of points \((\mathbb{P}^2)^{[n]}.\) Finally, the authors prove that the Hilbert scheme \((\mathbb{A}^2)^{[n]}\) has a symplectic structure induced by the hyperkähler metric. They construct Poisson and symplectic structures on the analogues moduli spaces and study the resulting Poisson and symplectic geometry. The algebras \(S\in\underline{AS}_3\) have been classified in terms of geometric data. When \(\text{qgr}-S\ncong\text{coh}(\mathbb{P}^2),\) \(S\) is determined by the commutative data \((E,\sigma)\) where \(E\hookrightarrow\mathbb{P}^2\) is a plane cubic curve and \(\sigma\in\text{Aut}(E)\) is a nontrivial automorphism. Thus the authors may write \(S=S(E,\sigma).\) A key fact is that \(\text{coh}(E)\simeq\text{qgr}(S/gS)\subset\text{qgr}-S\) for an element \(g\in S_3\) unique up to scalar multiplication, and this inclusion has a left adjoint of restricion to \(E\). The authors incorporate the categories \(\text{qgr}-S\simeq\text{coh}(\mathbb{P}^2)\) by letting \(\sigma\) be trivial and \(E\) any cubic. The generic example is the \textit{Sklyanin algebra} \(\text{Skl}(E,\sigma)\) which is determined by a smooth elliptic curve \(E\) and an automorphism \(\sigma\) given by translation under the group law. Noncommutative projective planes have all the basic properties of \(\text{coh}(\mathbb{P}^2)\) and therefore admit natural definitions of vector bundles and torsion-free sheaves as well as invariants like Euler characteristic and Chern classes. This gives meaning to the general results in the article. One key element is the thoroughgoing use of the cohomological tools from commutative algebraic geometry, primarily cohomology and base change.
The authors are going through the background material in a detailed and understandable way, making it possible to understand the rest of the article. One has to mention the section where the authors show that an analogue of the Beilinson spectral sequence also works for vector bundles in a noncommutative \(\mathbb{P}^2\), and that this can be used to construct a projective moduli space as a GIT quotient of a subvariety of a product of Grassmannians. The authors construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogues to those of moduli spaces of sheaves over the usual plane \(\mathbb{P}^2.\) In case of the Sklyanin algebra \(S=\text{Skl}(E,\sigma)\), the fine moduli space of line bundles over \(S\) with first Chern class zero and Euler characteristic \(1-n\) provides a symplectic variety that is a deformation of the Hilbert scheme of \(n\) points on \(\mathbb{P}^2\backslash E.\) moduli spaces; noncommutative projective geometry; symplectic structure Nevins T.A., Stafford J.T., Sklyanin algebras and Hilbert schemes of points, Adv. Math., 2007, 210(2), 405--478 Noncommutative algebraic geometry, Parametrization (Chow and Hilbert schemes), Fine and coarse moduli spaces, Free, projective, and flat modules and ideals in associative algebras, Rings arising from noncommutative algebraic geometry, Grothendieck categories, Symplectic structures of moduli spaces Sklyanin algebras and Hilbert schemes of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Marc Giusti classified the simple isolated complete intersection singularities of positive dimension and showed that besides the simple hypersurface singularities there are only simple singularities of curves in complex 3-space. He also gave a list of adjacencies between singularities. Later V. V. Goryunov found some additional adjacencies, but the complete list remained unknown.
The authors of this paper show the complete list of adjacencies between the simple space curve singularities which are not hypersurface singularities and complete Giusti's list by two new deformations. deformation; space curve; singularity DOI: 10.1007/PL00004672 Deformations of complex singularities; vanishing cycles, Deformations of singularities The deformations of the simple space curve singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert scheme of \(n\) points in the projective plane parameterizes degree \(n\) zero-dimensional subschemes of the projective plane. We examine the dual cones of effective divisors and moving curves on the Hilbert scheme. By studying interpolation, restriction, and stability properties of certain vector bundles on the plane, we fully determine these cones for just over three-fourths of all values of \(n\). A \textit{general Steiner bundle} on \(\mathbb P^N\) is a vector bundle \(E\) admitting a resolution of the form
\[
0\rightarrow\mathcal O_{\mathbb P^N}(-1)^s\overset {M}\rightarrow\mathcal O_{\mathbb P^N}^{s+r}\rightarrow E\rightarrow 0,
\]
where the map \(M\) is general. We complete the classification of slopes of semistable Steiner bundles on \(\mathbb P^N\) by showing every admissible slope is realized by a bundle which restricts itself to a balanced bundle on a rational curve. The proof involves a basic question about multiplication of polynomials on \(\mathbb P^1\) which is interesting in its own right. Jack Huizenga, Restrictions of Steiner bundles and divisors on the Hilbert scheme of points in the plane, Int. Math. Res. Not. IMRN 21 (2013), 4829 -- 4873. Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Minimal model program (Mori theory, extremal rays), Parametrization (Chow and Hilbert schemes) Restrictions of Steiner bundles and divisors on the Hilbert scheme of points in the plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The purpose of this paper is to describe the generating series of classes of Hilbert schemes of points on complex orbifolds. The computation takes place in the Grothendieck ring of complex quasi-projective varieties and uses the notion of `power structure' in this ring (a method of giving meaning to the power of a power series whose coefficients come from a commutative ring). The authors do on excellent job of making the paper economical and direct. Hilbert schemes of points; orbifolds; Grothendieck ring; generating series Parametrization (Chow and Hilbert schemes), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Topology and geometry of orbifolds Generating series of classes of Hilbert schemes of points on 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 We study the structure of the relative Hilbert scheme for a family of nodal (or smooth) curves via its natural cycle map to the relative symmetric product. We show that the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We discuss some applications and connections, notably with birational geometry and intersection theory on Hilbert schemes of smooth surfaces. Revised version corrects some minor errors. Z. Ran, \textit{Cycle map on Hilbert schemes of nodal curves}, in \textit{Projective Varieties with Unexpected Properties}, Walter De Gruyter, Berlin, 2005, pp. 363-380. Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) Cycle map on Hilbert schemes of nodal 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 Suppose that \(H^3_{d,g}\) is the Hilbert scheme of smooth curves of degree d and genus g in \({\mathbb{P}}^ 3\). If \(g\leq (d^{3/2}/(6\cdot 2^{1/2})+\text{ lower terms}),\) the author constructs a component V of \(H^ 3_{d,g}\) satisfying the following properties:
1. If C is a general curve of V, then \(H^ 1(N_{C/{\mathbb{P}}^ 3})=0\). Hence V is generically smooth and \(\dim (V)=4d.\)
2. The dimension of the image of V in the \(M_ g\), the moduli space of curve, is equal to \(\text{Min}(3g-3,3g-3-(\rho(d,g,3))\).
The method of proof is by smoothing reducible nodal curves. Hilbert scheme; moduli space of curve; smoothing reducible nodal curves Pareschi, G, Components of the Hilbert scheme of smooth space curves with the expected number of moduli, Manuscr. Math., 63, 1-16, (1989) Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) Components of the Hilbert scheme of smooth space curves with the expected number of moduli | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 0626.00011.]
This paper is an excellent survey, written mainly for non-experts, concerning the recent developments in the theory of Hilbert schemes of points. Four main themes are discussed, namely: (1) Punctual Hilbert scheme of a surface and of a curve lying on it, (2) The local punctual Hilbert scheme and the mapping germs, (3) The geometry of \(Hilb^ nY\) and foundations, (4) Extensions to modules, applications and vector bundles.
The author succeeded to give some feeling for this subject, illustrating the theory by many examples. The paper can also be considered as reflecting the actual state of affairs in the field. An extensive bibliography is included. Bibliography; zero cycles; symmetric products; Hilbert schemes [I3] Iarrobino, A.: Hilbert Scheme of Points: Overview of Last Ten Years. Proc. of Symp. in Pure Math. Vol.46 Part 2, Algebraic Geometry, Bowdoin 1987, 297--320 Parametrization (Chow and Hilbert schemes), Algebraic cycles, History of algebraic geometry, History of mathematics in the 20th century Hilbert scheme of points: Overview of last ten years | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Using \(K\)-theory, we construct a map \(\pi:T_Y\mathrm{Hilb}^p(X)\rightarrow H_y^p(\Omega_{X/\mathbb Q}^{p-1})\) from the tangent space to the Hilbert scheme at a point \(Y\) to the local cohomology group. We use this map \(\pi\) to answer (after slight modification) a question by [\text it{M. Green} and \text it{P. Griffiths}, On the tangent space to the space of algebraic cycles on a smooth algebraic variety. Princeton, NJ: Princeton University Press (2005; Zbl 1076.14016)] on constructing a map from the tangent space \(T_Y\mathrm{Hilb}^p(X)\) to the Hilbert scheme at a point \(Y\) to the tangent space to the cycle group \(TZ^p(X)\). deformation of cycles; tangent spaces to cycle groups; \(K\)-theory; Chern character; tangent spaces to Hilbert schemes; Koszul complex; Newton class; absolute differentials Applications of methods of algebraic \(K\)-theory in algebraic geometry, Algebraic cycles, Parametrization (Chow and Hilbert schemes), \(K\)-theory in geometry \(K\)-theory, local cohomology and tangent spaces to 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 The author studies certain numerically Gorenstein elliptic singularities of normal surfaces with \(\mathbb{C}^*\) action. The weighted dual graph of the exceptional divisor in a minimal good resolution is determined. A condition for such a singularity to be maximally elliptic (in the sense of Yau) is also obtained. For a definition (somewhat technical) of maximally elliptic singularity, the reader may refer to \textit{S. S.-T. Yau}'s paper [Trans. Am. Math. Soc. 257, 269-329 (1980; Zbl 0343.32009)]. elliptic singularities of normal surfaces; weighted dual graph of the exceptional divisor Singularities of surfaces or higher-dimensional varieties, Group actions on varieties or schemes (quotients) On numerically Gorenstein quasi-simple elliptic singularities with \(\mathbb{C}^*\)-action | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review gives an explicit example of a pair of Calabi--Yau threefolds \(X\) and \(Y\) such that (1) \(X\) and \(Y\) are birational and rigid, (2) \(X\) and \(Y\) are not homeomorphic, but (3) \(X\) and \(Y\) are connected by a projective flat deformation over some connected scheme.
The two Calabi--Yau threefolds satisfying the above properties are constructed as follows. Let \(\zeta=e^{2\pi i/3}\) and let \(E_\zeta = \mathbb{C}/\mathbb{Z} + \zeta\mathbb{Z}\) be the elliptic curve with period \(\zeta\). Let \(Q_0 = 0\), \(Q_1=(1-\zeta)/3\) and \(Q_2=-(1-\zeta)/3\) be the fixed points in \(E_\zeta\) of the scalar multiplication by \(\zeta\) on \(E_\zeta\). Let \(E_\zeta^3/<\zeta >\) be the quotient of the threefold product \(E_\zeta^3\) by the scalar multiplication by \(\zeta\). For \({i, j, k}\in\{0, 1, 2\}\), let \(Q_{ijk} = (Q_i, Q_j, Q_k)\in E_\zeta^3\) and \(\bar{Q}_{ijk}\) to be its image in \(E_\zeta/<\zeta >\). The quotient \(E_\zeta^3/<\zeta >\) has singularities of type \(\frac13 (1, 1, 1)\) at \(\bar{Q}_{ijk}\).
Blowing up at these 27 singular points, one gets a rigid Calabi--Yau threefold of \textit{A. Beauville} [Prog. Math. 39, 1--26 (1983; Zbl 0537.53057)], and this is the rigid Calabi--Yau threefold \(X\).
Denote by \(E_{ijk}\) the exceptional divisor lying over \(\bar{Q}_{ijk}\), then \(E_{ijk}\) is isomorphic to \(\mathbb{P}^2\). Let \(p : X \to E_\zeta^2/<\zeta >\) be the morphism induced by the projection \(pr_{12} :E_\zeta^3\to E_\zeta^2\). Then \(p^{-1}\bar{Q}_{ij} = l_{ij}\cup E_{ij0}\cup E_{ij1}\cup E_{ij3}\) where \(l_{ij}\) is a smooth rational curve meeting \(E_{ijk}\) transversally. Performing the elementary transformation along \(\bigcup_{i,j} l_{ij}\), one gets a smooth rigid Calabi--Yau threefold of \textit{K. Oguiso} [Math. Z. 221, No. 3, 437--448 (1996; Zbl 0852.14012)], and this is the rigid Calabi--Yau threefold \(Y\).
It is shown that \(X\) and \(Y\) are non-homeomorphic, but are connected by a projective flat deformation. Calabi-Yau threefold; rigid; non-homeomorphic; deformation Lee N H, Oguiso K. Connecting certain rigid birational non-homeomorphic Calabi-Yau threefolds via Hilbert scheme. Comm Anal Geom, 2009, 17: 283--303 Calabi-Yau manifolds (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Birational automorphisms, Cremona group and generalizations Connecting certain rigid birational non-homeomorphic Calabi-Yau threefolds via Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S\) be a smooth projective surface and denote with \(S^{[n]}\) the Hilbert scheme that parameterizes length \(n\) subschemes of \(S\). There exists a natural map from \(S^{[n]}\) to the symmetric power \(S^{(n)}\), whose fibers over multiplicity \(n\) cycles define the \textit{punctual Hilbert scheme} \(P_n\). In other words, \(P_n\) parameterizes subschemes of length \(n\) supported at one point. The authors study the tangent space \(T_\xi\) to \(P_n\) at a scheme \(\xi\in P_n\). The dimension \(\dim(T_\xi) \) is bounded below by the corank of the normal map \(\alpha_{n,\xi}\) which sends \(H^0(S, T_{S|\xi})\) to Hom\((\mathcal I_\xi,\mathcal O_\xi)\). The authors prove that when the ideal \(I_\xi\) of \(\xi\) is a monomial ideal, then \(\dim(T_\xi) \) is indeed equal to the corank of \(\alpha_{n,\xi}\). They also show how, for schemes \(\xi\) defined by monomials, the corank of \(\alpha_{n,\xi}\) can be computed from the Young diagram associated to \(I_\xi\). Hilbert scheme Parametrization (Chow and Hilbert schemes) The tangent space of the punctual 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 Motivated by the minimal model program of the moduli spaces of Bridgeland-semistable sheaves on surfaces, this paper computes the nef cones of the nested Hilbert scheme \(X^{[n+1,n]} \subset X^{[n+1]} \times X^{[n]}\) and the universal family \(Z=X^{[n,1]}\) of the Hilbert scheme \(X^{[n]}\) of \(n\)-points, where \(X\) is the projective plane, a Hirzebruch surface, or a \(K3\) surface of Picard rank \(1\). Several interesting questions are also concluded in the last section. nested Hilbert schemes; nef cones Parametrization (Chow and Hilbert schemes), Rational and birational maps, Rational and ruled surfaces, \(K3\) surfaces and Enriques surfaces Nef cones of nested Hilbert schemes of points on surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article strengthens the connections between Cherednik algebras and geometry by showing that they can be regarded as noncommutative deformations of Hilbert schemes of points in the plane. The article explicitly defines the rational Cherednik algebra \(H_c\) of type \(A_{n-1}\), and its spherical subalgebra \(U_c\). Let \(W=\mathfrak S_n\) be the symmetry group on \(n\) letters, regarded as the Weyl group of type \(A_{n-1}\) acting on its \((n-1)\)-dimensional representation \(\mathfrak h\in\mathbb{C}^n\) by permutations. Then \(H_c\) may be regarded as a deformation of the twisted group ring \(D(\mathfrak h)\ast W\), where \(D(\mathfrak h)\) is the ring of differential operators on \(\mathfrak h \) with the natural action of the symmetric group \(W=\mathfrak S_n.\) The algebra \(U_c\) is then the corresponding deformation of the fixed ring \(D(\mathfrak h)^W.\)
\(U_c\) and \(H_c\) both have a natural filtration by order of differential operators, with associated graded rings \(\operatorname{gr}U_c\cong\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]^W\) and \(\operatorname{gr}H_c\cong\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]\ast W\), and so \(U_c\) may be regarded as a deformation of \(\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]^W.\) The article is mostly concerned about \(U_c\), but the authors prove that \(U_c\) and \(H_c\) are Morita equivalent.
It is known that the map \(\tau:\text{Hilb}^n\mathbb{C}\rightarrow \mathbb{C}^{2n}/W\) is a resolution of singularities, and Haiman has described \(\text{Hilb}^n\) as ``Proj'' of Rees rings. This article proves that for \(\text{Hilb}(n)=\tau^{-1}(\mathfrak h\oplus\mathfrak h^\ast/W)\), \(\tau:\text{Hilb}(n)\rightarrow \mathfrak h\oplus\mathfrak h^\ast/W \) is a crepant resolution of singularities. The ring \(U_c\) has finite global homological dimension so one should expect the properties of a smooth deformation of \(\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]^W,\) that is, its properties should be more closely related to those of \(\text{Hilb}(n)\) than to \(\mathfrak h\oplus\mathfrak h^\ast/W\). The main ideal of this article is to formalize this idea by showing that there exists a second way of passing to associated graded objects that maps \(U_c\)-modules precisely to coherent modules on \(\text{Hilb}(n).\)
The main result of this article gives an affirmative answer to the question of whether the following diagram can be completed:
\[
\begin{tikzcd} {?} \ar[d,"\mathrm{gr}" '] & U_c \ar[l,"\sim" ']\ar[d,"\mathrm{gr}"] \\ \mathcal O_{\mathrm{Hilb}(n)} & \mathcal O(\mathfrak h\oplus\mathfrak h^\ast/W) \ar[l,"\tau" ']\rlap{\,.} \end{tikzcd}
\]
Given a graded ring \(R\), write \(R\)-qgr for the quotient category of noetherian graded \(R\)-modules modulo those of finite length. The main result then says that
There exists a graded ring \(B\), filtered by order of differential operators, such that
(1) there is an equivalence of categories \(U_c\text{-mod}\cong B\text{-qgr};\)
(2) there is an equivalence of categories \(\text{gr }B\text{-qgr}\cong\text{Coh}(\text{Hilb}(n))\).
The construction of \(B\) is not the same as Haiman's construction of the Rees ring, but a noncommutative deformation. This leads to the theory of \(\mathbb{Z}\)-algebras and Poincaré series in this setting. The authors then need results concerning Morita equivalence of the Cherednik algebras. The authors give explicit definitions and results about the Cherednik algebras and its representations. They study thoroughly the Morita equivalence of these algebras. Then Haiman's constructions of the Hilbert scheme and the resulting formulas for the Poincaré series are derived, and a geometric interpretation is given. Finally, \((\mathfrak h\oplus\mathfrak h^\ast)/W)\) is blown up.
In the noncommutative case, the Rees rings have to be replaced by noncommutative analogs. These are called \(\mathbb{Z}\)-algebras and a nice introduction is given here. The main theorem is then proved by introducing the noncommutative Poincaré series, and this is higly nontrivial.
This article is a very nice introduction to the theory and problems of noncommutative (graded) projective geometry. \(\mathbb Z\)-algebras; resolution of singularities Opdam, E.: Complex reflection groups and fake degrees (1998) arXiv:math/9808026\textbf{(preprint)} Parametrization (Chow and Hilbert schemes), Modifications; resolution of singularities (complex-analytic aspects), Deformations of associative rings, Module categories in associative algebras, Noncommutative algebraic geometry Rational Cherednik algebras and Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X,Y\) be curve singularities, i.e. one-dimensional, reduced and closed subschemes of \((k^ n,0) = \text{Spec} (R)\). The contact between \(X\) and \(Y\), i.e. \((X.Y) = \dim_ k (R/ (I(X) + I(Y)))\), is expressed, in the case of plane curves, by Noether's formula in terms of infinitely near points. In this paper the author generalizes Noether's formula to space curve singularities, giving an upper bound of the contact of a pair of curve singularities in terms of the degrees of minimal standard bases of the ideals. The proofs are based on some results on the Hilbert function of torsion free modules of rank one. intersection; contact; infinitely near points; space curve singularities; Hilbert function Singularities of curves, local rings, Plane and space curves, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry The zero dimensional scheme defined by the intersection of two 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 It is known that any component of the Hilbert scheme which parametrizes locally Cohen-Macaulay curves in \(\mathbb{P}^3\) has dimension \(\geq 4d\), where \(d\) is the common degree of the curves [see \textit{L. Ein}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 469-478 (1986; Zbl 0606.14003)]. In the present paper the authors show that the lower bound of \(4d\) is valid for any irreducible component of the Hilbert scheme of space curves.
In the first section, the authors show that there are components where the general point represents a nonlocally Cohen-Macaulay curve which is not a disjoint union of a locally Cohen-Macaulay curve and a zero-dimensional scheme. Despite of the existence of such components, in the second section the authors show the quoted bound. To this purpose, they ask if there is an example of a component as before, with dimension exactly equal to \(4d\). Taking into account the work of \textit{H. Kleppe} [J. Algebra 53, 84-92 (1978; Zbl 0384.14004)], one can improve this lower bound when the curve corresponding to the generic point of the component has an embedded component which is a Gorenstein zero-dimensional scheme. deformation theory; obstruction; Hilbert scheme; locally Cohen-Macaulay curves; Gorenstein zero-dimensional scheme Parametrization (Chow and Hilbert schemes), Plane and space curves, Formal methods and deformations in algebraic geometry, Deformations and infinitesimal methods in commutative ring theory, Local deformation theory, Artin approximation, etc., Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Dimension of the Hilbert scheme of curves in \(\mathbb{P}^ 3\) | 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 The authors compute a cell decomposition of the Hilbert scheme \(\text{Hilb}^d(\mathbb{P}^2)\)
induced by the theorem of \textit{A. Bialynicki-Birula} [Ann. Math.~(2)~98, 480--497 (1973; Zbl 0275.14007)] when applied to the natural torus action.
The existence of such a decomposition was already used by the authors to compute the Betti numbers of
\(\text{Hilb}^d(\mathbb{P}^2)\) [Invent. Math. 87, 343--352 (1987; Zbl 0625.14002)] and can be explicitly given by a slight modification of the argument used there.
Each cell is in some \(\text{Hilb}^d(\mathbb{A}^2)\) and is constructed via invariant ideals of colength \(d\) having a prescribed resolution (in this way, they are parametrized by the partitions of \(d\)). The authors construct to each partition a corresponding quasifinite and flat family of subschemes. It should be noted that this family is not always finite over their base (as used in the paper), however, the main results still hold due to the corrections of \textit{M. Huibregtse} [Invent. Math. 160, 165--172 (2005; Zbl 1064.14005)]. G. Ellingsrud et S. Strømme , On a cell decomposition of the Hilbert scheme of points in the plane , Inv. Math. 91 (1988) 365-370. Parametrization (Chow and Hilbert schemes), (Equivariant) Chow groups and rings; motives, Fine and coarse moduli spaces On a cell decomposition of the Hilbert scheme of points in the plane. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{E. Witten} [Commun. Math. Phys. 121, No. 3, 351--399 (1989; Zbl 0667.57005)] interpreted the Jones invariants of links in \(S^3\) (such as the Jones polynomial) in terms of his topological quantum field theory using Chern-Simons theory (see \textit{K. Marathe} [in: The mathematics of knots. Theory and application. Banagl, Markus (ed.) et al., Berlin: Springer. Contributions in Mathematical and Computational Sciences 1, 199--256 (2011; Zbl 1221.57023)] for a recent survey). This theory depends on a rank \(N\) and level \(k\), the quantum Hilbert space being identified with level \(k\) highest weight representations of the Lie algebra corresponding to \(\text{SU}(N)\) and acted on by the group \(\text{SL}(2,\mathbb Z)\) via matrices \(S\) and \(T\). Recently \textit{M. Aganagic} and \textit{Sh. Shakirov} proposed a refinement of the \(\text{SU}(N)\) Chern-Simons theory for links in 3-manifolds having \(S^1\) symmetry [String-Math 2011, Proc. Symp. Pure Math. 85, Amer. Math. Soc., Providence RI 2012, 3--31 (2012)] in which the matrices \(S\) and \(T\) are replaced by matrices used by \textit{I. Cherednik} [Invent. Math. 122, No. 1, 119--145 (1995; Zbl 0854.22021)] and \textit{A. A. Kirillov, jun.} [J. Am. Math. Soc. 9, No. 4, 1135--1169 (1996; Zbl 0861.05065)]. In the refined theory, the Hilbert space is identified with the MacDonald polynomials of type \(\text{SU}(N)\) with parameters \(q,t\) satsifying \(q^k t^N = 1\).
In the paper under review, the author computes the limit of the matrix \(S\) as \(N \to \infty\) for the refined theory. Starting with the explicit form of \(S\) given by \textit{M. Aganagic} and \textit{Sh. Shakirov} [``Knot homology from refined Chern-Simons theory'', Preprint, 2011, \url{arXiv:1105.5117}], he replaces \(t^N\) by a variable \(u\) to obtain a stable version expressed in terms of the modified MacDonald polynomials of \textit{A. Garsia} et al. [Sémin. Lothar. Comb. 42, B42m, 45 p. (1999; Zbl 0920.05071)]. To compute the kernel function, he starts with the Cherednik-MacDonald-Mehta identity in the form used by Garsia, Haiman and Tesler [loc. cit.] and using the relation between MacDonald polynomials and Hilbert schemes of \(n\) points in \(X=\mathbb C^2\) due to \textit{M. Haiman} [J. Am. Math. Soc. 14, No. 4, 941--1006 (2001; Zbl 1009.14001), Invent. Math. 149, No. 2, 371--407 (2002; Zbl 1053.14005)] he expresses two of the terms as power series in \(u\): the coefficient of \(u^n\) in each case is an equivariant Euler characteristics of certain sheaves on the Hilbert scheme \(X^{[n]}\) for the group action of \(\mathbb C^* \times \mathbb C^*\) - in one case the structure sheaf \({\mathcal O}_{X^{[n]}}\) and in the other case the tensor product of arbitrary Schur functors \(s_{\lambda}\) and \(s_{\mu}\) applied to a universal sheaf. Hilbert scheme of points in the plane; Chern-Simons theory Hiraku Nakajima, Refined Chern-Simons theory and Hilbert schemes of points on the plane, Perspectives in representation theory, Contemp. Math., vol. 610, Amer. Math. Soc., Providence, RI, 2014, pp. 305 -- 331. Parametrization (Chow and Hilbert schemes), Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Refined Chern-Simons theory and Hilbert schemes of points on the plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors consider various natural compactifications of the space \(\mathbf X_0\) of smooth rational cubic curves in \(\mathbb P^r\), \(r\geq 3\). These are: the GIT quotient \(\mathbf X\), the Hilbert scheme compactification \(\mathbf H\), the moduli space of stable maps compactification \(\mathbf M\), the moduli space of stable sheaves compactification \(\mathbf S\), the Chow scheme compactification \(\mathbf C\), the net of quadrics compactification \(\mathbf N\) (for \(r=3\)) [\textit{G. Ellingsrud, R. Piene} and \textit{S. A. Strømme}, Space curves, Proc. Conf., Rocca di Papa/Italy 1985, Lect. Notes Math. 1266, 84--96 (1987; Zbl 0659.14027)]. In the case \(r=3\) it was shown (1) by \textit{R. Piene} and \textit{M. Schlessinger} [Am. J. Math. 107, 761-774 (1985; Zbl 0589.14009)] that \(\mathbf H\) is smooth, (2) by Ellingsrud, Piene and Strømme [loc. cit.] that \(\mathbf H\) is the blowup of \(\mathbf N\) along the locus of nets of quadrics defining point-plane incidences, (3) by \textit{H. G. Freiermuth} and \textit{G. Trautmann} [Am. J. Math. 126, No. 2, 363--393 (2004; Zbl 1069.14012)] that \(\mathbf H=\mathbf S\), (4) by \textit{D. Chen} [Int. Math. Res. Not. 2008, Article ID rnn067, 17 p. (2008; Zbl 1147.14009)] that \(\mathbf H\) is the flip of \(\mathbf M\) over \(\mathbf C\).
\textit{Y.-H. Kiem} and \textit{H.-B. Moon} [Int. J. Math. 21, No. 5, 639--664 (2010; Zbl 1191.14015)] showed that the birational map from \(\mathbf X\) to \(\mathbf M\) is the composition of three blow-ups followed by two blow-downs. In the present paper it is shown that one gets \(\mathbf S\) from \(\mathbf M\) by a sequence of three blow-ups followed by three blow-downs. This allows for a calculation of the Betti numbers of \(\mathbf S\), which agrees with that of Ellingsrud, Piene and Strømme [loc.cit.] in the case \(r=3\). twisted cubic curve; Hilbert scheme; moduli space; stable maps; stable sheaves; Betti numbers Greuel, G.M., Pfister, G. et al: Singular 3.1.1, a computer algebra system for polynomial computations. Center for Computer Algebra, University of Kaiserslautern (2010). (http://www.singular.uni-kl.de) Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Minimal model program (Mori theory, extremal rays) Hilbert scheme of rational cubic curves via stable maps | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((A\mathfrak m,k)\) be a noetherian local ring of characteristic \(p>0\). Then the concept of Hilbert-Kunz multiplicity (denoted by \(e_{HK}(I)\)) was defined by Kunz and Monsky. \(\ldots\)
In this paper we give a formula for \(e_{HK}(I)\) for any integrally closed ideals of a 2-dimensional cyclic quotient singularity. Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Multiplicity theory and related topics, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities in algebraic geometry Hilbert-Kunz multiplicity of integrally closed ideals of 2-dimensional cyclic quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main theorem states that the cohomology of \(\text{ Hilb}^{[n]}\), the Hilbert scheme of \(n\)-points on a \(K3\)-surface, is the \(S_n\)-invariant part of the \(S_n\)-Frobenius algebra associated to the symmetric product of the cohomology of the surface twisted by a discrete torsion. Here \(S_n\) denotes the symmetric group of \(n\) letters. The main idea is to use the notion of \(G\)-Frobenius algebras for a finite group \(G\), which arise from the stringy study of objects with a global \(G\)-action.
In section one, the author presents the general functorial setup for extending functors to Frobenius algebras to those with values in \(G\)-Frobenius algebras. Section two contains the basic definitions of \(G\)-Frobenius algebras. Section three introduces intersection Frobenius algebras which are adapted to the situation in which one can take successive intersections of fixed point sets. Section four reviews the analysis of discrete torsion. In section five, the author recalls the results on the structure of \(S_n\)-Frobenius algebras. Section six assembles these results in the case of any \(S_n\)-Frobenius algebra twisted by a specific discrete torsion. The result applied to the situation of the Hilbert scheme yields the above main theorem. Kaufmann, R. M.: Discrete torsion, symmetric products and the Hilbert scheme. Frobenius manifolds, quantum cohomology, and singularities (2004) Noncommutative algebraic geometry, Parametrization (Chow and Hilbert schemes) Discrete torsion, symmetric products and the Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a complex smooth projective surface. The Hilbert scheme of \(n\)-points \(X^{[n]}\) parametrizes the closed finite subschemes of length \(n\) on \(X\). It is a crepant resolution of the \(n\)-fold symmetric product \(X^{(n)}\) of \(X\).
\textit{Y. Ruan} [Contemp. Math. 312, 187--233 (2002; Zbl 1060.14080)] conjectured that the orbifold cohomology \(n\)-fold symmetric product is isomorphic to the cohomology ring of the Hilbert scheme of \(n\)-points. In this interesting and well-written paper a deep result is proved for the cohomology ring of \(X^{[n]}\) (when \(X\) has numerically trivial canonical bundle), which implies Ruan's conjecture.
In section 2, the authors prove various technical results about graded Frobenius algebras (we recall that a graded Frobenius algebra \(A\) is a graded vector space endowed with a graded commutative and associative multiplication and a linear map \(T: A \to {\mathbb Q}\) such that \(T(ab)\) is a nondegenerate symmetric bilinear form). In particular, a sequence of endomorphisms \(A\to A^{[n]}\) is constructed in the category of graded Frobenius algebras. The cohomology ring \(H^\ast(X^{[n]}, {\mathbb Q})[2n]\) can be given the structure of a graded Frobenius algebra by setting \(T(a) = (-1)^n \int_{X^{[n]}} a\), where the integral denotes the evalution on the fundamental class of \(X^{[n]}\).
In section 4, the authors prove their main theorem, which provides an intrinsic and explicit description of the cup product structure of the cohomology ring of the Hilbert scheme of \(n\)-points of a surface \(X\) having numerically trivial canonical bundle (so, \(X\) is a K3, or an Enriques, or an abelian, or a bielliptic surface). Namely, there is a canonical isomorphism of graded rings
\[
(H^\ast(X, {\mathbb Q})[2])^{[n]} \to H^\ast(X^{[n]}, {\mathbb Q})[2n]\,.
\]
This result generalizes previous results obtained by \textit{E. Vasserot} [C. R. Acad. Sci., Paris, Sér. I, Math. 332, 7--12 (2001; Zbl 0991.14001)] and by \textit{M. Lehn} and \textit{Ch. Sorger} [Duke Math. J. 110, 345--357 (2001; Zbl 1093.14008)]. cohomology ring; graded Frobenius algebras; cup product; cohomology of Hilbert scheme Lehn, M; Sorger, C, The cup product of Hilbert schemes for \(K3\) surfaces, Invent. Math., 152, 305-329, (2003) Parametrization (Chow and Hilbert schemes), Minimal model program (Mori theory, extremal rays), \(K3\) surfaces and Enriques surfaces, Products and intersections in homology and cohomology The cup product of Hilbert schemes for K3 surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In Publ. Res. Inst. Math. Sci. 26, No. 1, 15-78 (1990; Zbl 0713.17014)] \textit{K. Saito} introduced the ``flat structure'' for the extended affine root system in order to construct the inverse mapping of the period mapping for the primitive form for the semi-universal deformation of a simple elliptic singularity. In this paper the flat theta invariants are constructed explicitly in the case of \(E_ 6\) using the Jacobi form introduced by Wirtmüller. Combining this with the explicit description of the flat coordinates by Kato and Noumi, this gives an answer to Jacobi's inversion problem of the period mapping for a simple elliptic singularity of type \(\widetilde{E}_ 6\). period mapping; deformation; elliptic singularity; Jacobi form; type \(\widetilde{E}_ 6\) I. Satake: Flat structure for the simple elliptic singularity of type \(\tilde{E}_{6}\) and Jacobi form , Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 247-251. Deformations of complex singularities; vanishing cycles, Period matrices, variation of Hodge structure; degenerations, Local structure of morphisms in algebraic geometry: étale, flat, etc., Infinite-dimensional Lie (super)algebras, Singularities in algebraic geometry Flat structure for the simple elliptic singularity of type \(\widetilde {E}_ 6\) and Jacobi form | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 uses Shintani decomposition to give an effective construction of Ehler's basic fan, the toroidal variety of this fan is the resolution of cusps of Hilbert modular variety. toroidal variety; fan; resolution of cusps of Hilbert modular variety G K SANKARAN, Effective resolution of cusps on Hubert modular varieties, Math Proc Cam Phi Soc 99 (1986), 51-57 Global theory and resolution of singularities (algebro-geometric aspects), Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Special surfaces Effective resolution of cusps on Hilbert modular varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R = {\mathbb C}[x_1,\ldots, x_n]\) be the polynomial ring graded by a non-negative \(d\times n\)-matrix \(A = (a_1, \ldots, a_n)\) of non-negative integers such that \(\deg x_i = a_i \in {\mathbb N}^d\) is given by a vector. This defines a decomposition \(R = \bigoplus_{b \in {\mathbb N}A} R_b,\) where \({\mathbb N}A\) denotes the subsemigroup of \({\mathbb N}^d\) generated by the vectors \(a_1, \ldots, a_n\) and \(R_b\) is the \({\mathbb C}\)-span of an element \(b\) of the subsemigroup. The toric Hilbert scheme parametrizes all the \(A\)-homogeneous ideals \(I \subset R\) with the property that the graded component \((R/I)_b\) is a 1-dimensional \({\mathbb C}\)-vector space. This concept was summarized by \textit{B. Sturmfels} in the first chapter of his book ``Gröbner bases and convex polytopes'', Univ. Lect. Ser. 8 (1996; Zbl 0856.13020)]. In the paper under review, the authors illustrate the use of Macaulay 2 for exploring the structure of toric Hilbert schemes. It is known that all the components of the scheme are toric varieties. Among them there is a fairly well understood component, the coherent component. The results contribute to the open problem whether toric Hilbert schemes are always connected. In their investigations the authors encounter algorithms from commutative algebra, polyhedral theory and geometric combinatorics. Macaulay2; toric Hilbert scheme; semigroup algebra Stillman, M., Sturmfels, B., Thomas, R.: Algorithms for the toric Hilbert scheme. In: Computations in Algebraic Geometry using Macaulay 2, D. Eisenbud et al. (eds.), Algorithms and Computation in Mathematics Vol 8, Springer, 2002, pp. 179--213 Computational aspects of higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), Software, source code, etc. for problems pertaining to algebraic geometry, Symbolic computation and algebraic computation Algorithms for the toric Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a reductive linear algebraic group, let \(X\) be an affine \(G\)-variety and let \(h\colon \mathrm{Irr}(G) \to \mathbb{Z}_{\geq 0}\) be a function that associates to any irreducible representation of \(G\) some non-negative integer. The \emph{invariant Hilbert scheme} \(\mathcal{H}:=\mathrm{Hilb}_h^G(X)\), associated with the triple (\(G,X,h\)), is a moduli space, constructed by \textit{V. Alexeev} and \textit{M. Brion} [J. Algebr. Geom. 14, No. 1, 83--117 (2005; Zbl 1081.14005); Adv. Lect. Math. (ALM) 24, 64--117 (2015; Zbl 1322.14001)], whose points correspond to \(G\)-stable closed subschemes \(Z\) of \(X\) such that the coordinate ring \(\mathbb{C}[Z]\) decomposes into irreducible \(G\)-modules accordingly to the function \(h\). Also, for a well-chosen function \(h\), the scheme \(\mathcal{H}\) is equipped with a projective morphism \(\gamma \colon \mathcal{H} \to X/\!/G=\mathrm{Spec}(\mathbb{C}[X]^G)\) which is an isomorphism over a dense open subset of \(X/\!/G\).
The main contribution of the author of this article is the study of a new family of examples where \(G=\mathbb{G}_m \times \mu_m\) acts on some hypersurface \(X \subset \mathbb{C}^5\) such that \(X/\!/G=E_{l,m}\) is a quasi-homogeneous \(\mathrm{SL}(2)\)-variety. \textit{V. L. Popov} [Math. USSR, Izv. 7, 793--831 (1974; Zbl 0286.14013)] gives a complete classification of affine normal quasi-homogeneous \(\mathrm{SL}(2)\)-threefolds: they are uniquely determined by a pair of numbers \((l,m) \in (\mathbb{Q} \cap ]0;1]) \times \mathbb{Z}_{>0}\). Also, Popov proves that the variety \(E_{l,m}\) (corresponding to the pair \((l,m)\)) is smooth if and only if \(l=1\), and otherwise it contains a unique singular point. When singular, if follows from the work of \textit{V. Batyrev} and \textit{F. Haddad} [Mosc. Math. J. 8, No. 4, 621--646 (2008; Zbl 1221.14052)] that the variety \(E_{l,m}\) is furthermore dominated by a canonical weighted blow-up \(E'_{l,m} \to E_{l,m}\), where \(E'_{l,m}\) that is smooth if and only if \(E_{l,m}\) is toric (\(\Leftrightarrow \) \(q-p\) divides \(m\) with \(l=\frac{p}{q}\)).
Since any affine normal quasi-homogeneous \(\mathrm{SL}(2)\)-threefold is of this form, it is natural to ask whether a canonical \(\mathrm{SL}(2)\)-equivariant desingularization of \(E_{l,m}\) can be obtained from the invariant Hilbert scheme. In this article the author gives a positive answer by describing \(\mathcal{H}\), which is shown to be connected and smooth, and the Hilbert-Chow morphism \(\gamma \colon \mathcal{H} \to X/\!/G=E_{l,m}\) using a wide range of techniques from classical invariant theory to Luna-Vust theory of spherical embeddings. More precisely, she proves the following (Main Theorem):
\begin{enumerate}
\item If \(l=1\), then \(\gamma\colon \mathcal{H} \to E_{l,m}\) is an isomorphism.
\item If \(l<1\) and \(E_{l,m}\) is toric, then \(\gamma\colon \mathcal{H} \to E_{l,m}\) coincides with the weighted blow-up \(E'_{l,m} \to E_{l,m}\).
\item If \(l<1\) and \(E_{l,m}\) is non-toric, then \(\gamma\colon \mathcal{H} \to E_{l,m}\) coincides with the composition of the weighted blow-up \(E'_{l,m} \to E_{l,m}\) followed by the minimal desingularization \(\widetilde{E'_{l,m}} \to E'_{l,m}\).
\end{enumerate} invariant Hilbert schemes; quasihomogeneous varieties; equivariant resolutions; toric varieties, spherical varieties, invariant theory Group actions on varieties or schemes (quotients), Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, Toric varieties, Newton polyhedra, Okounkov bodies, Compactifications; symmetric and spherical varieties Invariant Hilbert scheme resolution of Popov's \(\mathrm{SL}(2)\)-varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a reduced closed subscheme of the smooth scheme \(Y\) (of finite type over a field of characteristic \(0\)) and \(\pi:\widetilde{Y}\to Y\) be a log resolution of \(X\) in \(Y\), \(E\) the reduced pre-image of \(X\) in \(\widetilde{Y}\). It is proved that \(X\) has Du Bois singularities if and only if the canonical map \(\mathcal O_X\to R\pi_\ast \mathcal O_E\) is a quasi-isomorphism. The result is used to prove a conjecture of Kollár: \(\log\) canonical singularities which are local complete intersection are Du Bois. singularity; rational; log canonical; Du Bois Schwede, K., \textit{A simple characterization of du bois singularities}, Compositio Math., 143, 813-828, (2007) Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) A simple characterization of Du Bois singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors study the degrees of the generators of the ideal of a projected Veronese variety \(v_2(\mathbb P^3)\subset \mathbb P^9\) to \(\mathbb P^6\), depending on the center of the projection. The interest of the authors in these projections is related to understanding the geometry of constructions of some Calabi-Yau threefolds in \(\mathbb P^6\). The main result of the paper gives several equivalences to the fact that the projected Veronese variety is contained in a cubic hypersurface. Some of these equivalences unexpectedly involve the smoothability of some zero-dimensional schemes or Cremona transforms. Hilbert scheme; Cremona map; smoothability; Tonoli Calabi-Yau; special projection Calabi-Yau manifolds (algebro-geometric aspects), Birational automorphisms, Cremona group and generalizations, Parametrization (Chow and Hilbert schemes), Families, moduli, classification: algebraic theory Smoothable zero dimensional schemes and special projections of algebraic varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((V,0)\) be a normal surface singularity, \((M,A) \to (V,0)\) a minimal good resolution and \(T^1_V\) the space of first order infinitesimal deformations of \((V,0)\). An invariant is the irregularity \(q\) [\textit{S. S.-T. Yau}, Ann. J. Math. 104, 1063-1100 (1982; Zbl 0523.14002)]. If \((V,0)\) is Gorenstein, \(\dim T^1_V \geq q\) and if \((V,0)\) is a cone over a curve \(A\), then \(q = \sum^\infty_{n = 1} \dim H^0 (A,K_A (nA))\).
We now assume that the exceptional set \(A\) of the resolution is a smooth curve. The first result of this paper is that \(q\) is less than the stated sum formula. When \((V,0)\) is Gorenstein, i.e. when \(K_M = - mA\) for some integer \(m\), the author gives some numerical condition on the formal neighborhood of \(A\) in \(M\) which implies that \(q > 0\). For example, if \(m = 3\) and \(A\) is not hyperelliptic, then \(\dim T^1_V > q > 0\). surface singularity; infinitesimal deformations; irregularity Singularities of surfaces or higher-dimensional varieties, Deformations of singularities, Global theory of complex singularities; cohomological properties A note on irregularity of two dimensional simple hyperbolic singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S = \mathbb A^2\) be the affine plane and denote by \(S^{[n]}\) the Hilbert scheme parametrizing zero dimensional subschemes of length \(n\). The Chow ring \(A^*(S^{[n]}, \mathbb Q)\) has been studied from different vantage points over the last twenty years. For example, [\textit{H. Nakajima}, Lectures on Hilbert schemes of points on surfaces. University Lecture Series. 18. Providence, RI: American Mathematical Society (1999; Zbl 0949.14001)] and [\textit{M. Lehn}, Invent. Math. 136, No. 1, 157--207 (1999; Zbl 0919.14001)] gave a basis and described the ring structure by considering linear operators on the direct sum \(\bigoplus_{n \in \mathbb N} A^*(S^{[n]}, \mathbb Q)\) and their commutativity relations. \textit{G. Ellingsrud} and \textit{S.-A. Strømme} gave a basis [Invent. Math. 87, 343--352 (1987; Zbl 0625.14002)] by using the action of the 2-torus \(T = (k^*)^2\) and the theorem of \textit{A. Białnicki-Birula} [Some properties of the decompositions of algebraic varieties determined by the actions of a torus, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24, 667--674 (1976; Zbl 0355.14015)].
Here the authors restrict to fixed points for the action of the torus \(T\) on \(S = \mathbb A^2\), using operators (creation/destruction operators \(q_i\), the boundary operator \(\partial\) and an auxiliary operator \(\rho\)) on the equivariant Chow ring \(\bigoplus_{n \in \mathbb N} A_T^* (S^{[n]}) \otimes_{A_T^* (\mathrm{pt})} K\), where \(K\) is the fraction field of \(A_T^* (\mathrm{pt})\). By manipulating linear combinations of Young diagrams they recover equivariant analogs of some commutativity relations obtained by \textit{Lehn} and \textit{Nakajima} [loc. cit.]. They also show that the Chow ring bases of \textit{Nakajima} and \textit{Ellingsrud-Strømme} [loc. cit.] are equal up to sign and a normalizing constant. As part of their process, they compute the tangent space to the Hilbert schemes \(S^{[n,n+1]}\) of flags \(z_n \subset z_{n+1}\) of zero dimensional subschemes of length \(n\) and \(n+1\). While the spaces \(S^{[n,n+1]}\) are known to be irreducible by work of \textit{J. Cheah} [Pac. J. Math. 183, No. 1, 39--90 (1998; Zbl 0904.14001)], the authors show that the space \(S_0^{[n,n+1]}\) consisting of subschemes supported at the origin is irreducible. Examples show that the spaces \(S_0^{[p,q]}\) are not irreducible in general. equivariant cohomology; Hilbert schemes; Chow ring [5] Pierre-Emmanuel Chaput &aLaurent Evain, &On the equivariant cohomology of Hilbert schemes of points in the plane&#xhttp://arxiv.org/abs/1205.5470 (Equivariant) Chow groups and rings; motives, Parametrization (Chow and Hilbert schemes) On the equivariant cohomology of Hilbert schemes of points in the plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{V. Ginzburg} and \textit{D. Kaledin} [Adv. Math. 186, No. 1, 1--57 (2004; Zbl 1062.53074)] posed the problem of comparing the McKay correspondence, the dual McKay correspondence and the multiplicative McKay correspondence for a finite dimensional \(\mathbb C\)-vector space \(V\), with an action of a finite subgroup \(G\) of SL\((V)\). The vector space \(V\) is assumed to be equipped with a symplectic form, which is preserved by \(G\). Moreover a crepant resolution of singularities \(Y\to V/G\) is fixed. Ginzburg and Kaledin proposed to compute explicitly the Poincaré isomorphism and the Chern character isomorphism (see Problems 1.4 and 1.5 of the above quoted article).
In the paper under review, the author solves the problem in the special case when \(V=\mathbb C^n\otimes \mathbb C^2\), with the action by permutations of the symmetric group \(S_n\) and the canonical symplectic form. In this situation \(Y\) is the Hilbert scheme Hilb\(^n(\mathbb C^2)\). The author gives explicit formulae for the Poincaré and the Chern character isomorphisms and uses them to prove the main theorem of the paper (Theorem 1.2). It says that the McKay correspondence is compatible with the topological filtration of the Grothendieck group \(K(\text{Hilb}^n(\mathbb C^2))\) and with the decreasing filtration of the space of symmetric functions \(\Lambda^n\). It is then proved that the graded McKay correspondence so obtained coincides with both the multiplicative and the dual McKay correspondence. symmetric functions; equivariant cohomology; Macdonald polynomials S. Boissière, On the McKay correspondences for the Hilbert scheme of points on the affine plane, arXiv:math.AG/0410281. Parametrization (Chow and Hilbert schemes), Symmetric functions and generalizations, Symmetric groups, Equivariant homology and cohomology in algebraic topology, Global theory and resolution of singularities (algebro-geometric aspects) On the McKay correspondences for the Hilbert scheme of points on the affine plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) a smooth quasi-projective algebraic surface and \(E\) a line bundle on \(X\). Consider \(X^{[n]}\), the Hilbert scheme of \(n\) points on \(X\), and the tautological bundle \(E^{[n]}\) on it naturally associated to \(E\). In the present paper the author relates the cohomology of \(X^{[n]}\) with values in \((E^{[n]})^{\otimes 2}\) with the cohomologies of \(X\) with values in \(E^{\otimes 2}\), \(E\) and \({\mathcal O}_X\).
This calculation is done using recent results on the McKay correspondence [\textit{T. Bridgeland, A. King} and \textit{M. Reid}, J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] adapted to the case of an isospectral Hilbert scheme in [\textit{M. Haiman}, J. Am. Math. Soc. 14, No. 4, 941--1006 (2001; Zbl 1009.14001)], which give a Fourier-Mukai equivalence \(\Phi\) between the derived category of the Hilbert scheme \(X^{[n]}\) and the \(S_n\)-equivariant derived category of \(X^n\). These results allow indeed to calculate the cohomologies of \(X^{[n]}\) with values in the tensor powers \((E^{[n]})^{\otimes k}\) of the tautological bundle as the hypercohomologies of \(S^n X\) with values in the invariants \(\Phi((E^{[n]})^{\otimes k})^{S_n}\). The latter groups can be calculated using polygraphs and the calculation is explicitely performed for \(k=2\) by means of a spectral sequence in order to obtain the main result. Hilbert scheme; tautological bundles; McKay correspondence; Fourier-Mukai functor Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Cohomology of the Hilbert scheme of points on a surface with values in the double tensor power of a tautological bundle | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 a striking result which characterizes singularities in terms of the jet schemes. This paper is the leading paper for forthcoming results [\textit{M. Mustaţă}, J. Am. Math. Soc. 15, No. 5, 599--615 (2002; Zbl 0998.14009), \textit{L. Ein}, \textit{M. Mustaţă} and \textit{T. Yasuda}, Invent. Math. 153, No. 3, 519--535 (2003; Zbl 1049.14008); \textit{L. Ein}, \textit{R. Lazarsfeld} and \textit{M. Mustaţă}, Compos. Math. 140, No. 5, 1229--1244 (2004; Zbl 1060.14004)]. The author shows that a locally complete intersection variety has rational singularities if and only if the jet schemes corresponding to the variety are all irreducible. He also proves that if a locally complete intersection, normal variety has pure-dimensional jet schemes, then it has log canonical singularities. To prove them the author makes use of motivic integration. There, the appearance of certain terms of the motivic integration plays an important role. motivic integration M. Mustaţă, Jet schemes of locally complete intersection canonical singularities, Invent. Math. 145 (2001), 397--424. Singularities in algebraic geometry, Infinitesimal methods in algebraic geometry, Birational geometry Jet schemes of locally complete intersection canonical singularities. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors construct for each braid a 2-periodic complex of quasi-coherent \(\mathbb{C}^{\ast} \times \mathbb{C}^{\ast}\)-equivariant sheaves on the non-commutative nested Hilbert scheme. They obtain a knot invariant: a triply graded vector space (hypercohomology) associated with the closure of the braid. It is proved to be a categorification of the HOMFLY-PT polynomial. This triply graded space is related to sheaves on the Hilbert scheme. The push-forward of the sheaf (with respect to the projection from the nested Hilbert scheme to the ordinary one) is proved to be an invariant of the conjugacy class of the braid. triply graded knot homology; sheaves on Hilbert schemes; braid group; braid closure; HOMFLY-PT polynomial; matrix factorizations Oblomkov, A., Rozansky, L.: Affine Braid group, JM elements and knot homology. Transformation Groups (to appear) Knots and links in the 3-sphere, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Knot homology and sheaves on the Hilbert scheme of points on the plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Smooth Fano threefolds \(V\) with very ample anticanonical divisor \(-K_V\) lead to points \([V]\in\mathcal H_d\) in the Hilbert scheme of degree \(d:=-K_V^3\) subvarieties of \(\mathbb{P}(|-K_V|)\). These point lie on unique irreducible components.
Maximal families of those Fanos, i.e.\ the set of those components of \(\mathcal H_d\) are completely classified.
For small degree \(d\), these families can be distinguished by \((d,b_2,b_3)\) where \(b_i\) denote the \(i\)-th Betti number of \(V\).
The dimensions of the corresponding components, i.e.\ \(h^0(\mathcal N_{V})\), are known, too.
On the other hand, toric Fano threefolds \(X\) of degree \(d\) with Gorenstein singularities correspond to three-dimensional reflexive polytopes. They are classified, too.
They do also provide points \([X]\in\mathcal H_d\) which might, however, sit in several components. Presenting this incidence for \(d\leq 12\) is the main point of the paper. Thus, it provides a complete overview about toric degenerations of the above \(V\) which is important from the view point of mirror symmetry.
One of the tools is the identification of certain Stanley-Reisner schemes \(S\) wich provide points \([S]\in\mathcal H_d\), too.
These are the most degenerate versions. Degenerations from \(X\) to \(S\) are understood by unimodular triangulations of the reflexive polytopes.
Moreover, deformation theory of \(X\) is used because the components of \(\mathcal H_d\) containing \([X]\) are identified by the components of the tangent cone of \(\mathcal H_d\) in \([X]\).
The equations of the latter are obtained via explicit computeralgebra computations. toric degenerations; Fano threefolds; reflexive polytopes; Hilbert schemes Parametrization (Chow and Hilbert schemes), Families, moduli, classification: algebraic theory, Fano varieties, Toric varieties, Newton polyhedra, Okounkov bodies Hilbert schemes and toric degenerations for low degree Fano threefolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We use algebraic methods to study systems of linear partial differential equations with constant coefficients. equations and systems with constant coefficients; Hilbert schemes General theory of PDEs and systems of PDEs with constant coefficients, Overdetermined systems of PDEs with constant coefficients, Parametrization (Chow and Hilbert schemes) Linear differential operators with constant coefficients and Hilbert schemes of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Isolated line singularities are hypersurface singularities whose singular locus is one dimensional and smooth, and such that transverse to a general point of the singular locus it has a Morse singularity. Simple isolated line singularities were classified by Siersma.
In this article, the author gives two characterizations of simple isolated line singularities in three spaces. The first is by the possible singularities which might occur on the strict transform of the zero set and the exceptional divisor upon blowing up the singular locus of the surface singularity. The second characterization is for quasihomogeneous singularities only. The author shows that the simple isolated line singularities are those with inner modality zero. simple isolated line singularities Zaharia, A., Characterizations of simple isolated line singularities, Canad. Math. Bull., 42, 499-506, (1999) Complex surface and hypersurface singularities, Singularities in algebraic geometry Characterizations of simple isolated line 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 \(H=H(d,g,n)\) be the closure in the Hilbert scheme \(Hilb({\mathbb{P}}^ n)\) of the open subset corresponding to smooth, nondegenerate curves of degree \(n\) and genus \(g\) in \({\mathbb{P}}^ n\) and \(p: H\to {\mathcal M}_ g\) be the natural map into the moduli variety. Let \(r=r(d,g;n)=g-(n+1)(n-d+g)\) be the Brill-Noether number. One says that an irreducible component \(Z\subset H\) has the right number of moduli, if \(\dim (p(Z))=\min (3g- 3,3g-3+r)\). - In the article under review, the following theorem is proved:
(1) There exists a function v: \({\mathbb{N}}\to {\mathbb{R}}\) such that \(\lim_{g\to \infty}(v(g))=(n-2)/n\), and, for \(d\geq g\cdot v(g)\) and \(n\geq 5\), there exists an irreducible component \(Z\subset H\) such that for its general curve C the equality \(h^ 1(N_ C)=0\) (and analogously \(h^ 1(N_ C(-1))=0)\) is valid, where \(N_ C\) is the normal bundle to \(C\subset {\mathbb{P}}^ n.\)
(2) There exists a sequence of pairs \((d_ k,g_ k)\) such that \(\lim_{k\to \infty}g_ k/d_ k=n/(n-2)\) and there exists an irreducible component \(W_ k\subset H(d_ k,g_ k,n)\), \(n\geq 5\), such that \(W_ k\) has the right number of moduli and, for the general curve C, \([C]\in W_ k\), the equality \(h^ 1(N_ C)=0\) is fulfilled. An analogous result in the case \(n=4\) is obtained. Hilbert scheme; degree; genus; Brill-Noether number; number of moduli Hartshorne, R., Hirschowitz, A.: Droites en position générale dans \(\mathbb{P}^n\), Algebraic Geometry. In: Proceedings, La Rábida 1981, 169-188, Lect. Notes in Math. 961, Springer, Berlin (1982) Plane and space curves, Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus Bonnes petites composantes des schémas de Hilbert de courbes lisses de \({\mathbb{P}}^ n\). (Good small components of Hilbert schemes of smooth curves in \({\mathbb{P}}^ n)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K\) be an algebraically closed field. The existence of the moduli space \(H_{N,p}\) parametrizing the embedded curve singularities in \((\mathbb C^N,0)\) with admissible Hilbert polynomial \(p\) is proved. \(H_{N,p}\) is not locally of finite type. It is a projective limit of schemes of finite type. The tangent space ot \(H_{N,p}\) at a closed point is computed. moduli space; space curves; embedded curve singularities Singularities of curves, local rings, Deformations of singularities, Fine and coarse moduli spaces A moduli scheme of embedded curve singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the paper the authors use the vertex algebra techniques to determine the generators of \({\mathbb H}_{n}=H^{*}(X^{[n]};{\mathbb Q})\) the cohomology ring of the Hilbert scheme of points on a smooth projective surface \(X\) over the field of complex numbers. The set of \(n\cdot \dim H^{*}(X ;{\mathbb Q})\) generators for the cohomology ring \({\mathbb H}_{n}\) has been found and the relations among them have been interpreted in terms of certain operators belonging to \({\mathbb H}\), where \({\mathbb H}= {\bigoplus}_{n\geq 0}{\mathbb H}_{n}.\) More precisely, let
\[
G(\gamma , n)=p_{1*}(\text{ch} {\mathcal O}_{{\mathcal Z}_{n}}\cdot p_{2}^{*} \text{td}(X)\cdot p_{2}^{*} \gamma) \in H^{*}(X^{[n]};{\mathbb Q}),
\]
where \({\gamma}\in H^{*}(X; {\mathbb Q}),\) \(n\geq 0,\) \({{\mathcal Z}_{n}}\) is the universal codimension-2 subscheme of \(X^{[n]}\times X ,\) td\((X)\) is the Todd class of the sheaf \({\mathcal O}_{{\mathcal Z}_{n}},\) and \(p_{1},\) \(p_{2}\) are the natural projections of \(X^{[n]}\times X \) to \(X^{[n]}\) and \(X\) respectively. Let \({\mathcal B}(\gamma) = {\bigoplus}_{n\geq 0} G({\gamma} , n)\) be an element of \(\text{End}(\mathbb H)\), where \(G({\gamma} , n)\) acts on \({\mathbb H}_{n}\) by cup product. For \(i \in {\mathbb Z},\) \({\gamma}\in H^{s}(X ; {\mathbb Q})\) let \(G_{i}({\gamma} , n)\) denote the component of \(G({\gamma} , n)\) in \(H^{s+2i}(X^{[n]};{\mathbb Q})\) and \({\mathcal B}_{i}(\gamma) = {\bigoplus}_{n\geq 0} G_{i}({\gamma} , n).\) The main theorem of the paper is the following (theorem 5.30):
Theorem: For \(n\geq 1,\) the cohomology ring \({\mathbb H}_{n}=H^{*}(X^{[n]};{\mathbb Q})\) is generated by
\[
G_{i}({\gamma}, n) = {\mathcal B}_{i}(\gamma)(1_{X^{[n]}})
\]
where \(0\leq i < n\) and \(\gamma\) runs over a linear basis of \(H^{*}(X; {\mathbb Q}).\) Moreover, the relations among these generators are precisely the relations among the restrictions \({\mathcal B}_{i}(\gamma)|_{{\mathbb H}_{n}}\) of the corresponding operators \({\mathcal B}_{i}(\gamma)\) to \({\mathbb H}_{n}.\)
It is worth mentioning that the above result generalises the work of \textit{G. E. Ellingsrud} and \textit{S. A. S. Stromme} [J. Reine Angew. Math. 441, 33-44 (1993; Zbl 0814.14003)], where the case \(X={\mathbb P}^2\) was considered. As the authors acknowledge the paper was inspired by the work of Ellingsrud and Stromme as well as of \textit{M. Lehn} [Invent. Math. 136, 157-207 (1999; Zbl 0919.14001)]. Hilbert scheme; vertex algebra; cohomology generators; cohomology ring W.-P Li, Z Qin, and W Wang, Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces , Math. Ann. 324 (2002), 105--133. \CMP1 931 760 Parametrization (Chow and Hilbert schemes), Vertex operators; vertex operator algebras and related structures, Étale and other Grothendieck topologies and (co)homologies Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The goal of the article is to explore the structure of singularities that occur in generic fibres in positive characteristic. The author determines which rational double points do and which do not occur on generic fibres.
Let \(k\) be an algebraically closed ground field of characteristic \(p>0\), and suppose \(f:S \rightarrow B\) is a morphism between smooth integral schemes. Then the generic fiber \(S_\eta\) is a regular scheme of finite type over the function field \(E=\kappa(\eta).\) In characteristic \(0,\) this implies that \(S_\eta\) is smooth over \(E\). The absolute Galois group \(G=\operatorname{Gal}(\bar E/E)\) acts on the geometric generic fiber \(S_{\bar\eta}\) with quotient isomorphic to \(S_\eta\), so to understand the generic fiber it suffices to understand the geometric generic fiber which is again smooth over an algebraically closed field, together with its Galois action.
The situation is more complicated in characteristic \(p>0\). The reason is that over nonperfect fields the notion of regularity is weaker than the notion of geometric regularity, which coincides with formal smoothness. Here it easily happens that the geometric generic fiber \(S_{\bar\eta}\) acquires singularities. As is proved by Bombieri and Mumford, there are quasi-elliptic fibrations of \(p=2\) and \(p=3\), which are analogous to elliptic fibrations but have a cusp on the geometric generic fiber. A proper morphism \(f:S\rightarrow B\) of smooth algebraic schemes is called a \textit{quasifibration} if \(\mathcal O_B=f_\ast(\mathcal O_S)\) and if the generic fiber \(S_\eta\) is not smooth. Quasi-fibrations involve some fascinating geometry and offer new freedom to achieve geometrical constructions that are impossible in characteristic \(0\).
Nonsmoothness of the generic fiber \(S_\eta\) leads to unusual complications. However, singularities appearing on the geometric generic fiber \(S_{\bar\eta}\) are not arbitrary. First, they are locally of complete intersection; hence many powerful methods from commutative algebra apply. They also satisfy far more restrictive conditions, and the goal of the paper is to analyze these. Hirokado started an analysis, characterizing those rational double points in odd characteristic that appear on geometric fibres. His approach was to study the closed fibres \(S_b\) \((b\in B)\) and their deformation theory. In this article, the author looks at the generic fibre \(S_\eta\) and work over the function field \(\kappa(\eta).\) The author works in the following abstract setting: Given a field \(F\) in characteristic \(p>0\) and a subfield \(E\) such that the field extension \(E\subset F\) is purely inseparable. Then the author considers \(F\)-schemes \(X\) of finite type that descend to regular \(E\)-schemes \(Y\), that is, \(X\backsimeq Y\otimes_E F.\) The first results of such schemes are: In codimension 2, the local fundamental groups are trivial and the torsion of the local class groups are \(p\)-groups. Moreover, the Tjurina numbers are divisible by \(p\), the stalks of the Jacobian ideal have finite projective dimension, and the tangent sheaf \(\Theta_X\) is locally free in codimension \(2\). These conditions give strong conditions on the singularities. The first main result of the article is a surprising restriction on the cotangent sheaf: If an \(F\)-scheme \(X\) descends to a regular scheme then, for each point \(x\in X\) of codimension \(2\), the stalk \(\Omega^1_{X/F,x},\) contains an invertible direct summand.
As an application of these results the author determines which rational double points appear on surfaces descending to regular schemes and which do not. It turns out that the situation is most challenging in characteristic 2: Besides the \(A_n\)-singularities, which behave as in characteristic \(0\), there are the following isomorphism classes: \(D_n^r\) with \(0\leq r\leq \lfloor n/2\rfloor-1\) and \(E_6^0, E_6^1,E_7^0,\dots,E_7^3,E_8^0,\dots,E_8^4.\) Notice that all the members of the list have a tangent sheaf that are locally free, although there are other rational double points with locally free tangent sheaf.
The author sets up notation and gives some elementary examples and results. He analyzes \(F\)-schemes that \(X\) descend to regular \(E\)-schemes \(Y\), and treats the local fundamental groups. He proves that integer-valued invariants are multiples of \(p\), and he treats the finite projective dimension of sheaves obtained from the cotangent sheaf \(\Omega^1_{X/F}.\)
This article is written in an easy to understand language, it is more or less self contained with good references when needed. Finally, it treats a computable field of singularity theory which is of importance to the study in positive characteristic. singularities on generic fibers; absolute Galois group; positive characteristic; descent; descends to regular scheme; quasi-fibration Stefan Schröer, Singularities appearing on generic fibers of morphisms between smooth schemes, Michigan Math. J. 56 (2008), no. 1, 55 -- 76. Arithmetic ground fields (finite, local, global) and families or fibrations, Fibrations, degenerations in algebraic geometry Singularities appearing on generic fibers of morphisms between smooth 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 author proves that if \(S\) is a Poisson surface, i.e. a smooth projective surface with a Poisson structure, then the Hilbert scheme of points of \(S\) has a natural Poisson structure, induced by the one of \(S\). This generalizes previous results obtained by \textit{A. Beauville} [J. Differ. Geom. 18, 755-782 (1983; Zbl 0537.53056)] and \textit{S. Mukai} [Invent. Math. 77, 101-116 (1984; Zbl 0565.14002)] in the symplectic case, i.e. when \(S\) is an Abelian or K3 surface. Finally, the author applies his results to give some examples of integrable Hamiltonian systems naturally defined on these Hilbert schemes. In the case \(S= \mathbb{P}^2\) he obtains a large class of integrable systems, which includes the ones studied by \textit{P. Vanhaecke} [Prog. Math. 145, 187-212 (1997; Zbl 0873.58038)]. Poisson surface; Hilbert scheme; Poisson structure; integrable Hamiltonian systems; Hilbert schemes F. BOTTACIN, Poisson structures on Hilbert schemes of points of a surface and integrable systems, Manuscripta Math., 97 (1998), pp. 517-527. Zbl0945.53049 MR1660136 Poisson manifolds; Poisson groupoids and algebroids, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Parametrization (Chow and Hilbert schemes), Special surfaces Poisson structures on Hilbert schemes of points of a surface and integrable systems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We continue the study of maximal families \(W\) of the Hilbert scheme, \( {{\mathrm{H}}}(d,g)_{sc}\), of smooth connected space curves whose general curve \(C\) lies on a smooth degree-\(s\) surface \(S\) containing a line. For \(s \geq 4\), we extend the two ranges where \(W\) is a unique irreducible (resp. generically smooth) component of \( \mathrm{H}(d,g)_{sc}\). In another range, close to the boarder of the nef cone, we describe for \(s=4\) and 5 components \(W\) that are non-reduced, leaving open the non-reducedness of only 3 (resp. 2) families for \(s \geq 6\) (resp. \(s=5\)), thus making progress to recent results of the first author and \textit{J. C. Ottem} in [Int. J. Math. 26, No. 2, Article ID 1550017, 30 p. (2015; Zbl 1323.14005)]. For \(s=3\) we slightly extend previous results on a conjecture of non-reduced components, and in addition we show its existence in a subrange of the conjectured range. space curves; quartic surfaces; cubic surfaces; Hilbert scheme; Hilbert-flag scheme; relative Picard scheme Parametrization (Chow and Hilbert schemes), Divisors, linear systems, invertible sheaves, Picard schemes, higher Jacobians, \(K3\) surfaces and Enriques surfaces, Plane and space curves The Hilbert scheme of space curves sitting on a smooth surface containing a line | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 possible Hilbert functions of reduced arithmetically Cohen-Macaulay (ACM) subschemes of projective space of any codimension is well known. What can be the Hilbert function of a reduced t and irreducible ACM subvariety of projective space? In codimension two this is again well-known. Answering this question for reduced and irreducible ACM subvarieties in codimension \(\geq 3\) is an open question of great interest. There is a strong connection with the Hilbert function of points in uniform position, due to Harris, but in codimension \(\geq 3\) it is not even known if the answers are the same, never mind a complete classification result in either case.
In codimension \(c\), the standard determinantal subschemes (i.e.\ the subschemes of codimension \(c\) defined by the maximal minors of a \(t \times (t+c-1)\) homogeneous matrix) are all ACM, and when \(c=2\) the notions of ACM and ``standard determinantal'' coincide. The authors here first prove some folklore facts about the Hilbert function of standard determinantal schemes of codimension \(c\), both in the general case and in the irreducible case.
The possible Hilbert functions of arithmetically Gorenstein (AG) subschemes of projective space are also not known, except in codimension \(\leq 3\); much less is it known what are the Hilbert functions of reduced, irreducible AG subschemes. A nice class of AG subschemes was studied by \textit{Kleppe} et al. [Gorenstein liaison, complete intersection liaison invariants and unobstructedness. Mem. Am. Math. Soc. 154 (2001; Zbl 1005.14018)], namely those that are twisted anticanonical divisors on an ACM subscheme satisfying certain mild local conditions. The authors of the present paper study when such divisors on a reduced, irreducible ACM subscheme \(S\) can again be reduced and irreducible, and then they describe the Hilbert functions that occur in terms of the Hilbert function of \(S\). Then as an application, they consider the case where \(S\) is a reduced, irreducible standard determinantal subscheme, and combine their folklore results with the latter results to obtain a large class of Hilbert functions that arise for reduced, irreducible AG subschemes of any codimension. Hilbert function; standard determinantal scheme; degree matrix; irreducible arithmetically Gorenstein scheme; divisor; regularity Budur, N.; Casanellas, M.; Gorla, E.: Hilbert functions of irreducible arithmetically Gorenstein schemes. J. algebra 272, No. 1, 292-310 (2004) Determinantal varieties, Divisors, linear systems, invertible sheaves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Hilbert functions of irreducible arithmetically Gorenstein schemes. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider graded Gorenstein quotients of \(R=k[X_1,X_2, \dots,X_s]\) of codimension \(r\). If \(s=r\), we let \(PGor(H)\) be the space parametrizing all graded Gorenstein \(R\)-algebras \(R/I\) with Hilbert function \(H\) \((H(i)=\dim(R/I)_i)\), with a scheme structure induced by the vanishing of the relevant catalecticant minors [as explained by \textit{S. J. Diesel}, Pac. J. Math. 172, No. 2, 365-397 (1996; Zbl 0882.13021)]. If \(s>r\) we define \(PGor(H)\) similarly and endow it with an apparently different scheme structure (remark 1.6). The main theorems of this note are concerned with \(PGor(H)\) for \(s=r=3\) in which case we prove that \(PGor (H)\) is a smooth irreducible scheme and we compute its dimension. For \(s>r=3\) a corresponding theorem is proved by \textit{J. O. Kleppe} and \textit{R. M. Miró-Roig} [J. Pure Appl. Algebra 127, No. 1, 73-82 (1998)]. As a corollary we prove a conjecture of Geramita, Pucci, and Shin on the Hilbert function of \(R/I^2\). If \(s=r>3\), we find conditions for \(PGor(H)\) to be smooth and we prove a useful linkage result for computing its dimension. We also prove some results on the codimension of \(PGor(H)\) embedded in some natural strata of the punctual Hilbert scheme. In particular if \(s=r=3\), we compute the dimension of \(ZGor(H)\) of all (not necessarily graded) Gorenstein quotients of \(k[[X_1,X_2, \dots, X_s]]\) with symmetric Hilbert function \(H\) at a graded quotient \(R\to A\), leading to a criterion for \(A\) to be non-alignable. Our method of proof applies the theorem of \textit{A. Iarrobino} and \textit{V. Kanev} [``The length of a homogeneous form, determinantal loci of catalecticants and Gorenstein algebras'' (preprint)] where they determine the tangent space of \(PGor(H)\). punctual Hilbert scheme; graded Gorenstein quotients; Hilbert function; dimension Kleppe, JO, The smoothness and the dimension of \({\mathrm PGor}(H)\) and of other strata of the punctual Hilbert scheme, J. Algebra, 200, 606-628, (1998) Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The smoothness and the dimension of \(PGor(H)\) and of other strata of the punctual 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 the entire collection see Zbl 0721.00009.]
These notes represent a survey of the main points of the authors' explicit proof of canonical desingularization of an algebraic subvariety (resp. analytic subspace) \(X\) of an algebraic (resp. analytic) manifold \(M\), in characteristic zero. The full details of this result are planned for a forthcoming work `Canonical desingularization in characteristic zero: a simple constructive proof'. The authors' result is essentially a new proof of Hironaka's theorem, although they also give an explicit resolution algorithm. The centres of the blowings-up used in the desingularization are determined by a local invariant of the singularity of \(X\), defined over a sequence of blowings-up.
The first section of this paper describes the general strategy of the proof and ends with a precise statement of the main theorem. --- The second section recalls the definitions and basic notions involved in resolution of singularities (analytic space, blowing-up, strict transform, normal crossings, etc.). --- The third section introduces the local invariant of a singularity of \(X\) and begins an analysis of this invariant. --- The fourth section gives a key part of the proof that it is in fact invariant. Much of these notes deal with the special case where \(X\) is a hypersurface, and the general case involves a reduction to this case. canonical desingularization of an algebraic subvariety; resolution algorithm Zhou, X. Y., Zhu, L. F.: An optimal \(L\)\^{}\{2\} extension theorem on weakly pseudoconvex Kähler manifolds. To appear in J. Differential Geom. Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Effectivity, complexity and computational aspects of algebraic geometry A simple constructive proof of canonical resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We give a characterization of Atiyah's and Hitchin's transverse Hilbert schemes of points on a symplectic surface in terms of bi-Poisson structures. Furthermore, we describe the geometry of hyperkähler manifolds arising from the transverse Hilbert scheme construction, with particular attention paid to the monopole moduli spaces. Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Parametrization (Chow and Hilbert schemes), Holomorphic symplectic varieties, hyper-Kähler varieties Transverse Hilbert schemes, bi-Hamiltonian systems and hyperkähler 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 \(X\) be a complex analytically irreducible quasi-ordinary (q.o) singularity, defined by \(f\in \mathbb C\{x_1,x_2\}[z]\). It can be parametrized in the form \(x_1=x_1\), \(x_2=x_2\), \(z=\zeta(x_1,x_2)\) with \(\zeta\in \mathbb C\{x_1,x_2\}[z]\). [\textit{Y.-N. Gau}, Mem. Am. Math. Soc. 388, 109--129 (1988; Zbl 0658.14004)] as shown: A finite set of exponents in the support of the series \(\zeta\) -- they are called the characteristic exponents -- are complete invariants of the topological type of the singularity.
In the paper under review, the authors look for invariants for {\em all} types of singularities. They consider the set of jet schemes of \(X\). For \(m\in\mathbb N\), they define a functor \(F_m\colon \mathbb C\text{-Schemes}\to \text {Sets}\) which is representable by a \({\mathbb C}\)-scheme \(X_m\), the \(m\)th jet scheme. There is a canonical projection \(\pi_m \colon X_m\to X\). In section 4 q.o.\ surfaces with one characteristic exponent are considered. The irreducible components of the \(m\)-jet schemes through the singular locus of a such a surface are described in Th.\ 4.14. A graph \(\Gamma\) is constructed which represents the decomposition of \((\pi^{-1}_m(X_{\text{Sing}}))_{\text{red}}\) for every \(m\). The graph \(\Gamma\) is equivalent to the topological type of the singularity. In section 5 these results are generalized to the general case. quasi-ordinary; surface singularities; jet schemes Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Jet schemes of quasi-ordinary surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We calculate the Picard group, over the integers, of the Hilbert scheme of smooth, irreducible, nondegenerate curves of degree \(d\) and genus \(g\geq 4\) in \(\mathbb{P}^r\), in the case when \(d\geq 2g +1\) and \(r\leq d -g\). We express the classes of the generators in terms of some ``natural'' divisor classes. Picard groups; Hilbert schemes of curves Picard groups, Parametrization (Chow and Hilbert schemes) Picard groups of Hilbert schemes of curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author works over the complex field \({\mathbb{C}}\). - Theorem 1: Let f,g: \(C\to {\mathbb{P}}^ 1\) be two coverings of \({\mathbb{P}}^ 1\) by a smooth irreducible curve C of genus \(\geq 1\). Assume that both have simple branching and that the branching occurs over the same set \(\Gamma \subset {\mathbb{P}}^ 1\). If \(\Gamma\) is sufficiently general, then f and g are isomorphic as coverings of \({\mathbb{P}}^ 1.\)
This theorem was previously known if \(d<(g-1)/2\) [\textit{E. Arbarello} and \textit{M. Cornalba}, Math. Ann. 256, 341-362 (1981; Zbl 0454.14023)]. - Examples showing that the conditions: genus\((C)\geq 1\) and \(\Gamma\) sufficiently general are necessary, are given. - The authors show that this theorem implies that the Tyrell conjecture (saying that the 40 elliptic curves tangent to 6 general concurrent lines are pairwise nonisomorphic) is true.
In the last section some geometric results on the monodromy of the covering \({\mathcal H}_{d,g}\to {\mathcal P}_ b\) (where \({\mathcal H}_{d,g}\) is the Hurwitz scheme of degree \(d\) branched covers of \({\mathbb{P}}^ 1\) by curves of genus \(g\geq 1\) and \({\mathcal P}_ b\) is the moduli space of b- pointed rational curves with \(b=2d-2+2g)\) are presented. Geometric interpretations of a Cohen result for \(d=3\) [\textit{D. B. Cohen}, J. Algebra 32, 501-517 (1974; Zbl 0343.20002)] and a Maclachlan remark for \(d=4\) [\textit{C. Maclachlan}, Mich. Math. J. 25, 235-244 (1978; Zbl 0366.20032), last paragraph] are given. isomorphic coverings; Tyrell conjecture; monodromy of the covering; Hurwitz scheme 10. Eisenbud, D., Elkies, N., Harris, J., Speiser, R.: On the Hurwitz scheme and monodromy. Compositio Math. 77, 95--117 (1991) Coverings of curves, fundamental group, Topological properties in algebraic geometry On the Hurwitz scheme and its monodromy | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Denote by \(H_{d,g}^{S}\) the Hilbert scheme parametrizing projective smooth connected curves of degree \(d\) and genus \(g\) in \({\mathbb P^3}\). \textit{D. Mumford} [Am. J. Math. 84, 642--648 (1962; Zbl 0114.13106)] proved that \(H_{d,g}^{S}\) may be non-reduced, showing the existence of an irreducible component \(W\subset H_{14,24}^{S}\) of dimension \(56\), whose general member \(C\) is contained in a smooth cubic surface, and such that the tangent space of \(H_{14,24}^{S}\) at \(C\) is \(57\).
In the paper under review, the author presents a generalization of Mumford's example. More precisely, let \(W\) be an irreducible closed subset of \(H_{d,g}^{S}\) whose general member \(C\) is contained in a smooth cubic surface. Assuming that \(W\) is maximal among all such subsets, that \(d>9\) and \(g\geq 3d-18\), and that the gap between \(\dim(W)\) and the dimension of the tangent space of \(H_{d,g}^{S}\) at \(C\) is equal to \(1\), the author proves that \(W\) is an irreducible component of \(H_{d,g}^{S}\), and that \(H_{d,g}^{S}\) is non-reduced along \(W\).
In the proof of this result, following an approach first used by \textit{D. Curtin} [Trans. Am. Math. Soc. 267, 83--94 (1981; Zbl 0477.14008)] in analyzing Mumford's example, the author shows the existence of a first order infinitesimal deformation of \(C\subset {\mathbb P^3}\) that cannot be lifted to any second order deformation. This implies that \(H_{d,g}^{S}\) is singular along \(W\). In this analysis one uses linear systems on the cubic surface containing \(C\). The author also compares his result with other generalizations of Mumford's example previously studied by many other authors. Hilbert scheme; deformation theory; obstruction theory; smooth cubic space surface Nasu, H, Obstructions to deforming space curves and non-reduced components of the Hilbert scheme, Publ. Res. Inst. Math. Sci., 42, 117-141, (2006) Parametrization (Chow and Hilbert schemes), Plane and space curves, Formal methods and deformations in algebraic geometry Obstructions to deforming space curves and non-reduced components of the Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert scheme of \(m\) points \(\text{Hilb}_m(S)\) of a quasiprojective surface \(S\) parametrizes \(m\)-point subschemes of \(S\). The main purpose of this paper is to characterize the ring structure of the small equivariant quantum cohomology rings of \(\text{Hilb}_m(\mathcal{A}_n)\) for all \(m\) and \(n\), where \(\mathcal{A}_n\) is the crepant resolution of the \(A_n\) singularity \(\mathbb{C}/\mathbb{Z}_{n+1}\). This generalizes the work of Okounkov-Pandharipande for \(\text{Hilb}_m(\mathbb{C}^2)\).
The classical equivariant cohomology ring \(H^*_T(\mathcal{A}_n,\mathbb{Q})\) is generated by the \(n\) exceptional divisors \(\omega_i\) and the identity class \(1\). Moreover, \(H^*_T(\text{Hilb}_m(\mathcal{A}_n),\mathbb{Q})\) has a basis, the Nakajima basis, labeled by cohomology weighted partitions \(\overrightarrow{\mu}=\{((\mu^{(1)},\gamma_{i_1}),\dots, (\mu^{(l)},\gamma_{i_l}))\}\) of \(m\), where \(\mu^{(1)}+\dots+\mu^{(l)}=m\) and \(\gamma_{i}\in H^*_T(\mathcal{A}_n,\mathbb{Q})\). Two divisors \(D:=-\{(2,1),(1,1),\dots,(1,1)\}\), a multiple of the boundary divisor of two point collisions, and \((1,\omega_i):=\{(1,\omega_i),(1,1),\dots,(1,1)\}\) play a central role in the paper.
The Fock space \(\mathcal{F}_{\mathcal{A}_n}\) modeled on \(\mathcal{A}_n\) is graded isomorphic to
\[
\mathcal{F}_{\mathcal{A}_n}=\bigoplus\limits_{m\geq0}H^*_T(\text{Hilb}_m(\mathcal{A}_n),\mathbb{Q})\,,
\]
and the results of this paper are naturally stated in terms of this Fock space after extension of coefficients. The underlying reason is that \(\mathcal{F}_{\mathcal{A}_n}\otimes\mathbb{Q}(t_1,t_2)\) is isomorphic to a subspace in the basic representation of the affine Lie algebra \(\widehat{\mathfrak{gl}}(n+1)\). Quantum product on
\[
QH^*_T(\text{Hilb}_m(\mathcal{A}_n)):=H^*_T(\text{Hilb}_m(\mathcal{A}_n),\mathbb{Q}) \otimes\mathbb{Q}(t_1,t_2)((q))[[s_1,\dots,s_n]]
\]
is a deformation of the classical cup product defined via three-point genus \(0\) Gromov-Witten invariants of the Hilbert scheme.
Two-point invariants can be encoded into an operator \(\Theta\) on the Fock space. To generate them, the authors define explicit representation-theoretic operators \(\Omega_0\), \(\Omega_+\) and prove that \(\Theta=(t_1+t_2)(\Omega_0+\Omega_+)\). This leads to their main result, explicit formulas for quantum multiplication operators \(M_D\) and \(M_{(1,\omega_i)}\) in terms of their classical counterparts \(M_D^{cl}\) and \(M_{(1,\omega_i)}^{cl}\):
\[
M_D=M_D^{cl}+(t_1+t_2)\,q\frac{\partial}{\partial q}(\Omega_0+\Omega_+)
\]
\[
M_{(1,\omega_i)}=M_{(1,\omega_i)}^{cl}+(t_1+t_2)\,s_i\frac{\partial}{\partial s_i}\Omega_+\,.
\]
One consequence of these formulas is that the multiplication operators are rational in \(q,s_1,\dots,s_n\) in contrast to the case of general surfaces, where only rationality in \(q\) is expected. They also imply a correspondence between multiplication by divisors in \(QH^*_T(\text{Hilb}_m(\mathcal{A}_n),\mathbb{Q})\) and in the equivariant Gromov-Witten theory of \(\mathcal{A}\times\mathbb{P}^1\) relative to the fibers at \(0\), \(1\) and \(\infty\). Under an additional conjecture that all joint eigenspaces of \(M_D\) and \(M_{(1,\omega_i)}\) are one-dimensional, and hence \(D\), \((1,\omega_i)\) generate \(QH^*_T(\text{Hilb}_m(\mathcal{A}_n),\mathbb{Q})\), the authors prove complete Gromov-Witten/Hilbert correspondence for \(\mathcal{A}_n\). If true, this is another special feature of these surfaces, which fails already for \(\mathbb{P}^2\), at least if no change of curve class variables is allowed. At the end of the paper the authors briefly discuss the quantum differential system \(q\frac{\partial}{\partial q}\psi=M_D\psi\), \(\,s_i\frac{\partial}{\partial s_i}\psi=M_{(1,\omega_i)}\psi\) and its monodromy, and indicate how their formulas can be extended to crepant resolutions of \(D\) and \(E\) singularities. Hilbert scheme of points; \(A_n\) singularity; small quantum cohomology; Nakajima basis; Gromov-Witten/Hilbert correspondence Maulik, D; Oblomkov, A, Quantum cohomology of the Hilbert scheme of points on \({\mathcal{A}}_n\)-resolutions, J. Am. Math. Soc., 22, 1055-1091, (2009) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology of the Hilbert scheme of points on \(\mathcal {A}_n\)-resolutions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author gives an outline of his construction of representations of affine Lie algebras in cohomology of a certain Lagrangian subvariety in the space of framed ASD connections on the ALE spaces associated to minimal resolutions of simple surface singularities. A similar construction for finite-dimensional simple Lie algebras was discussed in an earlier paper [\textit{H. Nakajima}, Int. Math. Res. Not. 1994, No.~2, 61-74 (1994; Zbl 0832.58007)]. affine Lie algebras; resolutions of simple surface singularities Modifications; resolution of singularities (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Applications of global analysis to structures on manifolds Gauge theory on resolutions of simple singularities and affine Lie algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a complex \(K3\) surface with an effective symplectic action of a group \(G\). Then \(G\) is a subgroup of the Mathieu group \(M_{23}\subset M_{24}\) [\textit{S. Mukai}, Invent. Math. 94, No. 1, 183---221 (1988; Zbl 0705.14045)], and all possible actions are classified by the 82 topological types of the quotient \(X/G\) [\textit{G. Xiao}, Ann. Inst. Fourier 46, No. 1, 73---88 (1996; Zbl 0845.14026)].
The \(G\)-invariant Hilbert scheme of \(X\) (of length \(n\)) parametrizes the \(G\)-invariant length \(n\) subschemes \(Z\subset X\) of \(X\). It can be identified with the \(G\)-fixed point locus in the Hilbert scheme of points \(\operatorname{Hilb}^n(X)^G\subset \operatorname{Hilb}^n(X)\).
In the paper under review, the authors define the \(G\)-\textit{fixed partition function} of \(X\) as the function \(Z_{X,G}\colon \mathbb{H}\to \mathbb{C}\) of the upper-half plane \(\mathbb{H}\) sending \(\tau\in\mathbb{H}\) to
\[
Z_{X,G}(q):=\sum_{n=0}^\infty e(\operatorname{Hilb}^n(X)^G)q^{n-1},
\]
where \(q=\operatorname{exp}(2\pi i \tau)\) and \(e(-)\) is the topological Euler characteristic.
The authors prove that \(Z_{X,G}(q)^{-1}\) is a modular cusp form of weight \(\frac{1}{2}e(X/G)\) with respect to the congruence subgroup \(\Gamma_0(|G|)\).
They also give an explicit formula of \(Z_{X,G}(q)\) in terms of the Dedekind eta function. The formula is a product in which the factors are related to the singular points of \(X/G\). Using this result, the form \(Z_{X,G}(q)^{-1}\) is explicitly given for all 82 possible types of \((X,G)\). \(K3\) surfaces; modular forms; Hilbert schemes; group actions \(K3\) surfaces and Enriques surfaces, Holomorphic symplectic varieties, hyper-Kähler varieties, Parametrization (Chow and Hilbert schemes), Fourier coefficients of automorphic forms \(G\)-fixed Hilbert schemes on \(K3\) surfaces, modular forms, and eta products | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 étale homology theory is brought to bear with the \(K\)-theoretic and cohomological methods used by several authors in the last years in a way that yields more powerful tools to deal with \(A_ 0(X)_{tor}\), the torsion part of the group of rational classes of zero-cycles on an algebraic scheme \(X\) over a field \(k\). The main result establishes that for any prime number \(\ell\) which is different from the characteristic of \(k\) there is a canonical surjective map \(\varphi_ X:H_ 1(X,\mathbb{Q}_ \ell/\mathbb{Z}_ \ell)\to A_ 0(X)\{\ell\}\) from the étale homology group of \(X\) onto the group of elements in \(A_ 0(X)\) of finite order not divisible by \(\ell\), and that \(\varphi_ X\) is an isomorphism if there is an open immersion \(X\hookrightarrow Z\) with \(Z\) an algebraic proper scheme over \(k\) such that \(Z-X\) is connected and with the property (let us call it \(S)\) that an element in \(H^ 1(X,\mathbb{Q}_ \ell)\) is zero if its restriction to any regular subscheme of \(Z\) is zero. The main corollary is that if \(X\) is complete and satisfies \(S\), then \(A_ 0(X)\{\ell\}\simeq\pi_ 1^{ab}(X)\otimes\mathbb{Q}_ \ell/\mathbb{Z}_ \ell\). If \(X\) is in addition smooth, then this isomorphism yields Roitman's isomorphism \(A_ 0(X)\{\ell\}\simeq\text{Alb}(X)\{\ell\}\). The relation of M. Levine's generalization of Roitman's isomorphism to the main corollary is also clarified. étale homology theory; torsion part of the group of rational classes of zero-cycles S. Saito, ''Torsion zero-cycles and étale homology of singular schemes,'' Duke Math. J., vol. 64, iss. 1, pp. 71-83, 1991. Étale and other Grothendieck topologies and (co)homologies, Algebraic cycles Torsion zero-cycles and étale homology of singular schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R\) be a polynomial ring over a field, occasionally (but not usually) of characteristic zero. Let \(M\) be a finitely generated graded \(R\)-module. Let \(\mathcal A\) be a homogeneous presentation matrix for \(M\), and assume that the ideal \(I_t (\mathcal A)\) of maximal minors of \(\mathcal A\) has maximal codimension in \(R\). Let \(X = \text{Proj}(R/I_t(\mathcal A))\). Assuming that \(X\) is smooth in a sufficiently large open subset, \(\dim X \geq 1\), and a weak technical condition, the author shows that the local graded deformation functor of \(M\) is isomorphic to the local Hilbert (scheme) functor at \(X \subset \text{Proj}(R)\). The author shows that the Hilbert scheme is smooth at \((X)\), and obtains an explicit formula for the dimension of its local ring. As a corollary he proves a conjecture of his with Miró-Roig that the closure of the locus of standard determinantal schemes with fixed degrees of the entries in a presentation matrix is a generically smooth component of the Hilbert scheme, and he proves another conjecture of the same authors about the dimension of this component if \(\dim X \geq 1\). Finally, he obtains results about the vanishing of the normal sheaf of \(X\) in \(\text{Proj}(R)\), and shows how his results can be extended from \(R\) to any Cohen-Macaulay quotient of \(R\). Hilbert scheme; determinantal scheme; parametrization; deformation; module of maximal grade; normal sheaf; André-Quillen cohomology; vanishing of Ext J. O. Kleppe, Deformations of modules of maximal grade and the Hilbert scheme at determinantal schemes. \textit{J. Algebra} 407 (2014), 246 -274. MR3197160 Zbl 1328.14080 Determinantal varieties, Parametrization (Chow and Hilbert schemes), Homological functors on modules of commutative rings (Tor, Ext, etc.), Deformations and infinitesimal methods in commutative ring theory, Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) Deformations of modules of maximal grade and the Hilbert scheme at determinantal 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 authors develop a classification of simple parametrized curves over algebraically closed fields of characteristic \(p>0\); the case of simple multigerms of curves over the field of complex numbers was treated by \textit{P. A. Kolgushkin} and \textit{R. R. Sadykov} [Rev. Mat. Complut. 14, No. 2, 311--344 (2001; Zbl 1072.14501)]. In this paper, the authors treat the first part of this classification, of pairs of curves with regular first component.
The classification is based upon first finding a list of non-simple confining singularities, then finding a weak normal form using only left-equivalence, independent of the characteristic, and finally using right-equivalence for normal forms, characteristic dependent. \(\mathcal{A}\)-equivalence; characteristic \(p\); parameterized curves; simple singularities of multigerms Singularities of curves, local rings, Singularities in algebraic geometry, Plane and space curves Simple singularities of reducible curves in characteristic \(p > 0\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert scheme \(\mathrm{Hilb}^n_{p(t)}\) parametrizes closed subschemes \(X \subset \mathbb P^n_k\) with Hilbert polynomial \(p(t)\), and hence the corresponding graded ideals in \(J \subset k[x_0, \dots, x_n]\). Algebraists study the geometry of the Hilbert scheme through special ideals, such as Borel-fixed ideals. For example, \textit{A. Reeves} and \textit{M. Stillman} showed that the Hilbert scheme is smooth at a lexicographic point [J. Algebr. Geom. 6, 235--246 (1997; Zbl 0924.14004)]. \textit{P. Lella} and \textit{M. Roggero} showed that smoothness of the Hilbert scheme at certain ideals implies that the component on which they lie is rational [Rend. Semin. Mat. Univ. Padova 126, 11--45 (2011; Zbl 1236.14006)]. Recently \textit{A. P. Staal} showed that with an appropriate probability measure, more than half of all Hilbert schemes are smooth and irreducible [Math. Z. 296, 1593--1611 (2020; Zbl 1451.14010)].
Here the authors say two saturated Borel-fixed ideals \(J,J^\prime\) defining points in \(\mathrm{Hilb}^n_{p(t)}\) are \textit{Borel-adjacent} if for \(r \gg 0\), the respective monomial bases \(F, F^\prime\) of \(J_r, J_r^\prime\) have the property that the sets \(F \setminus F^\prime\) and \(F^\prime \setminus F\) have the same linear syzygies. The authors prove that in this case there is a rational curve on \(\mathbf{Hilb}^n_{p(t)}\) passing through the corresponding points, so that \(J, J^\prime\) lie on the same component. They then form the Borel Graph of \(\mathrm{Hilb}^n_{p(t)}\) by taking the vertices to be Borel-fixed ideals with edges between the Borel-adjacent ideals. This graph is a subgraph of the \(T\)-graph introduced by \textit{K. Altmann} and \textit{B. Sturmfels} [J. Pure Appl. Algebra 201, 250--263 (2005; Zbl 1088.13012)]. Each term order induces an orientation of the Borel graph, the corresponding directed graphs are called degeneration graphs, which the authors classify by means of a polyhedral fan called the Gröbner fan associated to \(\mathbf{Hilb}^n_{p(t)}\). Constructing minimal spanning trees for some of these degeneration graphs, the authors recover the connectedness theorem of \textit{R. Hartshorne} [Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)]. Furthermore, they conjecture that the number of irreducible components of \(\mathrm{Hilb}^n_{p(t)}\) is at least the maximum number of vertices with no incoming edge in any degeneration graph. The paper has many helpful examples and pictures. Hilbert schemes; strongly stable ideals; Gröbner degenerations; polyhedral fans; connectedness Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Combinatorial aspects of commutative algebra The Gröbner fan of the Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Differential operators on smooth schemes have played a central role in the study of embedded desingularization. \textit{J. Giraud} [Math. Z. 137, 285--310 (1972; Zbl 0275.32003) and Ann. Sci. Éc. Norm. Supér. 8, 201--234 (1975; Zbl 0306.14004)] provided an alternative approach to the form of induction used by Hironaka in his desingularization theorem (over fields of characteristic zero). In doing so, Giraud introduced technics based on differential operators. This result was important for the development of algorithms of desingularization in the late 80's (i.e., for constructive proofs of Hironaka's theorem).
More recently, differential operators appear in the work of \textit{J. Wlodarczyk} [J. Am. Math. Soc. 18, No. 4, 779--822 (2005; Zbl 1084.14018)] and also on the notes of \textit{J. Kollár} [Lectures on resolution of singularities, Ann. Math. Stud. 166 (2007; Zbl 1113.14013)]. The form of induction used in Hironaka's desingularization theorem, which is a form of elimination of one variable, is called maximal contact. Unfortunately it can only be formulated over fields of characteristic zero. In this paper we report on an alternative approach to elimination of one variable, which makes use of higher differential operators. These results open the way to new invariants for singularities over fields of positive characteristic [\textit{O. Villamayor}, Adv. Math. 213, No. 2, 687--733 (2007; Zbl 1118.14016), preprint \url{arXiv:math.AG/0606796}]. Villamayor, O.: Differential operators on smooth schemes and embbeded singularities. Rev. Un. Mat. Argentina 46 (2005), no. 2, 1-18. Global theory and resolution of singularities (algebro-geometric aspects) Differential operators on smooth schemes and embedded 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 Some connections of the ordinary intersection numbers of the Hilbert scheme of points on surfaces to the Hurwitz numbers for \(\mathbb{P}^1\) as well as to the relative Gromov-Witten invariants of \(\mathbb{P}^1\) are established. Li W.-P., Qin Z., Wang W.: Hilbert scheme intersection numbers, Hurwitz numbers, and Gromov-Witten invariants. Contemp. Math. 392, 67--81 (2005) Parametrization (Chow and Hilbert schemes), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Hilbert scheme intersection numbers, Hurwitz numbers, and Gromov-Witten invariants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a smooth projective variety \(X\), the Hilbert scheme \(X^{[n]}\) of \(n\) points on \(X\) is again a smooth projective variety of dimension \(\dim X^{[n]} = n\, \dim X\). Each vector bundle \(E\) on \(X\) defines the \emph{vector bundle} \(E^{[n]} = p_{2,*}p_1^*E\). Here \(p_k\) is the projection to the \(k\)-factor from the universal subscheme \(\Pi_n \subset X \times X^{[n]}\). The bundle \(E^{[n]}\) is called the Fourier-Mukai transform of \(E\) (with respect to \(\Pi_n\)). By work of Lehn and Lehn-Sorger, these transforms are important tools to study the topology and geometry of Hilbert schemes. Conversely, they are useful to study bundles on \(X\) itself e.g. by work of Voison, Ein-Lazarsfeld and Agostini.
The present article enhances the Fourier-Mukai transform to so-called V-cotwisted Hitchin pairs \((E, \theta)\). Here \(E\) and \(V\) are vector bundles on \(X\) and \(\theta\colon E \otimes V \to E\) is a section. The outcome of the enhanced Fourier-Mukai transform are \(V^{[n]}\)-cotwisted Hitchin pairs \((E^{[n]}, \theta^{[n]})\) on \(X^{[n]}\). Note here that if \(V = T_X\), then \(V^{[n]}\cong T_{X^{[n]}}(-\log B_n)\) (by a result of Stapleton) where \(B_n\subset X^{[n]}\) is the locus of non-reduced sub-schemes of \(X\). In particular, the enhanced Fourier-Mukai transforms of Higgs bundles (i.e. \(T_X\)-cotwisted Hitchin pairs) are logarithmic Higgs bundles on \(X^{[n]}\).
After establishing basic results on the enhanced Fourier-Mukai transform, which are of independent interest, the authors prove various interesting results on the relationship between Hitchin pairs on \(X\) and their enhanced Fourier-Mukai transforms (similar results for vector bundles were already obtained by the second author), for example:
\begin{itemize}
\item If \((E, \theta)^{[n]} \cong (F, \eta)^{[n]}\) on \(X^{[n]}\), then \((E,\theta) \cong (F, \eta)\) on \(X\) where \(X\) is any smooth projective curve of genus \(\geq 1\) or any smooth quasi-projective variety of \(\dim X \geq 2\);
\item relationship between the stability conditions for \((E, \theta)\) on \(X\) and \((E, \theta)^{[n]}\) on \(X^{[n]}\) for any smooth projective curve \(X\).
\end{itemize} logarithmic Higgs bundle; Hilbert scheme; Fourier-Mukai transformation; stability Stacks and moduli problems, Algebraic moduli problems, moduli of vector bundles, Coverings of curves, fundamental group, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Parametrization (Chow and Hilbert schemes) Fourier-Mukai transformation and logarithmic Higgs bundles on punctual 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 The main result of this paper is a quite general procedure for constructing smooth connected curves in projective 3-space which are singular points on the Hilbert scheme, sitting on a reduced component. One starts by constructing a flat family of curves where the special fiber is contained in a surface of smaller degree than expected. Using liaison, this jumping behaviour is inherited by the cohomology of the normal bundles of the linked curves.
It is still open whether there exists such an example which lies in only one irreducible component of the Hilbert scheme. singularity of space curves; postulation deficiency; liaison; cohomology of the normal bundles of the linked curves; Hilbert scheme Ph. Ellia and M. Fiorentini : Défaut de postulation et singularités du Schéma de Hilbert . Annali Univ. di Ferrara 30 (1984) 185-198. Families, moduli of curves (algebraic), Singularities of curves, local rings, Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Défaut de postulation et singularités du schéma de Hilbert | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(K3\) surface whose Picard group equals the integers. If the Picard group is generated by a smooth curve with self-intersection number \(2(g-1)n^2\), then the author shows that the Hilbert scheme of \(g\) points on \(S\) is a Lagrangian fibration. \textit{D. Markushevich} [Manuscr. Math. 120, No.~2, 131--150 (2006; Zbl 1102.14031)] has obtained similar results, and \textit{M. G. Gulbrandsen} [Lagrangian fibrations on generalised Kummer varieties, \texttt{math.AG/0510145}] has used similar techniques for generalized Kummer varieties. Lagrangian fibration; Hilbert scheme Justin Sawon, Lagrangian fibrations on Hilbert schemes of points on \?3 surfaces, J. Algebraic Geom. 16 (2007), no. 3, 477 -- 497. Algebraic moduli problems, moduli of vector bundles, Parametrization (Chow and Hilbert schemes), Fibrations, degenerations in algebraic geometry, \(K3\) surfaces and Enriques surfaces Lagrangian fibrations on Hilbert schemes of points on \(K3\) surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Fix an integer \(n\). Let \(X_n\) be the Hilbert scheme parametrizing subschemes of \(\mathbb{C}^2\) of finite length \(n\). Let \(Z_n\) be the center of the algebra \(\mathbb{C} [S_n]\), where \(S_n\) denotes the symmetric group. The author proves that there is a natural filtration \(F\) on \(Z_n\), compatible with the product, such that the graded rings \(H^{2*} (X_n,\mathbb{C}^2)\) and \(Gr_*^F(Z_n)\) are isomorphic. Moreover the author conjectures that there exists an analogous isomorphism in the more general case of a crepant resolution \(M\to \mathbb{C}^{2n}/G\), where \(G\) is a finite subgroup of \(Sp_{2n} (\mathbb{C})\). equivariant cohomology ring; filtration; crepant resolution; group action; Hilbert scheme Vasserot, E, Sur l'anneau de cohomologie du schéma de Hilbert de \({\mathbb{C}}^2\), C. R. Acad. Sci. Paris Sér. I Math., 332, 7-12, (2001) Parametrization (Chow and Hilbert schemes), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) On the cohomology ring of the Hilbert scheme of \(\mathbb{C}^2\) | 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 In this article the author generalizes the notions of smooth, unramified and étale morphisms to the category of fine logarithmic schemes and gives criteria which are analogs of criteria in the category of schemes.
First he recalls the definition of logarithmic schemes, which have been introduced by \textit{K. Kato}: a log structure \((M_X , \alpha)\) on a scheme \(X\) is a sheaf of monoids \(M_X\) on the étale site on \(X\) and a homomorphism \(\alpha : M_X \to {\mathcal O}_X\) such that \(\alpha ^{-1} ({\mathcal O} _X ^*) \buildrel {\scriptstyle \simeq} \over \longrightarrow {\mathcal O} _X ^*\); a chart \((P \to M_X)\) of \(M_X\) is a homomorphism \(P_X \to M_X\), where \(P_X\) is the constant sheaf of monoids of value \(P\), with \(P\) finitely generated and integral.
Then, as in the category of schemes, we can give the following definitions: A morphism \(f : X \to Y\) of fine log schemes is formally smooth (resp. formally unramified, resp. formally étale) if for any strict closed immersion of affine fine log schemes \(Z_0 \buildrel {\scriptstyle i} \over \hookrightarrow Z\) and any morphism \(Z \to Y\) the map \(\text{Hom}_Y (Z,X) \to \text{Hom}_Y (Z_0 ,X)\) induced by \(i\) is surjective (resp. injective, resp. bijective); and we get the same basic properties for these morphisms of fine log schemes when we consider composition, base change, etc... A morphism \(f : X \to Y\) of fine log schemes is called smooth (resp. unramified, resp. étale) if it is formally smooth (resp. formally unramified, resp. formally étale) and the underlying morphism of schemes is locally of finite presentation. We have again the same properties as in the category of schemes, and the author gives the following characterization:
Theorem: Let \(f : X \to Y\) be a morphism of fine log schemes such that the underlying morphism of schemes is of finite presentation; the following properties are equivalent:
(a) \(f\) is unramified.
(b) The diagonal \(\Delta : X \to X \times _Y X\) is étale.
(c) The differential module \(\Omega _{X/Y} ^1\) is zero.
(d) For any \(y \in Y\) the fibre \(X_y = f^{-1} (y)\) provided with the induced log structure is unramified over Spec\(k(y)\).
(e) Étale locally on \(X\) there exists a chart \( (Q \to M_X , P \to M_Y, P \to Q)\) extending a given chart on \(P \to M_Y\) such that: (i) \(P^{gp} \to Q^{gp}\) is injective with finite cokernel of order invertible on \(X\), (ii) the induced morphism of schemes \(X \to Y \times _{\text{Spec}(\mathbb Z [P ])} \text{Spec}(\mathbb Z [Q ])\) is unramified.
In the last part the author gives criteria for a morphism to be smooth, flat or étale. First he shows that the morphism \(f\) is smooth if and only if locally it can be factorized over an étale map into the standard log affine space over \(Y\). Then he recalls the following theorem of Kato:
Theorem: Let \(f : X \to Y\) be a morphism of fine log schemes; the following properties are equivalent:
(a) \(f\) is smooth (resp. étale).
(b) Étale locally on \(X\) there exists a chart \( (Q \to M_X , P \to M_Y, P \to Q)\) of \(f\) extending a given chart on \(P \to M_Y\) such that: (i) \(P^{gp} \to Q^{gp}\) is injective and the torsion part of its cokernel (resp. its cokernel) is finite of order invertible on \(X\), (ii) the induced morphism of schemes \(X \to Y \times _{\text{Spec}({\mathbb Z} [P ])} \text{Spec}(\mathbb Z [Q ])\) is smooth (resp. étale).
The author gives the following definition: A morphism of fine log schemes \(f : X \to Y\) is called ``flat'' if fppf locally on \(X\) and \(Y\), there exist a chart \( (Q \to M_X ,P \to M_Y, P \to Q) \) of \(f\) such that: (i) \(P^{gp} \to Q^{gp}\) is injective, (ii) the induced morphism of schemes \(X \to Y \times _{\text{Spec}(\mathbb Z [P ])} \text{Spec}(\mathbb Z [Q ])\) is flat.
With this definition he can generalize the usual fibre criterion for flatness: Let \(f : X \to Y\) be an \(S\)-morphism of fine log schemes, with \(X/S\) flat, then \(f\) is flat if and only if the induced maps on the fibres \(f_s : X_s \to Y_s\) are flat.
He can also generalize criteria for étale and smooth morphisms: \(f : X \to Y\) is étale if and only if \(f\) is flat and unramified; \(f : X \to Y\) is smooth if and only if \(f\) is flat and the induced maps on the fibres \(f_y : X_y \to y\) are smooth. category of fine logarithmic schemes; log structure; morphism of fine log schemes; fibre criterion for flatness Werner Bauer, On smooth, unramified étale and flat morphisms of fine logarithmic schemes, Math. Nachr. 176 (1995), 5 -- 16. Local structure of morphisms in algebraic geometry: étale, flat, etc., Schemes and morphisms, Étale and other Grothendieck topologies and (co)homologies On smooth, unramified, étale and flat morphisms of fine logarithmic 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 This article deals with the exposition of a new approach to the problem of choice of suitable centers in a Hironaka style desingularization, i.e. in a resolution of singularities by a sequence of blowing ups. Although constructive proofs of resolution of singularities in characteristic zero have been known and refined for some time now, there are major obstructions to generalizing them to the open case of positive characteristic, e.g. the key inductive argument on the descent in ambient dimension involves `hypersurfaces of maximal contact' which do not exist in positive characteristic. The approach explained in this article replaces the restriction to such hypersurfaces by local projections which exist in any characteristic and which also allow related resolution invariants with significantly easier globalization. It is shown that in characteristic zero this construction provides the same information as the traditional one. A drawback and still open problem of this approach is that it only simplifies the singularities to a monomial case; this in turn is not as straightforward as in Hironakas classical approach and its resolution is not shown in all generality, but only under some additional conditions. As a sample application the case of 2-dimensional embedded schemes is treated. constructive resolution of singularities; desingularization; choice of center; projections ,\textit{Techniques for the study of singularities with applications to resolution of 2-dimensional} \textit{schemes}, Math. Ann. 353 (2012), no. 3, 1037--1068. http://dx.doi.org/10.1007/s00208-011-0709-5.MR2923956 Global theory and resolution of singularities (algebro-geometric aspects) Techniques for the study of singularities with applications to resolution of 2-dimensional schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S\) be a smooth projective algebraic surface \(S\) over an algebraically closed field of characteristic 0. One assumes the polarization \(H\) to be chosen so that \(gcd(r,c_{1}H,(1/2)c_{1}(c_{1}-K_{S})-c_{2})=1\), where \(K_{S}\) is the canonical class of the surface \(S\). A new compactification of the variety of moduli of Gieseker stable vector bundles of rank \(r=2\) with Chern classes \(c_{1}\) and \(c_{2}\) is constructed for the case in which the universal family of stable sheaves is defined and there are no strictly semistable sheaves.
This compactification is a subvariety in the Hilbert scheme of subschemes of a Grassmann manifold with fixed Hilbert polynomial. It is obtained from the variety of bundle moduli by adding points corresponding to locally free sheaves on surfaces which are modifications of the initial surface. Moreover, a morphism from the new compactification of the moduli space to its Gieseker-Maruyama compactification is constructed. Gieseker stable vector bundles; stable sheaves; projective surface; compactification of moduli space; Hilbert scheme; Grassmann manifold; universal family N. V. Timofeeva, ''Compactification in Hilbert scheme of moduli scheme of stable 2-vector bundles on a surface,'' Mat. Zametki 82(5), 756--769 (2007) [Math. Notes 82 (5), 677--690 (2007)]. Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Parametrization (Chow and Hilbert schemes) A compactification of the moduli variety of stable vector 2-bundles on a surface in the Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author studies tautological vector bundles and their tenor products over the Hilbert scheme of points on a smooth quasi-projective algebraic surface, with a particular interest in their cohomology. In section one, standard notations and preliminary results are recalled. For a smooth quasi-projective algebraic surface \(X\), let \(X^{[n]}\) be the Hilbert scheme parametrizing length-\(n\) \(0\)-dimensional closed subschemes of \(X\). It is well-known that there exists a derived equivalence
\[
\Phi: \text{D}^b(X^{[n]}) \to \text{D}_{S_n}^b(X^n)
\]
where \(\text{D}^b(X^{[n]})\) is the derived category of bounded complexes of coherent sheaves on \(X^{[n]}\), and \(S_n\) acts on \(X^n\) by permutation. In section two, let \(L^{[n]}\) be the tautological rank-\(n\) bundle on \(X^{[n]}\) corresponding to a line bundle \(L\) on \(X\). The author identifies the image \(\Phi(L^{[n]})\) in terms of a complex \(\mathcal C_L^\bullet\) of \(S_n\)-equivariant sheaves on \(X^n\), and characterize the image \(\Phi(L^{[n]} \otimes \cdots \otimes L^{[n]})\) in terms of the hyperderived spectral sequence associated to the derived tensor products of the complex \(\mathcal C_L^\bullet\). In sections three and four, the derived direct images of \(L^{[n]} \otimes L^{[n]}\) and \(\Lambda^k L^{[n]}\) under the Hilbert-Chow morphism are obtained. In section five, the author computed the cohomology of \(X^{[n]}\) with values in \(L^{[n]} \otimes L^{[n]}\) and \(\Lambda^k L^{[n]}\). Hilbert schemes of points; tautological bundles; cohomology Scala L: Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles. Duke Math. J 2009,150(2):211--267. 10.1215/00127094-2009-050 Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Representations of finite symmetric groups Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The minimal model program for the Hilbert schemes \(\text{Hilb}^n \mathbb{P}^2\) of \(n\)-points on the complex projective plane \(\mathbb{P}^2\) and the correspondence between wall-crossings in effective cones of divisors and wall-crossings in Bridgeland stability manifolds are explored in detail in [\textit{D. Arcara} et al., Adv. Math. 235, 580--626 (2013; Zbl 1267.14023)]. The paper under review studies the relevant problems in ibid. in a broader setting.
The Sklyanin algebra \(S=\text{Skl}(E, \mathscr{L}, \lambda)\) is in general a noncommutative algebra, constructed from the data: an elliptic curve \(E\), a degree \(3\) line bundle \(\mathscr{L}\) on \(E\) and a translation \(\lambda\) on \(E\). When \(\lambda\) is chosen to be the identity, \(S\simeq\mathbb{C}[x_0, x_1, x_2]\), the homogeneous coordinate ring of \(\mathbb{P}^2\). The category of \(\text{qgr-}S\) (see Section 2) corresponds to, in commutative case, the category of coherent sheaves on \(\mathbb{P}^2\). One can introduce Chern classes for the objects in \(\text{qgr-}S\) and hence the notion of slope stability. The moduli space \(\text{Hilb}^n S\) of semistable objects with variants \((\text{rk}, c_1, \chi)=(1, 0, 1-n)\) gives a generic deformation of \(\text{Hilb}^n \mathbb{P}^2\) by [\textit{T. A. Nevins} and \textit{J. T. Stafford}, Adv. Math. 210, No. 2, 405--478 (2007; Zbl 1116.14003)] and [\textit{N. Hitchin}, Mosc. Math. J. 12, No. 3, 567--591 (2012; Zbl 1267.32010)].
In a similar fashion to Bridgeland, Arcara-Bertram's construction for stability conditions on projective surfaces, the authors construct stability conditions \(\sigma=\sigma(s, t)\) on the derived category \(D^b(\text{qgr-}S)\), for \((s, t)\) in the upper-half plane. Let \(\mathfrak{M}_{\sigma}(n)\) be the moduli space of stable objects of invariants \((\text{rk}, c_1, \chi)=(1, 0, 1-n)\) with respect to \(\sigma\). The authors show that \(\mathfrak{M}_{\sigma}(n)\) coincides with \(\text{Hilb}^n S\) for \(s<0\) and \(t\gg 0\). The proof involves an interpretation for the moduli space as a moduli space of quiver representations using G.I.T.
One main theorem is that for general \(\sigma\) not lying on destabilizing walls, \(\mathfrak{M}_{\sigma}(n)\) is a nonsingular, projective variety of dimension \(2n\) so long as it is nonempty, and that for \(\sigma\neq \sigma'\) not on any destabilizing wall, \(\mathfrak{M}_{\sigma}(n)\) and \(\mathfrak{M}_{\sigma'}(n)\) are birationally equivalent.
Fix \(s\in \mathbb{R}\), then for the heart of the bounded \(t\)-structure corresponding to \(s\), one can let the parameter \(t\) decrease. When \(t\) crosses a destabilizing wall in the upper- half plane, there are induced birational transforms between the moduli spaces corresponding to \(\sigma(s, t)\) on the two sides of the wall. By the variation of G.I.T., the authors establish for \(\text{Hilb}^n S\) a one-to-one correspondence between the destabilizing walls in the upper-half plane of stability conditions and the stable base locus walls in the effective cone of divisors. In particular, for \(\text{Hilb}^n \mathbb{P}^2\), an explicit correspondence between the destabilizing walls in the second quadrant and the stable base locus walls is confirmed, as conjectured by Arcara, Bertram, Coskun and Huizenga in [Zbl 1267.14023]. birational geometry; stability condition; geometric invariant theory Algebraic moduli problems, moduli of vector bundles, Minimal model program (Mori theory, extremal rays), Parametrization (Chow and Hilbert schemes) The minimal model program for deformations of Hilbert schemes of points on the projective plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The arithmetic Hilbert-Samuel theorem in the context of Arakelov geometry was firstly derived by \textit{H. Gillet} and \textit{C. Soulé} [Invent. Math. 110, No. 3, 473--543 (1992; Zbl 0777.14008)]. It relates the covolumes of lattices of global sections of powers of a hermitian ample line bundle on an arithmetic variety to the height of this variety. In the paper under review, the authors proved a general arithmetic Hilbert-Samuel theorem for powers of line bundles twisted by the canonical bundle, which are endowed with the semi-positive metrics of finite energy. Such metrics are defined for line bundles on non-compact varieties and are supposed to have suitable singularities along the boundary of the compactification, so that the height can be defined and is a finite real number. To describe the main result in the paper under review, let \(\mathcal{X}\to \mathbb{Z}\) be an integral flat projective scheme of relative dimension \(n\), with smooth generic fibre \(\mathcal{X}_{\mathbb{Q}}\). Suppose there exists an invertible sheaf \(\mathcal{K}\) with \(\mathcal{K}_{\mathbb{Q}}=K_{\mathcal{X}_{\mathbb{Q}}}\), which is a model of the canonical sheaf of \(\mathcal{X}_{\mathbb{Q}}\). Let \({\bar \mathcal{L}}\) be a semi-positive line bundle of finite energy. Assume that \(\mathcal{L}_{\mathbb{Q}}\) is ample and \(\mathcal{L}\) is nef on vertical fires. Then there is an asymptotic expansion
\[
\widehat{\text{deg}}H^0(\mathcal{X},\mathcal{L}^{\otimes k}\otimes \mathcal{K})_{L^2}=h_{\bar \mathcal{L}}(\mathcal{X})\frac{k^{n+1}}{(n+1)!}+o(k^{n+1})\quad \text{as}\quad k\to +\infty.
\]
The class of metrics the author considered includes the log-singular metrics of Burgos-Kramer-Kühn [\textit{J. I. Burgos Gil} et al., Doc. Math., J. DMV 10, 619--716 (2005; Zbl 1080.14028); J. Inst. Math. Jussieu 6, No. 1, 1--172 (2007; Zbl 1115.14013)], and specifying to these log-singular metrics, the authors can relax the ampleness assumption and replace \(\mathcal{K}\) by any other log-singular hermitian line bundle. As an application of there main result, the authors investigate the Hilbert modular surfaces and establish an arithmetic analogue of the classical result expressing the dimensions of spaces of cusp forms in terms of special values of Dedekind zeta functions. Arakelov theory; heights; cusp forms; pluripotential theory; Monge-Ampere operators; finite energy functions R. Berman, G. Freixas i Montplet, An arithmetic Hilbert-Samuel theorem for singular Hermitian line bundles and cusp forms. Compos. Math. 150, 1703--1728 (2014) Arithmetic varieties and schemes; Arakelov theory; heights, Modular and Shimura varieties, Holomorphic bundles and generalizations, Complex Monge-Ampère operators An arithmetic Hilbert-Samuel theorem for singular Hermitian line bundles and cusp 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 The author studies and classifies simple singularities of space curves. A singularity is called simple if it has only finitely many non-equivalent deformations. More precisely, if \((X,0)\subset ({\mathbb C}^n,0)\) is an analytic subvariety, then one may act by the obvious group of automorphisms on the space of such \((X,0)\). If such a variety has a neighbourhood (in the set of all such) which intersects atmost finitely many orbits by the above group, one says that the singularity is simple. Simple complete intersection space curves were classified by \textit{M. Giusti} [in: Singularities, Summer Inst., Arcata 1981, Proc. Symp. Pure Math. 40, Part I, 457-494 (1983; Zbl 0525.32006)]. In this paper, the author classifies such space curve singularites for non-complete intersections. The starting point is that such curves are defined by the \(r\times r\) minors of a \(r\times (r+1)\) matrix with coefficients in the germs of functions of 3-space at the origin. Further, we may assume that all entries vanish at the origin. This is the celebrated Hilbert-Burch theorem. The case \(r=1\) gives the complete intersection case, treated by Giusti. The author shows that if \(r\geq 2\) and if the singularity is simple, then \(r=2\). Using this the author classifies all such singularities. simple singularities; space curves; non-complete intersections; Hilbert-Burch theorem Frühbis-Krüger, A., Classification of simple space curves singularities, Commun. Algebra, 27, 3993-4013, (1999) Singularities of curves, local rings, Plane and space curves, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Local complex singularities Classification of simple space curve singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review deals with the construction of virtual classes on nested Hilbert scheme of points and curves of smooth projective surfaces. Let \(S\) be a smooth projective surface, \(r>0\), \(\boldsymbol{n}=n_1,\dots,n_r\in \mathbb{Z}_{\geq 0}\) and \(\boldsymbol{\beta}=\beta_1,\dots, \beta_{r-1}\in H_2(S,\mathbb{Z})\). The closed points of the nested Hilbert scheme \(S_{\boldsymbol{\beta}}^{[\boldsymbol{n}]}\) consist of tuples of subschemes of \(S\)
\[Z_1,\dots,Z_r,\quad C_1,\dots,C_{r-1} \]
where \(Z_i\) is 0-dimensional of length \(n_i\), \(C_i\) is a divisor of class \(\beta_i\) and the the following nesting condition holds
\[ I_{Z_i}(-C_i)\subset I_{Z_{i+1}}, \]
where \(I_{Z_i}\) is the ideal sheaf of \(Z_i\).
For \(S\) a smooth projective surface and \(r\geq 2\), the authors construct a virtual fundamental class \[
[S_{\boldsymbol{\beta}}^{[\boldsymbol{n}]}]^{\text{vir}}\in A_d(S_{\boldsymbol{\beta}}^{[\boldsymbol{n}]}), \quad d=n_1+n_r+\frac{1}{2}\sum_{i=0}^{r-1}\beta_i\cdot(\beta_i-c_1(\omega_S))
\]
and, under suitable hypotheses on \(S\), a reduced virtual fundamental class \([S_{\boldsymbol{\beta}}^{[\boldsymbol{n}]}]^{\text{vir}}_{\text{red}}\in A_*(S_{\boldsymbol{\beta}}^{[\boldsymbol{n}]})\).
This class, constructed in great generality, recovers various virtual classes already present in the literature. The main examples are for the Hilbert scheme of points \(S^{[n]}\), the Hilbert scheme of divisors \(S_\beta\), and the moduli space of stable pairs \(S^{[0,n]}_\beta\).
As an application, the authors compute a closed formula for an operator introduced by Carlsson-Okounkov on Hilbert schemes of points. Define a \(K\)-theory class on \( S^{[n_1]}\times S^{[n_2]}\)
\[
\mathrm{E}^{n_1,n_2}_M:=[\mathbf{R}\pi_*p^*M]-[\mathbf{R}\mathcal{H}om_\pi(\mathcal{I}^{[n_1]},\mathcal{I}^{[n_2]}\otimes p^*M)],
\]
where \(M\) is a line bundle on \(S\). Then
\[ \sum_{n_1\geq n_2\geq 0}(-1)^{n_1+n_2}\int_{S^[n_1\geq n_2]^{\text{vir}}}i^*c(\mathrm{E}^{n_1,n_2}_M)q_1^{n_1}q_2^{n_2}=\\
\prod_{n>0}\left(1-q_2^{n-1}q_1^n\right)^{\langle K_S, K_S-M\rangle}\left(1-q_2^{n}q_1^n\right)^{\langle K_S-M, M\rangle-e(S)},
\]
where \(\langle-,-\rangle\) is the Poincaré pairing and \(i:S^{[n_1\geq n_2]}\hookrightarrow S^{[n_1]}\times S^{[n_2]}
\). nested Hilbert scheme; projective surfaces; virtual fundamental class Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Nested Hilbert schemes on surfaces: virtual fundamental class | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, we study the tangent spaces of the smooth nested Hilbert scheme \(\mathrm{Hilb}^{n,n-1}(\mathbb A^2)\) of points in the plane, and give a general formula for computing the Euler characteristic of a \(\mathbb T^2\)-equivariant locally free sheaf on \(\mathrm{Hilb}^{n,n-1}(\mathbb A^2)\). Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables \(q\) and \(t\) with non-negative integer coefficients. We call this conjecturally positive polynomial as the ``nested \(q,t\)-Catalan series'', for it has many conjectural properties similar to that of the \(q,t\)-Catalan series. Atiyah-Bott Lefschetz formula; (nested) Hilbert scheme of points; tangent spaces; diagonal coinvariants Can, M.: Nested Hilbert schemes and the nested q,t-Catalan series Parametrization (Chow and Hilbert schemes), Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Nested Hilbert schemes and the nested \(q,t\)-Catalan series | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We show how the ``finite Quot scheme method'' applied to Le Potier's strange duality on del Pezzo surfaces leads to conjectures (valid for all smooth complex projective surfaces) relating two sets of universal power series on Hilbert schemes of points on surfaces: those for top Chern classes of tautological sheaves and those for Euler characteristics of line bundles. We have verified these predictions computationally for low order. We then give an analysis of these conjectures in small ranks. We also give a combinatorial proof of a formula predicted by our conjectures: the top Chern class of the tautological sheaf on \(S^{[n]}\) associated to the structure sheaf of a point is equal to \((-1)^n\) times the \(n\)th Catalan number. Parametrization (Chow and Hilbert schemes), Vector bundles on surfaces and higher-dimensional varieties, and their moduli Universal series for Hilbert schemes and 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 For all integers \(r\ge 3\), \(d\ge r\), \(g\ge 0\) Let \(\mathcal{H}^{L}_{d,g,r}\) denote the union of the irreducible component Hilbert scheme of \(\mathbb {P}^r\) whose general element is a smooth, connected and linearly normal curve \(C\) with degree \(d\) and genus \(g\) (over an algebraically closed field with characteristic zero). The paper always works in the Brill-Noether range \(\rho(d,g,r) =g-(r+1)(g-d+r)\ge 0\). Since \(C\) is linearly normal, \(h^1(C,\mathcal{O}_C(1)) =g-d+r\). In this range the wide-open Modified Assertion of Severi asks if \(\mathcal{H}^{L}_{d,g,r}\). The author had previously solved the cases \(g-d+r\le 3\). In the paper under review the author studies the case \(g-d+r=4\). He proves the existence part (with a few exceptions) and the irreducibility part for \(3\le r\le 8\). For the latter part he construct explicit families on certain surfaces. Hilbert scheme; algebraic curves; linearly normal; special linear series Parametrization (Chow and Hilbert schemes), Plane and space curves On the Hilbert scheme of linearly normal curves in \(\mathbb{P}^r\) with small index of speciality | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities According to our approach for resolution of singularities in positive characteristic (called the idealistic filtration program, IFP for short), the algorithm is divided into the following two steps:{
}Step 1. Reduction of the general case to the monomial case.{
}Step 2. Solution in the monomial case.{
}While we have established Step 1 in arbitrary dimension, Step 2 becomes very subtle and difficult in positive characteristic. This is in clear contrast to the classical setting in characteristic zero, where the solution in the monomial case is quite easy. In dimension 3, we provided an invariant, inspired by the work of Benito-Villamayor, which establishes Step 2. In this paper, we propose a new strategy to approach Step 2, and provide a different invariant in dimension 3 based upon this strategy. The new invariant increases from time to time (the well-known Moh-Hauser jumping phenomenon), while it is then shown to eventually decrease. Since the old invariant strictly decreases after each transformation, this may look like a step backward rather than forward. However, the construction of the new invariant is more faithful to the original philosophy of Villamayor, and we believe that the new strategy has a better fighting chance in higher dimensions. resolution of singularities; idealistic filtration program Global theory and resolution of singularities (algebro-geometric aspects), Positive characteristic ground fields in algebraic geometry A new strategy for resolution of singularities in the monomial case 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 Hilbert modular group; resolution of singularities at cusps F.Hirzebruch, The Hilbert Modular group, resolution of the singularities at the cusps and related problems. Sém. Bourbaki, exp. 396 (1971). Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Modifications; resolution of singularities (complex-analytic aspects), Characteristic classes and numbers in differential topology, Special surfaces The Hilbert modular group, resolution of the singularities at the cusps and related problems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 Hilbert scheme \({\text{Hilb}}_d (\mathbb P_k^n)\) parametrizing length \(d\) closed subschemes \(X \subset \mathbb P_k^n\) is irreducible for small values of \(d\) (because it is the closure of the open locus \({\mathcal R}\) corresponding to \(d\) distinct points), but for \(d \gg 0\) it was shown to be reducible by \textit{A. Iarrobino} [Invent. Math. 15, 72--77 (1972; Zbl 0227.14006)]. The first explicit example \({\text{Hilb}}_8 (\mathbb P_k^4)\) was given by \textit{A. Iarrobino} and \textit{J. Emsalem} [Compos. Math. 36, 145--188 (1978; Zbl 0393.14002)]; more recently Cartwright, Erman, Velasco and Viray showed that \({\text{Hilb}}_8 (\mathbb P_k^n)\) has exactly two irreducible components for \(n \geq 4\) [\textit{D. A. Cartwright} et al., Algebra Number Theory 3, No. 7, 763--795 (2009; Zbl 1187.14005)].
Naturally the open Gorenstein locus \({\text{Hilb}}^G_d (\mathbb P_k^n) \subset {\text{Hilb}}_d (\mathbb P_k^n)\) also contains \(\mathcal R\) and is irreducible for small \(d\), but \textit{Iarrobino and V. Kanev} showed \({\text{Hilb}}^G_{14} (\mathbb P_k^6)\) is reducible and conjectured irreducibility for \(d < 14\) [\textit{A. Iarrobino} and \textit{V. Kanev}, Power sums, Gorenstein algebras, and determinantal loci. With an appendix `The Gotzmann theorems and the Hilbert scheme' by Anthony Iarrobino and Steven L. Kleiman. Lecture Notes in Mathematics. 1721. Berlin: Springer. (1999; Zbl 0942.14026)]. The authors of the paper under review earlier showed that \({\text{Hilb}}^G_{d} (\mathbb P_k^n)\) is irreducible for \(n \leq 3\) or \(d \leq 10\) [J. Pure Appl. Algebra 213, No. 11, 2055--2074 (2009; Zbl 1169.14003) and J. Pure Appl. Algebra 215, No. 6, 1243--1254 (2011; Zbl 1215.14009)] and here they extend the result to \(d=11\).
The problem reduces to studying zero dimensional local Gorenstein \(k\)-algebras \(A\) of length \(d = 11\), which via Macaulay's theory of inverse systems can be written \(k[[x_1, \dots x_n]]/\text{Ann}(F)\) for suitable \(F \in k[y_1, \dots y_n]\) via the action of \(k[[x_1, \dots x_n]]\) on \(k[y_1, \dots y_n]\) given by \(x_i = \partial/\partial y_i\). Using a filtration of \(\mathrm{gr}(A)\) appearing in work of \textit{T. Iarrobino} [Mem. Am. Math. Soc. 514, 115 p. (1994; Zbl 0793.13010)], the authors write the equation for \(F\) in a special form with an integer invariant \(f_3\). In general they show that if \(f_3 \leq 3\), then the scheme corresponding to the algebra \(A\) is in \({\overline {\mathcal R}}\) and \(f_3 > 3\) leads to particular Hilbert functions (especially \(H_A = (1,4,4,1,1)\)) which they handle with special arguments. They also give an almost complete description of the singular locus of \({\text{Hilb}}^G_{11} (\mathbb P_k^n)\). local Artinian Gorenstein algebras; punctual Hilbert schemes ----, Irreducibility of the Gorenstein locus of the punctual Hilbert scheme of degree \(11\) , J. Pure Appl. Alg. 218 (2014), 1635-1651. Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the Gorenstein locus of the punctual Hilbert scheme of degree 11 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 deformation of restrictions to \(\mathbb{P}^3\) and \(\mathbb{P}^4\) of Tango's rank 2 bundle on \(\mathbb{P}^5\) (which exists in characteristic 2). Using this, we construct an example of a family of rank two bundles on \(\mathbb{P}^3\) (in characteristic 2) with changing \(\alpha\)-invariant and an example of a component of the Hilbert scheme of smooth surfaces in \(\mathbb{P}^4\) which exists in characteristic 2 but not in any other characteristic. Tango characteristic 2 bundle Mohan Kumar, N., Peterson, C., Rao, A.P.: Hilbert scheme components in characteristic 2. Commun. Algebra 28, 5735--5744 (2000) Parametrization (Chow and Hilbert schemes), Finite ground fields in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Hilbert scheme components in characteristic 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 From the abstract: Given a homogeneous ideal \(I \) in a polynomial ring over a field, one may record, for each degree \(d\) and for each polynomial \(f\in I_d\), the set of monomials in \(f\) with nonzero coefficients. These data collectively form the \(tropicalization\) of \(I\) . Tropicalizing ideals induces a ``matroid stratification'' on any (multigraded) Hilbert scheme. Very little is known about the structure of these stratifications. In this paper, we explore many examples of matroid strata, including some with interesting combinatorial structure, and give a convenient way of visualizing them. We show that the matroid stratification in the Hilbert scheme of points (\(\mathbb{P}^1)^{[k]}\) is generated by all Schur polynomials in \(k\) variables. We end with an application to the \(T\) -graph problem of (\(\mathbb{A}^2)^{[n]}\); classifying this graph is a longstanding open problem, and we establish the existence of an infinite class of edges. Schur polynomials; tropical ideals; multigraded Hilbert schemes Parametrization (Chow and Hilbert schemes), Algebraic combinatorics The matroid stratification of the Hilbert scheme of points on \(\mathbb{P}^1\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0626.00011.]
A result of \textit{M. Reid} [same proceedings, Proc. Symp. Pure Math. 46, 345-414 (1987; Zbl 0634.14003)] gives a formula for the plurigenera of a 3-fold X with canonical singularities in terms of the characteristic numbers of X and an explicit formula on an ``equivalent basket'' of cyclic quotient singularities.
Here the explicit formula is manipulated to a more convenient form, and it is shown that vanishing of plurigenera restricts the number and type of singularities in the basket. It is shown that if \(\chi\) (\({\mathcal O}_ X)<0\) then \(P_ 2\geq 4\); if \(\chi\) (\({\mathcal O}_ X)=0\), \(P_ 2\geq 1\) and \(P_ 4>2\); and if \(\chi\) (\({\mathcal O}_ X)=1\) then \(P_{12}\geq 1\) and \(P_{24}\geq 2\). These results rest on extensive numerical computations. plurigenera of a 3-fold; characteristic numbers; quotient singularities A. R. Fletcher, Contributions to Riemann-Roch on projective \(3\)-folds with only canonical singularities and applications , Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 221-231. \(3\)-folds, Singularities of surfaces or higher-dimensional varieties, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Singularities in algebraic geometry Contributions to Riemann-Roch on projective 3-folds with only canonical singularities 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 For a complex, integral, locally planar curve \(C\), let \(C^{[n]}\) denote the Hilbert scheme of \(n\) points on \(C\).
\textit{J. V. Rennemo} [J. Eur. Math. Soc. (JEMS) 20, No. 7, 1629--1654 (2018; Zbl 1409.14011)] built on work of \textit{H. Nakajima} [Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)] and
\textit{I. Grojnowski} [Math. Res. Lett. 3, No. 2, 275--291 (1996; Zbl 0879.17011)] to construct a Weyl algebra
acting on the Borel Moore homology \(V = \oplus_{n \geq 0} H_* (C^{[n]})\) and described \(V\) in terms of
representation theory of Weyl algebras [J. Eur. Math. Soc. (JEMS) 20, 1629--1654 (2018; Zbl 1409.14011)].
The author extends Rennemo's approach to describe \(V = \bigoplus_{n \geq 0} H_* (C^{[n]}, \mathbb Q)\) when
\(C\) is reduced and locally planar, but not necessarily integral.
If \(C\) has \(m\) irreducible components, he constructs a bigraded action of
\[
A=A_m = [x_1, \dots, x_m,
\partial_{y_1}, \dots \partial_{y_m}, \sum_{i=1}^m y_i, \sum_{i=1}^m \partial_{x_i}]
\]
on \(V=\bigoplus V_{n,d}\) graded by the number of points \(n\) and homological degree \(d\).
After showing that the total spaces of relevant Hilbert flag schemes are smooth, he constructs operators on \(V\)
to set up the bivariant Borel-Moore homology formalism of \textit{W. Fulton} and \textit{R. MacPherson} [Categorical framework for the study of singular spaces. Providence, RI: American Mathematical Society (AMS) (1981; Zbl 0467.55005)]
and proves the commutation relations.
The end of the paper is devoted to the simplest case when \(m=2\) and \(C\) is the union of two
intersecting lines in \(\mathbb P^2\).
Starting with the geometric description of \(C^{[n]}\) given by \textit{Z. Ran} [J. Algebra 292, No. 2, 429--446 (2005; Zbl 1087.14005)], the author computes the
\(A\)-action, showing that \(\displaystyle V \cong \frac{\mathbb Q [x_1,x_2,y_2,y_2]}{\mathbb Q [x_1,x_2,y_1+y_2](x_1-x_2)}\) as an \(A\)-module. Hilbert scheme of points; representation theory Parametrization (Chow and Hilbert schemes) Hecke correspondences for Hilbert schemes of reducible locally planar 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 A. Grothendieck constructed the Hilbert scheme \(\text{Hilb}^n_X\) of \(n\) points on \(X\), for any quasi-projective scheme \(X\) on a noetherian base scheme \(S\). In the paper under review, the authors are interested in showing the existence of the Hilbert scheme of \(n\) points on \(\text{Spec}({\mathcal O}_{X,P})\), where \(P\) is a (non necessarily closed) point on such a scheme \(X\). A natural candidate would be \(\bigcap_{P\in U_\alpha}\text{Hilb}^n_{U_\alpha}\), where \(U_\alpha\) varies in the set of open subsets of \(X\) containing \(P\). But in general an infinite intersection of open subschemes of a scheme is not a scheme. It is a scheme if one takes only locally principal open subschemes.
The authors introduce and study the notion of generalized fraction rings and localized subschemes, of which \(\text{Spec}({\mathcal O}_{X,P})\) is a particular case. They prove that, if \(X\) is a scheme such that \(\text{Hilb}^n_X\) exists, then the functor of points of a localized scheme \({\mathcal S}^{-1}X\) is representable. As a particular case, they get the following result:
If \(X\rightarrow S\) is a projective morphism of Noetherian schemes and \(P\) is a point in \(X\), then the Hilbert scheme of \(n\) points on \(\text{Spec}({\mathcal O}_{X,P})\) exists and coincides with the intersection of the Hilbert schemes of \(n\) points of the open subschemes of \(X\) containing \(P\). localized schemes; determinants; fraction rings Parametrization (Chow and Hilbert schemes) Infinite intersections of open subschemes and the Hilbert scheme of points. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a K3 surface and let \(M(r,c_1,c_2)\) be the moduli of \(\operatorname {rank} r\) \(H\)-stable sheaves over \(X\) of Chern classes \(c_1\) and \(c_2\). The author shows that in case \(c_2=c_1^2/2+r/2 n^2 H^2+nc_1 H+(r+1)\) for some \(n\), then \(M(r,c_1,c_2)\) is birational to the Hilbert scheme \(X^{[l]}\) for some \(l\). This generalize earlier results of \textit{O'Grady, K. Zuo} and \textit{T. Nakashima}. moduli of vector bundles; K3 surfaces; Hilbert scheme Costa, L., \(K3\) surfaces: moduli spaces and Hilbert schemes. Dedicated to the memory of Fernando Serrano, Collect. Math., 49 (1998), 273-282. Families, moduli, classification: algebraic theory, \(K3\) surfaces and Enriques surfaces, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Characteristic classes and numbers in differential topology, (Equivariant) Chow groups and rings; motives K3 surfaces: Moduli spaces and 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 The aim of the authors (fully realized) is to reprove in a different way the theorem of \textit{C. G. Gibson} and \textit{C. A. Hobbs} [Math. Proc. Camb. Phil. Soc. 113, 297--310 (1993; Zbl 0789.58013)] giving the classifications of all unibranch simple singularities of germs of space curves. As a corollary of their proof they give a quick way to see that a formal parametrization does not parametrize a simple singularity for A-equivalence (Corollary 1). The reader may also see the following related papers: \textit{J. Stevens} [J. Singul. 12, 191--206 (2015; Zbl 1312.14078)]; \textit{M. Zhitomirskii} [Proc. Lond. Math. Soc. (3) 96, No. 3, 792--812 (2008; Zbl 1149.14004)]; \textit{F. K. Janjua} and \textit{G. Pfister} [Stud. Sci. Math. Hung. 51, No. 1, 92--104 (2014; Zbl 1299.14046)]; \textit{A. Frühbis-Kr\"ger} [Commun. Algebra 27, No. 8, 3993--4013 (1999; Zbl 0963.14011)]. simple singularities; space curves; parametrized curves Singularities of curves, local rings, Plane and space curves, Singularities in algebraic geometry A new proof for the classification of simple parameterized space 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 Let \(K\) be an algebraically closed field. A parametrization of an irreducible germ of a plane curve singularity is given by a pair \((x(t),y(t))\) of power series, \(x(t), y(t) \in K[[t]]\) such that dim\(_K K[[t]]/K[[x(t),y(t)]]=: \delta<\infty\).
Two parametrizations \((x(t),y(t))\) and \((\overline{x}(t),\overline{y}(t))\) are \(\mathcal{A}-\)equivalent (\((x(t),y(t))\thicksim_\mathcal{A} (\overline{x}(t),\overline{y}(t))\))
if there exist automorphisms
\[\psi:K[[t]]\rightarrow K[[t]] \text{ and } \phi=(\phi_1,\phi_2):K[[x,y]]\rightarrow K[[x,y]]\]
such that
\[(x(\psi(t)),y(\psi(t)))=(\phi_1(\overline{x}(t),\overline{y}(t)),\phi_2(\overline{x}(t),\overline{y}(t))).\]
A parametrization \((x(t),y(t))\) is called simple if there are only finitely many \(\mathcal{A}-\) equivalent classes in a deformation of \((x(t),y(t))\).
A bound for the determinacy in terms of the conductor is given.
Simple parametrizations are classified and compared with the corresponding classification of hypersurfaces of \textit{V. I. Arnol'd} [Funct. Anal. Appl. 6, 254--272 (1973; Zbl 0278.57011); translation from Funkts. Anal. Prilozh. 6, No. 4, 3--25 (1972)] (in characteristic zero) and \textit{G. M. Greuel} and \textit{H. Kröning} [Math. Z. 203, No. 2, 339--354 (1990; Zbl 0715.14001)] (in characteristic \(p>0\)). plane curve; parametrization; simple singularity; classification Singularities in algebraic geometry, Plane and space curves, Singularities of curves, local rings, Computational aspects of algebraic curves Parametrization simple irreducible curve singularities in arbitrary characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert scheme of \(d\) points in an \(m\)-dimensional affine space has been a central topic for some time. The goal of this article is to study a non-commutative analogue of these Hlbert schemes by their cohomological properties.
The Hilbert scheme of \(d\) points in \(\mathbb A^m\) parametrizes ideals of codimension \(d\) of the free polynomial algebra in \(m\) variables, so a direct generalization is to study the moduli space of left-ideals of codimension \(d\) in the free non-commutative algebra on \(m\) letters.
\textit{M. Reineke} [Algebr. Represent. Theory 8, No. 4, 541--561 (2005; Zbl 1125.14006)] proved that the non-commutative Hilbert scheme posses a cell decomposition where the cells are parametrized by by \(m\)-ary trees with \(d\) nodes. Then the Chow group and the singular cohomology is a free group with basis given by the closures of the cells. This implies that a formula for the Poincaré polynomial and the Euler characteristic can be found.
It is known that the non-commutative Hilbert scheme is a non-singular variety, and so the Chow group has a ring structure. Calculating the intersection product is a nontrivial task, and the basis of the Chow ring given by the cell decomposition is not well suited for computations. One main goal of this article is to provide another basis for the Chow ring so that the multiplicative structure can be studied by cohomology.
In the commutative case, \textit{M. Lehn} and \textit{C. Sorger} [Duke Math. J. 110, No. 2, 345--357 (2001; Zbl 1093.14008)] showed that the cohomology ring of the Hilbert scheme of \(d\) points in the affine plane is isomorphic to the ring of class functions of the symmetric group \(S_d\). Their technique is to use results of \textit{I. Grojnowski} [Math. Res. Lett. 3, No. 2, 275--291 (1996; Zbl 0879.17011)] and \textit{H. Nakajima} [Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)] where they prove that the direct sum over all \(d\) of all the cohomology groups has the structure of a vertex algebra which is isomorphic to the bosonic Fock space. In the non-commutative case, an analogue would be to give the cohomology of non-commutative Hilbert schemes a module structure over Kontsevich-Soibelman's Cohomological Hall algebra.
The two main results in the article are the following: First, the non-commutative Hilbert scheme is a (commutative) moduli space, and so has a universal bundle. The author constructs a basis of the Chow group consisting of monomials in the Chern classes of the universal bundle. This gives a description of the Chow ring as a quotient of a polynomial ring. Secondly, the the author realizes the cohomology of non-commutative Hilbert schemes, which equals their Chow ring after extending scalars to the rationals, as a quotient of the Cohomological Hall algebra. The kernel of the quotient map is described explicitly.
The first main result is a direct consequence of the fact that certain monomials in Chern classes, parametrized by \(m\)-ary trees, can be described as linear integer combinations of cell closures. The linear combinations have an upper uni-triangular base-change matrix. The theorem can be proved by expressing the filtration steps of the cell decomposition of Reineke as intersections of degeneracy loci and then proving that every irreducible component of this intersection has the necessary dimension.
The second main result is based on the description of the module structure over the Cohomological Hall algebra. Here is where the definition of \textit{CoHa} comes in. It is short for \textit{Cohomological Hall algebra}. Then the author proves that the kernel of the quotient map from the CoHa to the Chow rings of non-commutative Hilbert schemes can be described using the CoHa-multiplication. Also note that the proof of this result depends on the Harder-Narashiman stratification.
The author studies a commutative moduli of a non-commutative problem: For positive integers \(d,m\) and \(n\), for vector spaces \(V\) and \(W\) of dimension \(n\) and \(d\) respectively, \(\hat R\) is the vector space \(\text{Hom}(V,W)\oplus\text{End}(W)^m\) and \(G=\text{GL}(W)\) acting on \(\hat R\) by an explicit given action. For the stable points there exists a geometric \(G\)-quotient \(\pi:\hat{ R}^{\text{st}}\rightarrow H^{(m)}_{d,n}\) which is the object under study, called the \textit{non-commutative Hilbert Scheme}. The variety \(H^{(m)}_{d,n}\) is a \textit{framed quiver moduli space}, meaning that it can be stratified by its quiver representations.
The author gives a thorough treatment of the theory of quivers, i.e. \textit{words and forests} which leads to the cell decomposition of the representations: \(H_{d,m}\) is covered by open subsets \(U_{S_\ast}\) with \(S_\ast\in\mathcal F_{d,n}\), the corresponding forest, each of which is isomorphic to an affine space of dimension \(N=(m-1)d^2+nd\). Now, certain closed subsets of the \(U_{S_\ast}\), explicitly given, are the cells of the cell decomposition. This gives a basis of the Chow group, and makes out the prerequisites of the computations.
A connection between cell closures and Chern classes of the universal bundle is given, making it possible to choose a basis of the Chow group consisting of monomials in Chern classes. Then this basis is adjusted to make nice computations of the intersection products.
The author considers the \textit{Cohomological Hall algebra} for the \(m\)-loop quiver, and define a module structure on the Chow rings of non-commutative Hilbert schemes. This is done explicitly with two following, explicit examples, explaining the structure.
The article gives a very nice treatment of an algebaic geometric structure theory using representation theory. Chow group; cohomological Hall algebra; trees; forests; Chern classes; framed quiver; cell decomposition; non-commutative Hilbert scheme Franzen, H, On cohomology rings of non-commutative Hilbert schemes and coha-modules, Math. Res. Lett., 23, 804-840, (2016) Noncommutative algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Parametrization (Chow and Hilbert schemes) On cohomology rings of non-commutative Hilbert schemes and coha-modules | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper continues a long line of research by the author in which he investigates enumerative geometry of nodal curves via Hilbert schemes rather than the (now) more ubiquitous approach of stable maps pioneered by Kontsevich. This paper is mostly foundational, developing machinery to perform various intersection-theoretic computations in this setting. Along the way there are various nice concrete results, and the author concludes by re-deriving Harris-Mumford's formula for the fundamental class of the hyperelliptic locus in the moduli space of curves (at least, he presents this computation in the case of genus \(g=3\)) and he claims that this is a step toward a program to the compute the fundamental class of all gonal loci in the moduli space of curves, an ambitious project. The reader is suggested to read the introduction of the paper for more details on the results contained therein, though the reader is also warned that this paper builds heavily on the author's earlier work (a sequence of solo-authored papers) so it may be difficult to jump in and understand what's going on without first putting time into these earlier papers. Hilbert scheme; nodal curves; intersection theory; enumerative geometry Ran, Z.: Tautological module and intersection theory on Hilbert schemes of nodal curves. Asian J. Math. \textbf{17}, 193-264 (2013). arXiv:0905.2229v5 Projective and enumerative algebraic geometry, Curves in algebraic geometry Tautological module and intersection theory on Hilbert schemes of nodal 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 smooth complex surface and \(D\) be a smooth divisor. The authors compute certain top intersections on the Hilbert scheme of points on \(S\) relative to \(D\). More precisely, they consider elements of the \(K\)-group of the relative Hilbert scheme that are well behaved under degeneration. The main result of the paper computes the generating series for the integrals of the top Euler classes of such elements in terms of the corresponding integrals on the non-relative Hilbert scheme of points on \(S\). relative Hilbert schemes; intersection numbers; projective surface; Donaldson-Thomas invariants; modular forms Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Intersection numbers on the relative Hilbert schemes of points on surfaces | 0 |
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