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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 It remains an open problem to classify the Hilbert functions of double points in \(\mathbb{P}^2\). Given a valid Hilbert function \(H\) of a zero-dimensional scheme in \(\mathbb{P}^2\), we show how to construct a set of fat points \(Z\subseteq\mathbb{P}^2\) of double and reduced points such that \(H_Z\), the Hilbert function of \(Z\), is the same as \(H\). In other words, we show that any valid Hilbert function \(H\) of a zero-dimensional scheme is the Hilbert function of a set a positive number of double points and some reduced points. For some families of valid
Hilbert functions, we are also able to show that \(H\) is the Hilbert function of only double points. In addition, we give necessary and sufficient conditions for the Hilbert function of a scheme of a double points, or double points plus one additional reduced point, to be the Hilbert function of points with support on a star configuration of line. fat points; star configuration points; Hilbert functions Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Multiplicity theory and related topics, Configurations and arrangements of linear subspaces Hilbert functions of schemes of double and reduced 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 Simple singularities (also called ADE because the different types \(A_ k\), \(D_ k\) and \(E_ 6\), \(E_ 7\), \(E_ 8\) in which they are classified) appear in a natural way in problems of classification of singularities from different points of view. In dimension 1, simple singularities have been characterized in terms of their resolution procedure by \textit{W. P. Barth}, \textit{C. A. M. Peters} and \textit{A. J. H. M. Van de Ven} [``Compact complex surfaces'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 4 (1984; Zbl 0718.14023)] in characteristic 0 and in any characteristic by \textit{K. Kiyek} and \textit{G. Steinke} [Arch. Math. 45, 565-573 (1985; Zbl 0553.14012)] who obtained the normal forms of them.
In dimension 2 simple singularities are just the rational double points [see \textit{M. Artin}'s paper in Complex Anal. Algebr. Geom., Collect. Papers dedic. K. Kodaira, 11-22 (1977; Zbl 0358.14008)]. On the other hand in characteristic 0 and arbitrary dimension \textit{V. I. Arnol'd} showed [cf. Funct. Anal. 6(1972), 254-272 (1973); translation from Funkts. Anal. Prilozh. 6, No.4, 3-25 (1972; Zbl 0278.57011)] that they are exactly the hypersurface singularities of finite deformation type (i.e. singularities from which one can obtain, by deformation, only a finite number of nonequivalent singularities).
According to previous works of \textit{H. Knörrer} [Invent. Math. 88, 153- 164 (1987; Zbl 0617.14033)] and \textit{R.-O. Buchweitz}, \textit{G.-M. Greuel} and \textit{F.-O. Schreyer} [ibid. 165-182 (1987; Zbl 0617.14034)] it is known that the simple singularities are the hypersurface singularities with finite Cohen-Macaulay type (i.e. singularities for which there exists only a finite number of non isomorphic indecomposable maximal Cohen-Macaulay modules over their local ring). However, the relationship with the point of view of Arnol'd (using deformation theory) remains essentially unknown in arbitrary characteristic.
This point constitutes the main result in the paper, more precisely, if \(f\in k[[x_ 1,...,x_ n]]\), k being an algebraically closed field of arbitrary characteristic, the authors define f to be simple if and only if f is contact-equivalent with one of the normal forms given in dimension 1 by Kiyek and Steinke, in dimension 2 by Artin and for dimension greater than two by double suspension of curves or surfaces.
Then, the main theorem asserts that the following statements are equivalent: (a) f is simple; (b) f is of finite deformation type; and (c) f is of finite Cohen-Macaulay type. - In particular a complete list of the normal forms for simple singularities is given, and, although the calculations in order to obtain the normal form of a given hypersurface are not completely described, a useful list of the main subcases is included. Also the adjacencies between the different types of simple singularities are completely given except for the case of surface singularities in characteristic 2 in which only partial information appears [see also \textit{F. Knop}, Invent. Math. 90, 579-604 (1987; Zbl 0648.14002)]. The characterizations of the singularities \(A_{\infty}, D_{\infty}\) are also obtained by the same methods. The differences to the well known case of characteristic 0 are adequately explained and some useful examples are given in order to make clear the exceptions appearing only in positive characteristic. ADE singularities; hypersurface singularities of finite deformation type; hypersurface singularities with finite Cohen-Macaulay type; normal forms for simple singularities Greuel, G.-M., Kröning, H.: Simple singularities in positive characteristic. Math. Z. 203(2), 339-354 (1990). Zbl 0715.14001 Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Local deformation theory, Artin approximation, etc., Local ground fields in algebraic geometry, Complex surface and hypersurface singularities Simple singularities in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0626.00011.]
The author treats Severi's conjecture which asserts that the Hilbert scheme \(H_{d,g,n}\), consisting of smooth irreducible curves of genus \(g\) and degree \(d\) in \({\mathbb{P}}^ n_{{\mathbb{C}}}\), is irreducible if \(d\geq g+n\). In a previous paper [Ann. Sci. Norm. Supér., IV. Sér. 19, 469- 478 (1986; Zbl 0606.14003)], the author solved the conjecture affirmatively for \(n=3\). In this paper, by using a result of \textit{H. Kaji} [J. Lond. Math. Soc., II. Ser. 33, 430-440 (1986; Zbl 0565.14017)], he proves it for \(n=4\). He also gives a counterexample for \(n\geq 6\). Severi's conjecture; Hilbert scheme Ein, L.: The irreducibility of the Hilbert scheme of smooth space curves. Proc. sympos. Pure math. 46, 83-87 (1987) Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) The irreducibility of the Hilbert scheme of smooth space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert scheme \(\mathrm{Hilb}^{p(t)} (\mathbb P^n)\) parametrizing closed subschemes of \(\mathbb P^n\) with Hilbert polynomial \(p(t)\) has been of great interest every since Grothendieck constructed it in the early 1960s. Early results include the connectedness theorem of \textit{R. Hartshorne} [Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)] and smoothness of \(\mathrm{Hilb}^{p(t)} (\mathbb P^2)\) due to \textit{J. Fogarty} [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)]. \textit{A. Reeves} and \textit{M. Stillman} showed that every non-empty Hilbert scheme contains a smooth Borel-fixed point [J. Algebr. Geom. 6, No. 2, 235--246 (1997; Zbl 0924.14004)] and \textit{A. P. Staal} classified those with exactly one such fixed point, which are necessarily smooth and irreducible [Math. Z. 296, No. 3--4, 1593--1611 (2020; Zbl 1451.14010)].
The main result classifies Hilbert schemes with two Borel-fixed points over a field \(k\) of characteristic zero. To describe the result, express the Hilbert polynomial \(p(t)\) in the form used by \textit{Gotzmann}, namely
\[
p(t) = \sum_{i=1}^m \binom{t+\lambda_i-i}{\lambda_i-1}
\]
where \(\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_m \geq 1\) [\textit{G. Gotzmann}, Math. Z. 158, 61--70 (1978; Zbl 0352.13009)]. Writing \(\mathbf{\lambda} = (\lambda_1,\dots,\lambda_m)\), the theorem lists for exactly which \(\mathbf{\lambda}\) the Hilbert scheme \(\mathrm{Hilb}^{p(t)} (\mathbb P^n)\) has two Borel-fixed points and further determines when it is (a) smooth, (b) irreducible and singular or (c) a union of two components. In each case the irreducible components are normal and Cohen-Macaulay and the singularities of the Hilbert scheme appear as cones over certain Segre embeddings of \(\mathbb P^a \times \mathbb P^b\). Since the writing of his paper, (a) \textit{A. P. Staal} [``Hilbert schemes with two Borel-fixed points in arbitrary characteristic'', Preprint, \url{arXiv:2107.02204}] has shown that the theorem is valid in all characteristics with a small modification when char \(k=2\) and (b) \textit{R. Skjelnes} and \textit{G. G. Smith} [J. Reine Angew. Math. 794, 281--305 (2023; Zbl 07640144)] have classified the smooth Hilbert schemes are described their geometry.
Despite the difficulty of the content, the paper is readably written. Section 1 gives preliminaries on Borel-fixed (strongly stable) ideals and the resolution of \textit{S. Eliahou} and \textit{M. Kervaire} [J. Algebra 129, No. 1, 1--25 (1990; Zbl 0701.13006)], while Section 2 identifies the tuples \(\mathbf{\lambda} = (\lambda_1,\dots,\lambda_m)\) corresponding to Hilbert schemes with two components. Section 3 uses the comparison theorem of \textit{R. Piene} and \textit{M. Schlessinger} [Am. J. Math. 107, 761--774 (1985; Zbl 0589.14009)] to compute the tangent space of the non-lexicographic Borel-fixed ideal \(I(\mathbf{\lambda})\) and give a partial basis for the second cohomology group of \(k[x_0,\dots,x_n]/I(\mathbf{\lambda})\). These are used in Section 4 where the main theorem is proved to describe the universal deformation space of \(I(\mathbf{\lambda})\) and hence the nature of singularities of the Hilbert schemes. Finally in Section 5 the author gives examples of Hilbert schemes with three Borel-fixed points. The last three examples relate to Hilbert schemes studied in the literature [\textit{S. Katz}, in: Zero-dimensional schemes. Proceedings of the international conference held in Ravello, Italy, June 8-13, 1992. Berlin: de Gruyter. 231--242 (1994; Zbl 0839.14001); \textit{D. Chen} and \textit{S. Nollet}, Algebra Number Theory 6, No. 4, 731--756 (2012; Zbl 1250.14004); \textit{D. Chen} et al., Commun. Algebra 39, No. 8, 3021--3043 (2011; Zbl 1238.14012)]. Hilbert scheme; singularities; Borel-fixed points; deformations of ideals Syzygies, resolutions, complexes and commutative rings, Parametrization (Chow and Hilbert schemes), Fine and coarse moduli spaces, Singularities of surfaces or higher-dimensional varieties Hilbert schemes with two Borel-fixed points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author investigates the Hilbert scheme of points on a cusp curve \(X\). Specifically, let \(X\) be a projective cusp curve with a cusp locally defined by \(x^2=y^3\) and let \(H\) denote the Hilbert scheme parameterizing zero-dimensional length \(m\) subschemes \(X\), for a fixed \(m\). Also let \(H^0\) denote the subscheme of \(H\) parameterizing those subschemes supported only at the cusp. The author establishes the two nice theorems:
1. The reduced scheme \((H^0)^{\mathrm{red}}\) is isomorphic to \(\mathbb{P}^1\) for \(m \geq 2\).
2. \(H\) has one singular point along \(H^0\).
The proofs are explicit and thorough and will be of interest to those studying Hilbert schemes and in particular punctual Hilbert schemes. Hilbert schemes; cusp curve; punctual Hilbert schemes Parametrization (Chow and Hilbert schemes) A note on the Hilbert scheme of points on a cusp curve | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study a class of toric ideals obtained by using some geometric data of ADE trees which are the minimal resolution graphs of rational surface singularities. We compute explicit Gröbner bases for these toric ideals that are also minimal generating sets consisting of large number of binomials of degree \(\leq 4\). In particular, they give rise to squarefree initial ideals as well. toric ideals; semigroup of Lipman Toric varieties, Newton polyhedra, Okounkov bodies, Actions of groups on commutative rings; invariant theory, Modifications; resolution of singularities (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects) Toric ideals of simple 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 If \(X\) is a smooth quasiprojective scheme and \(G\) is a finite group acting faithfully on \(X\), the quotient space of orbits \(X/G\) is in general a singular scheme. A kind of more refined variant of a quotient of \(X\) by \(G\) is the \(G\)-Hilbert scheme of \((G,X)\) introduced in \textit{Y. Ito} and \textit{I. Nakamura} [Proc. Japan Acad., Ser. A 72, 135--138 (1996; Zbl 0881.14002)] by replacing the \(G\)-orbits with \(0\)-dimensional \(G\)-invariant subsets of \(X\). Again in general the \(G\)-Hilbert scheme is a singular variety but expectedly ``less'' singular than \(X/G\). The known cases of smooth \(G\)-Hilbert schemes appear as minimal resolutions of Klein singularities and crepant resolutions of the quotients of \({\mathbb{C}}^3\) by a finite subgroup of \(\text{SL}_3({\mathbb{C}})\) [cf. \textit{T. Bridgeland, A. King} and \textit{M. Reid}, J. Am. Math. Soc. 14, 535--554 (2001; Zbl 0966.14028)]. But the question for which finite subgroups \(G\) of \(\text{GL}_n({\mathbb{C}}), n \geq 4\) the \(G\)-Hilberts scheme of \(({\mathbb{C}}^n,G)\) is a crepant resolution of the quotient \({\mathbb{C}}^n/G\) still remains open. The only attempt was given by \textit{D. Dais, C. Haase} and \textit{G. Ziegler} [Tôhoku Math. J., II. Ser. 53, 95--107 (2001; Zbl 1050.14044)], and it is restricted to the 2-dimensional case. In the present paper a special example of an action of a finite subgroup of \(\text{GL}_4({\mathbb{C}})\) on \({\mathbb{C}}^4\)
is described, where the answer to the above question is positive. The group is the cyclic group \(\mu_{15}\) of order \(15\) with a generator \(\varepsilon = \text{exp}(2{\pi}i/15)\), which acts on \({\mathbb{C}}^4\) by weights \((1,2,4,8)\). The quotient \({\mathbb{C}}^4/{\mu}_{15}\) has a Gorenstein canonical singularity at the origin. The main result of the paper (theorem 2.9) affirms that the \(\mu_{15}\)-Hilbert scheme of \({\mathbb{C}}^4\) is smooth and gives a crepant resolution of the singularity of \({\mathbb{C}}^4/\mu_{15}\). quotient singularities; crepant resolutions; toric varieties Sebestean, M.: A smooth four-dimensional G-Hilbert scheme. Serdica math. J. 30, No. 2 -- 3, 283-292 (2004) Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Global theory and resolution of singularities (algebro-geometric aspects), \(4\)-folds A smooth four-dimensional \(G\)-Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper studies the Hilbert scheme of \(n\) points in affine \(d\) space, for \(n\) at most 8. In particular the authors focus on when this scheme is reducible. The main theorem classifies precisely when this happens, namely for \(n=8\) and \(d \geq 4\).
In general the Hilbert scheme is a difficult and complicated object to study and it is well known that it may contain arbitrarily ``bad'' singularities [\textit{R.~Vakil}, Invent. Math. 164, No. 3, 569--590 (2006; Zbl 1095.14006)] (however Hartshorne did prove that it is always connected). The authors do a wonderful and thorough study of these particular Hilbert schemes. The paper has many explicit calculations and examples. The study is comprehensive and includes a characterization of which points are smoothable (i.e. belongs to the smoothable component of the Hilbert scheme). The authors finish with some interesting open questions which would nicely extend their work in this paper. Hilbert scheme; zero-dimensional ideal; smoothable D. A. Cartwright, D. Erman, M. Velasco, B. Viray, Hilbert schemes of 8 points. \textit{Algebra Number Theory}\textbf{3} (2009), 763-795. MR2579394 Zbl 1187.14005 Parametrization (Chow and Hilbert schemes), Commutative Artinian rings and modules, finite-dimensional algebras Hilbert schemes of 8 points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert scheme \(S^{[n]}\) of points on an algebraic surface \(S\) is a simple example of a moduli space and also a nice (crepant) resolution of singularities of the symmetric power \(S^{(n)}\). For many phenomena expected for moduli spaces and nice resolutions of singular varieties it is a model case. Hilbert schemes of points have connections to several fields of mathematics, including moduli spaces of sheaves, Donaldson invariants, enumerative geometry of curves, infinite dimensional Lie algebras and vertex algebras and also to theoretical physics. This talk will try to give an overview over these connections. moduli spaces; vertex algebras; orbifolds; resolution of singularities; Donaldson invariants L Göttsche, Hilbert schemes of points on surfaces (editor T T Li), Higher Ed. Press (2002) 483 Parametrization (Chow and Hilbert schemes), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) 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 Through a geometric approach, we explain the origin of the crepant resolution conjecture of Y. Ruan. More precisely, we calculate the Chen-Ruan cohomology and the quantum corrections of Ruan for the cohomology of Hilbert schemes in the particular case of the two-fold symmetric product of \(\mathbb C\mathbb{P}^2\), which corresponds to the invariant part by the action of the symmetric group \(\mathfrak S_2\) on the blow-up of \(\mathbb C\mathbb{P}^2 \times \mathbb C\mathbb{P}^2\) along the diagonal. Chen-Ruan cohomology; Hilbert schemes; crepant resolution conjecture; symmetric product Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) On Hilbert schemes and Chen-Ruan cohomology | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth complex algebraic surface. The Hilbert scheme \(X^{[n]}\) of \(n\) points on \(X\) is a smooth variety of dimension \(2n\). \textit{E. Carlsson} studied the generating series for the intersection pairings between the total Chern class of the tangent bundle and the Chern classes of tautological bundles on \((\mathbb C^2)^{[n]}\), proving that the reduced series \(\langle \text{ch}_{k_1} \dots \text{ch}_{k_N} \rangle^\prime\) is a quasi-modular form [Adv. Math. 229, 2888--2907 (2012; Zbl 1255.14005)]. \textit{A. Okounkov} conjectured that these reduced series are multiple \(q\)-zeta values [Funct. Anal. Appl. 48, 138--144 (2014; Zbl 1327.14026)]. \textit{Z. Qin} and \textit{F. Yu} [Int. Math. Res. Not. 2018, 321--361 (2018; Zbl 1435.14007)] proved the conjecture modulo lower weight terms via the reduced series
\[
\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q) = (q;q)_\infty^{\chi (X)} \cdot \sum_n q^n \int_{X^{[n]}} (\Pi_{i=1}^N G_{k_i} (\alpha_i,n)) c(T_{X^{[n]}})
\]
where \(0 \leq k_i \in \mathbb Z\), \(\alpha_i \in H^* (X), (q;q)_\infty = \Pi_{n=1}^\infty (1-q^n)\) and \(G_{k_i}(\alpha_i, n) \in H^* (X^{[n]})\) are classes which play a role in the study of the geometry of \(X^{[n]}\) (see work of \textit{Z. Qin} [Hilbert schemes of points and infinite dimensional Lie algebras. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1403.14003)]).
In the paper under review, the authors further study the series \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\). Defining functions \(\Theta_k^\alpha (q)\) depending on \(\alpha \in H^* (X)\) and \(k \geq 0\), they fix \(0 \leq k_1, \dots, k_N \in \mathbb Z\) and \(\alpha_1, \dots, \alpha_N \in H^* (X, \mathbb Q)\) and prove the following:
(1) If \(\langle K_X^2,\alpha_i \rangle =0\) and \(2|k_i\) for each \(i\), then the leading term \(\Pi_{i=1}^N \Theta_k^\alpha (q)\) of \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\) is either \(0\) or a quasi-modular form of weight \(\sum (k_i+2)\).
(2) Suppose \(|\alpha_i|=4\) for each \(i\). If \(2|k_i\) for each \(i\), then \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\) is a quasi-modular form of weight \(\sum (k_i+2)\). if \(2 \not |k_i\) for some \(i\), then \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)=0\).
These results are proved by relating the leading term of \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\) for \(X\) to the leading term of \(\langle \text{ch}_{k_1} \dots \text{ch}_{k_N} \rangle^\prime\) for \(\mathbb C^2\) studied by Carlsson [loc. cit.]. Hilbert schemes of points on a surface; quasi-modular forms; multiple zeta value; generalized partition Parametrization (Chow and Hilbert schemes), Binomial coefficients; factorials; \(q\)-identities, Vertex operators; vertex operator algebras and related structures Hilbert schemes of points and quasi-modularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 smooth projective variety \(Y\), let \(\text{QCH}_r(Y)=CH_r(Y)\otimes \mathbb{Q}\) denote the rational Chow group of \(r\)-dimensional cycles. For a subscheme \(Y'\) in a projective space and a Cartier divisor \(Y\) in \(Y'\), an \(r\)-plane \(L\) in \(Y\) is said to be strong (with respect to \(Y'\)) if there exists an \((r+1)\)-plane \(L'\) in \(Y'\) containing \(L\) such that the set-theoretic intersection \(L'\cup Y\) is either \(L\) or \(L'\). One of the main results of this paper reads as follows: If, in the subvariety \(Y'\subset \mathbb{P}^n\), the Cartier divisor \(Y\in|\mathcal{O}_{Y'}(t)|\) is covered by strong \(r\)-planes, then the restriction map \(\text{QCH}_{r+1}(Y')\rightarrow \text{QCH}_r(Y)\) is surjective. In order to apply this result recursively to show the triviality of Chow groups for small values of \(r\), the authors say that an \(s\)-codimensional subvariety in a projective space has type \((d_1,\dots,d_s)\) when it is a union of irreducible components of a complete intersection of multi-degree \((d_1,\dots,d_s)\), and accordingly say that a pair \(Y\subset Y'\) has type \((d_1,\dots,d_s)\) if \(Y'\) has type \((d_1,\dots,d_{s-1})\) and \(Y\) is a divisor of degree \(d_s\) in \(Y'\). Thereafter they prove that if \(r\geq0\), \(s\geq1\), \(n\geq r+s\), \(2\leq d_1\leq\cdots\leq d_{s-1}<d_s\), and \((r+2)(n-r)-\sum_{i=1}^s\binom{d_i+r+1}{r+1}\geq n-s-r\), then for any pair \(Y\subset Y'\) of type \((d_1,\dots,d_s)\), \(Y\) is covered by strong \(r\)-planes. In particular, the triviality of \(\text{QCH}_{r+1}(Y')\) implies that of \(\text{QCH}_r(Y)\). complete intersections; Chow groups; Hilbert schemes Hirschowitz, A., Iyer, J.: Hilbert schemes of fat \(r\)-planes and the triviality of Chow groups of complete intersections. In: Proceedings of Vector Bundles and Complex Geometry, Contemporary Mathematics, vol. 522. American Mathematical Society, Providence (2010) Algebraic cycles, Variation of Hodge structures (algebro-geometric aspects), Fine and coarse moduli spaces, de Rham cohomology and algebraic geometry Hilbert schemes of fat \(r\)-planes and the triviality of Chow groups of complete intersections | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0614.00006.]
Let \(H(d,g)_ S\) be the open subscheme of the Hilbert scheme of curves of degree \(d\) and arithmetic genus g in \({\mathbb{P}}^ 3\) parametrizing smooth irreducible curves. The first author who pointed out the existence of irreducible non reduced components of \(H(d,g)_ S\), was \textit{D. Mumford} [Am. J. Math. 84, 642-648 (1962; Zbl 0114.131)] who found a non reduced component of \(H(14,24)_ S\), the general curve of which lies on a smooth cubic surface in \({\mathbb{P}}^ 3\). Mumford's example has been widely generalized by the author of the present paper in his thesis (``The Hilbert-flag scheme, its properties and its connection with the Hilbert scheme. Applications to curves in 3-space'', Preprint no. 5-1981, Univ. Oslo). Among other things it turns out from his analysis that if \(W\subseteq H(d,g)_ S\) is a closed irreducible subset whose general point corresponds to a curve C lying on a smooth cubic surface, W is maximal under this condition and \(d>9\), then W irreducible, non reduced component of \(H(d,g)_ S\) yields \(g\geq 3d-18\) and \(H^ 1({\mathcal J}_ C(3))\neq 0\) (the latter inequality implying that \(g\leq (d^ 2-4)/8.\)
The author conjectures that these necessary conditions are also sufficient for W to be a non reduced component of \(H(d,g)_ S\), and he proves this conjecture in the ranges \(7+(d-2)^ 2/8<g\leq (d^ 2-4)/8\), \(d\geq 18\) and \(-1+(d^ 2-4)/8<g\leq (d^ 2-4)/8\), 17\(\geq d\geq 14\). The proof consists in an interesting analysis of the tangent and obstruction space to the so called Hilbert-flag scheme (parametrizing pairs (curve, surface), the first contained in the latter) in particular for curves lying on surfaces of degree \(s\leq 4.\) space curves; Hilbert scheme; degree; arithmetic genus; obstruction space; Hilbert-flag scheme J. O. Kleppe, Nonreduced components of the Hilbert scheme of smooth space curves. In Space curves (Rocca di Papa, 1985), volume 1266 of Lecture Notes in Math. (Springer, Berlin, 1987), pp. 181-207. Zbl0631.14022 MR908714 Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Projective techniques in algebraic geometry Non-reduced components of the Hilbert scheme of smooth space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author proves the following theorem:
Let \(X\) be a \(C^\infty\) almost complex fourfold, \(X^{(k)}\) the symmetric power of \(X\), \(X_0^{(k)}\) the open subset of \(k\)-tuples of distinct points. For each \(k\) there exists a manifold \(\text{Hilb}^k(x)\) of real dimension \(4k\) endowed with a stable almost complex structure, and a continuous map
\[
c:\text{Hilb}^k(X)\to X^{(k)},
\]
which is a diffeomorphism over \(X_0^{(k)}\) and whose fibers over \(z\in X^{(k)}\) are naturally homeomorphic to the fibers of the Hilbert-Chow morphism \(c\) over \(z\) for any almost complex structure on \(X\) integrable in a neighborhood of \(\text{Supp} z\). Hilbert scheme; almost complex structure; pseudoholomorphic curves; desingularization; symmetric product; Hilbert-Chow morphism Voisin C.: On the Hilbert scheme of points of an almost complex fourfold. Ann. Inst. Fourier (Grenoble) 50(2), 689--722 (2000) Parametrization (Chow and Hilbert schemes), General geometric structures on manifolds (almost complex, almost product structures, etc.) On the Hilbert scheme of points of an almost complex fourfold | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Simple parametrizations of complex curve singularities of arbitrary embedding dimension are classified. A parametrization is called simple, if there are only finitely many isomorphism classes in the versal deformation. This is a more restricted definition of simpleness for parametrizations compared to the definitions of \textit{J. W. Bruce} and \textit{T. J. Gaffney} [J. Lond. Math. Soc., II. Ser. 26, 465--474 (1982; Zbl 0575.58008)] and \textit{C. G. Gibson} and \textit{C. A. Hobbs} [Math. Proc. Camb. Philos. Soc. 113, No. 2, 297--310 (1993; Zbl 0789.58013)], since it allows neighbouring singulairites with more irreducible components.
This leads to a smaller list than that obtained by looking at the neighbours in the space of multi-germs with a fixed number of branches. The list of simple parametrizations of plane curves is the A-D-E-list. For space curves the list of \textit{M. Giusti} [Proc. Symp. Pure Math. 40, 457--494 (1983; Zbl 0525.32006)] (complete intersections) together with the list of \textit{A. Frühbis-Krüger} [Commun. Algebra 27, No. 8, 3993--4013 (1999; Zbl 0963.14011)] (determinantal codimension 2 singularities) is obtained. In these cases the classification coincides with the classification of simple curves defined by equations [\textit{V. I. Arnol'd}, Proc. Steklov Inst. Math. 226, 20--28 (1999; Zbl 0991.32015); translation from Tr. Mat. Inst. Steklova 226, 27--35 (1999); Giusti, loc. cit.; Frühbis-Krüger, loc. cit.]. simple singularities; A-D-E-list; curve singularities; classification of simple singularities; parametric curves Singularities of curves, local rings, Local complex singularities Simple 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 We prove the algebraicity of the Hilbert functor, the Hilbert stack, the Quot functor, and the stack of coherent sheaves on an algebraic stack \(X\) with (quasi-)finite diagonal without any finiteness assumptions on \(X\). We also give similar results for Hom stacks and Weil restrictions. Hall, J., Rydh, D.: General Hilbert stacks and Quot schemes. ArXiv e-prints (2013), arXiv:1306.4118 Parametrization (Chow and Hilbert schemes), Stacks and moduli problems, Generalizations (algebraic spaces, stacks) General Hilbert stacks and Quot schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A determinantal subvariety of \(\mathbb P^n\) is the locus defined by the vanishing of the minors of given order of a homogeneous matrix of polynomials. Even classically, such varieties have been well studied, especially those defined by the maximal minors. A more modern setting is to consider the component \(\mathcal H\) of the Hilbert scheme corresponding to a determinantal variety, \(X\), defined by the minors of given size of a general homogeneous matrix of given type. Two natural questions are whether the corresponding point \([X]\) of \(\mathcal H\) is a smooth point (i.e., \(X\) is unobstructed), and whether there is an open subset of \(\mathcal H\) consisting of determinantal varieties of the same type. This was conjectured to be the case by Kleppe and Miró-Roig in a series of papers, and they proved many cases. The authors of the current paper prove this conjecture in the case of maximal minors, except when \(\dim X = 0\) or \(X\) is a hypersurface. determinantal subvariety; maximal minors; Hilbert scheme D. Faenzi, M. L. Fania, On the Hilbert scheme of varieties defined by maximal minors. \textit{Math. Res. Lett}. 21 (2014), 297-311. MR3247058 Zbl 1304.14063 Determinantal varieties, Parametrization (Chow and Hilbert schemes), Families, moduli, classification: algebraic theory, Complete intersections On the Hilbert scheme of varieties defined by maximal minors | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 prove that, for \(d \leq 9\), the Gorenstein locus of the Hilbert scheme parametrizing \(d\) points in projective space is irreducible. Moreover they give a complete description of the singular sub-locus. This description leads the authors to conjecture on the nature on this singular locus for any \(d\).
The paper begins with an introduction on the history of the study of Grothendieck's Hilbert scheme, quickly bringing the reader up to date on the most important advancements. The picture of interest -- the locus of Gorenstein subschemes -- is introduced along with the relevant state of knowledge, including the important fact that it is an open subset of the Hilbert scheme.
The paper proceeds with thorough and explicit calculations of rings with particular Hilbert functions. This information is then put together to establish the main theorems of the paper. Along the way some useful examples are given. punctual Hilbert scheme; Gorenstein Casnati, G.; Notari, R., On the Gorenstein locus of some punctual Hilbert schemes, \textit{J. Pure Appl. Algebra}, 213, 2055-2074, (2009) Parametrization (Chow and Hilbert schemes), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the Gorenstein locus of some 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 Let \(X\) be a smooth projective variety. The base locus of a divisor \(D \subset X\)
can vary erratically for divisors with similar equivalence classes, so one often studies instead
the \textit{stable base locus} given by the intersection of base loci of \(mD\)
taken over all \(m>0\). When \(X\) is a Mori dream space, work of \textit{L. Ein} et al. shows that there is an open set in \(N^1 (X)\) where the stable base locus
is well defined and locally constant [Ann. Inst. Fourier 56, 1701--1734 (2006; Zbl 1127.14010)].
The complement of the stable base locus form the walls of the stable base locus decomposition (SBLD).
Much work has been devoted to understanding SBLDs, for example \textit{A. Bayer} and \textit{E. Macrì}
[Invent. Math. 198, 505--590 (2014; Zbl 1308.14011); J. Am. Math. Soc. 27, 707--752 (2014; Zbl 1314.14020)].
Here the authors study the SBLD using augmented base loci and Severi varieties of Hilbert schemes
of points on smooth surfaces.
A birational morphism \(f:X \to Y\) of smooth projective surfaces induces a rational contraction
\(F: X^{[n]} \to Y^{[n]}\) on Hilbert schemes of points in the sense of \textit{Y. Hu} and \textit{S. Keel}
[Michigan Math. J. 48, 331--348 (2000; Zbl 1077.14554)] and an embedding \(F^* N^1(Y^{[n]}) \hookrightarrow N^1(X^{[n]})\)
of Néron-Severi groups.
The authors show that the effective cone of \(X^{[n]}\) yields the effective cone of
\(Y^{[n]}\) when restricted to the image of \(F^*\).
Assume further that \(f\) is a blow up at general points and \(X, Y\) are \textit{Mori surfaces}, meaning that their irregularity is zero
and their Hilbert schemes of \(n\) points have linear stable base locus decompositions for all \(n\).
Then they show that the linear augmented stable base locus decomposition of
\(\overline{\text{Eff}} (X^{[n]})\) gives the linear augmented stable base locus decomposition for \(\overline{\text{Eff}} (Y^{[n]})\) under \(F^*\)
(the \textit{augmented} base locus is obtained by perturbation by a small multiple of an ample divisor).
As an application, they show how walls in the stable base locus decomposition of
\(\overline{\text{Eff}} ({\mathbb F}_{r+1}^{[n]})\) induce walls in \({\mathbb F}_r^{[n]}\), where
\({\mathbb F}_r^{[n]}\) is the \(r\)th Hirzebruch surface.
The latter part of the paper deals with Severi divisors. If \(L\) is a divisor on a smooth surface \(X\),
the Severi variety \(V_n (L) \subset |L|\) is the closure of the locus of irreducible curves with exactly \(n\) nodes
as singularities, so there is a rational map \(f: V_n (L) \to X^{[n]}\) sending a general curve to its nodal singular set.
Assuming \(h^1({\mathcal O}_X)=0\) and \(C \subset X\) is a smooth curve with \(K_X + 3C\) effective and \(V_n(|C|)\) has expected dimension, the authors compute the class of the Severi divisor in \(X^{[n]}\).
Under the same assumptions they conclude that the map \(V_n(|C|) \to X^{[n]}\) is finite, extending work of
\textit{E. Arbarello} and \textit{M. Cornalba} [Math. Ann. 256, 341--362 (1981; Zbl 0454.14023)].
They close by exhibiting some Severi divisors on Hirzebruch surfaces and \(K3\) surfaces. Hilbert scheme; movable cone; Severi divisor; stable base locus decomposition Parametrization (Chow and Hilbert schemes), Minimal model program (Mori theory, extremal rays), Rational and birational maps, Fine and coarse moduli spaces On the birational geometry of Hilbert schemes of points and Severi divisors | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 birational geometry of the Hilbert scheme of points on a smooth, projective surface, with special emphasis on rational surfaces such as \(\mathbb P^2\), \(\mathbb P^1\times\mathbb P^1\) and \(\mathbb F_1\). We discuss constructions of ample divisors and determine the ample cone for Hirzebruch surfaces and del Pezzo surfaces with \(K^2\geq 2\). As a corollary, we show that the Hilbert scheme of points on a Fano surface is a Mori dream space. We then discuss effective divisors on Hilbert schemes of points on surfaces and determine the stable base locus decomposition completely in a number of examples. Finally, we interpret certain birational models as moduli spaces of Bridgeland-stable objects. When the surface is \(\mathbb P^1\times\mathbb P^1\) or \(\mathbb F_1\), we find a precise correspondence between the Mori walls and the Bridgeland walls, extending the results of \textit{D. Arcara} et al. [``The birational geometry of the Hilbert scheme of points on \(\mathbb P^2\) and Bridgeland stability'', \url{arxiv:1203.0316}] to these surfaces. A. Bertram and I. Coskun, ''The birational geometry of the Hilbert scheme of points on surfaces'' in Birational Geometry, Rational Curves, and Arithmetic, Springer, New York, 2013, 15--55. Minimal model program (Mori theory, extremal rays), Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Stacks and moduli problems The birational geometry of the Hilbert scheme 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 Let \(\text{Hilb}_P\) be the Hilbert scheme of subvarieties in the projective space with fixed Hilbert polynomial \(P\) (respectively, let \({\mathcal M}\) be a moduli space of varieties with fixed Chern numbers). It is known that \(\text{Hilb}_P\) (resp. \({\mathcal M}\)) has finitely many irreducible components and that the number of these components is bounded by some function of the Hilbert polynomial (resp. the Chern numbers). The next question to ask is whether the Hilbert scheme (resp. moduli space) is equidimensional if it is reducible. \textit{F. Catanese} [J. Differ. Geom. 19, 483-515 (1984; Zbl 0549.14012)] has shown that for \({\mathcal M}\), the moduli space of surfaces, the number of distinct dimensions can be arbitrarily large.
In this note we study the number of distinct dimensions of the components of the Hilbert scheme \(\text{Hilb}_P\) (resp. moduli space \({\mathcal M}\)) parametrizing subschemes with intersection numbers \(H^i K^{n-2-i}\) (resp. Chern numbers), where \(H\) is the hyperplane class and \(K\) is the canonical class.
Theorem. There is an infinite family of triples (resp. quadruples) of integers \((d,HK,K^2)\) (resp. \((d,H^2K, HK^2, K^3)\)) such that the Hilbert scheme of surfaces in \(\mathbb{P}^4\) (resp. 3-folds in \(\mathbb{P}^5\)) with intersection numbers \(d\), \(HK\), \(K^2\) (resp. \(d\), \(H^2K\), \(HK^2\), \(K^3\)) has irreducible components of at least \(O(y^{3/4})\) (resp. \(O(y))\) distinct dimensions, where \(y=K^2\) (resp. \(y=K^3\)).
In the statement above, replacing Hilbert scheme by moduli space and intersection numbers by Chern numbers, we have at least \(O(y^{5/4})\) many distinct dimensions for surfaces and \(O(y)\) for 3-folds, \(y\) being the self intersection number of the canonical class \(K\). Hilbert scheme; moduli space; Hilbert polynomial; Chern numbers M.-C. Chang, Inequidimensionality of Hilbert schemes. \textit{Proc. Amer. Math. Soc}. 125 (1997), 2521-2526. MR1389509 Zbl 0883.14001 Parametrization (Chow and Hilbert schemes), Enumerative problems (combinatorial problems) in algebraic geometry, Low codimension problems in algebraic geometry, Geometric invariant theory, Families, moduli, classification: algebraic theory Inequidimensionality of Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Hilbert schemes \(\text{Hilb}^n(X)\) of \(n\)-tuples on a complex projective manifold \(X\) are natural compactifications of their configuration spaces of unordered distinct \(n\)-tuples of points on \(X\). Their geometry is determined by the geometry of \(X\) itself and the geometry of ``punctual'' Hilbert schemes of all zero-dimensional subschemes on the affine plane that are supported at the origin. This geometry is most attractive when \(X\) is a surface, since the Hilbert schemes are themselves irreducible projective manifolds. This leads to the problem of the explicit determination of the geometry or the topological invariants of Hilbert schemes \(\text{Hilb}^n(X)\) from the corresponding data of the manifold \(X\) itself.
These lecture notes are one of the essential references for the understanding of the classical results and the recent developments of the topic, in particular the author's work [Invent. Math. 136, 157--207 (1999; Zbl 0919.14001)]. The present paper includes complete proofs and precise references to the literature for further readings.
The lectures treat the following points:
Configurations of unordered \(n\)-tuples. Set-theoretical discussion of the Hilbert scheme \(\text{Hilb}^n(X)\), the symmetric quotient \(\text{S}^n(X)\), the Hilbert-Chow morphism and the punctual Hilbert scheme.
The geometry of Hilbert schemes. Representability of the moduli functor \(\underline{\text{Hilb}}(X)\) (Theorem of Grothendieck), study of the tangent space, smoothness of \(\text{Hilb}^n(X)\) for \(X\) a smooth quasiprojective surface (Theorem of Fogarty), construction of the Hilbert-Chow morphism, induction schemes, irreducibility of the punctual Hilbert scheme (Theorem of Briançon).
The cohomology of \(\text{Hilb}^n(X)\). Computation of the Betti numbers (Theorem of Göttsche), structure of irreducible representation of a Heisenberg Lie algebra of the whole cohomology of Hilbert schemes of points on \(X\) for all values of \(n\) together (Theorem of Nakajima).
Vertex algebras. Exposition of the vertex algebra point of view for the understanding of the whole cohomology space of Hilbert schemes of points.
The ring structure. Study of the interplay between the vertex algebra structure induced by the Nakajima's operators and the ring structure of the cohomology investigated by construction of multiplication operators. This results in an explicit algorithm for the computation of the ring (Theorems of Lehn). A precise treatment is devoted to the Hilbert scheme on the affine plane with an explicit description of the cohomology ring (Theorem of Lehn-Sorger), and finally to the Hilbert scheme on a \(K3\) surface (Theorem of Lehn-Sorger) in relation with orbifold cohomology (Conjecture of Ruan). Hilbert schemes of points; vertex algebras Lehn, M.: Lectures on hilbert schemes. CRM Proc Lect Notes \textbf{38}, 1-30 Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Lectures 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 The principal component of the Hilbert scheme of \(n\) points on a scheme \(X\) is the closure of the open subset parameterizing \(n\) distinct points. In this article we construct the principal component as a certain blow-up of the symmetric product of \(X\). Our construction is based on a local explicit analysis of étale families, from which the appropriate universal property, needed to identify the principal component with the blow-up, is derived. Hilbert scheme of points; blow-up; symmetric product; étale families David Rydh and Roy Skjelnes, An intrinsic construction of the principal component of the Hilbert scheme, J. Lond. Math. Soc. (2) 82 (2010), no. 2, 459 -- 481. Parametrization (Chow and Hilbert schemes), Actions of groups on commutative rings; invariant theory An intrinsic construction of the principal component 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 authors prove that every two graded ideals in an exterior algebra over a infinite field \(k\), with the same Hilbert function, are connected by a sequence of Gröbner deformations; all the deformations are binomial except possibly the first and the last one. It follows that the Hilbert scheme that parametrizes all graded ideals with the same Hilbert function over an exterior algebra is connected. Minor modifications in the proof of the previous result enable the authors to obtain a new proof (in characteristic \(0\)) of a well known theorem of \textit{R. Hartshorne} [Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)], which states that the Hilbert scheme that parametrizes subschemes of \(\mathbb P^r\) with a fixed Hilbert polynomial is connected. exterior algebra; Gröbner deformation; Hilbert polynomial; generic initial ideal; lexicographic order Peeva I, Stillman M. Connectedness of Hilbert schemes. J Algebraic Geom, 2005, 14: 193--211 Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Connectedness of Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(H_{d,g}\) denote the Hilbert scheme of locally Cohen-Macaulay curves in \(\mathbb{P}^3\). For any \(d>4\) and \(g\leq {d-3\choose 2}\), \(H_{d,g}\) has two well-understood irreducible families: There is a component \(E\subset H_{d,g}\) corresponding to extremal curves [see \textit{M. Martin-Deschamps} and \textit{D. Perrin}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, No. 6, 757-785 (1996; Zbl 0892.14005), these are the curves with maximal Rao function] and \(S\), the family of subextremal curves [see \textit{S. Nollet}, Manuscr. Math. 94, No. 3, 303-317 (1997; Zbl 0918.14014), these have the next largest Rao function]. In this short note we show that \(S\cap E\neq \emptyset\) in \(H_{d,g}\) by constructing an explicit specialization (proposition 1). Our construction also works for ACM curves of genus \(g= {d-3\choose 2}+1\) (remark 2) and hence \(H_{d,g}\) is connected for \(g> {d-3\choose 2}\) (corollary 3). Hilbert scheme Nollet, S., A remark on connectedness in Hilbert scheme, Communications in Algebra, 28, 5745-5747, (2000) Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) A remark on connectedness in 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 [For the entire collection see Zbl 0667.00008.]
The authors study the deformations of space curves which lie on rational ruled surfaces. Let \(W(m,d,g,n)\) be the subscheme of the Hilbert scheme of projective n-space corresponding to those smooth degree \(d,\) genus \(g\) curves which are m-secant curves lying on a two dimensional rational scroll. Using the properties of ruled surfaces, the authors are able to give an upper bound on the Zariski tangent space at a general point of \(W(m,d,g,n)\). From that they are able to conclude that the closure of \(W(m,d,g,n)\) is a component of the Hilbert scheme, if d, g, n and m satisfy certain inequalities. This gives a generalization of an earlier example of \textit{J. Harris} which shows that \(H(d,g,n)\), the Hilbert scheme of smooth curves of degree \(d\) and genus \(g\) in \({\mathbb{P}}^ n (n\geq 6)\), is reducible even if \(d\geq g+n.\)
The authors also formulate many different conjectures concerning about the irreducibility of \(H(d,g,n)\). For instance they conjecture that if the Brill-Noether number \(\rho(d,g,n)\) is non-negative then the open set of \(H(d,g,n)\) corresponding to linearly normal curves is an irreducible variety. gonality; deformations of space curves; Hilbert scheme; ruled surfaces; Zariski tangent space; Brill-Noether number E. Mezzetti and G. Sacchiero, Gonality and Hilbert schemes of smooth curves, in Algebraic Curves and Projective Geometry, (Trento, 1988), 183--194, Lecture Notes in Math. 1389, Springer, Berlin, 1989. Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) Gonality and Hilbert schemes of smooth curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Gamma\) be a multigerm of a parametrized curve in \(\mathbb R^n\) (i.e. a collection \(\Gamma=(\Gamma_1, \ldots, \Gamma_d), \Gamma_i:(\mathbb R, 0)\to (\mathbb R^n, 0)\) \(C^\infty\)--maps). The singularity defined by \(\Gamma\) is called simple if there exists a \(k<\infty\) such that the singularities defined by multigerms with \(d\) components and the \(k\)--jet sufficiently close to the \(k\)--jet of \(\Gamma\) are exhausted by a finite number of singularities.
Let \(F:[a, b]\to \mathbb R^n\) be an arc representing \(\Gamma\). The singularity defined by \(\Gamma\) is called fully simple if there exists \(k<\infty\) such that the singularities of all arcs \(\widetilde{F}:[a,b]\to\mathbb R^n\) sufficiently \(C^k\)--close to the arc \(F\) at all points of their images sufficiently close to \(0\in\mathbb R^n\) are exhausted by a finite number of singularities.
All fully simple singularities of plane and space curves are classified. Any fully simple singularity is quasi--homogeneous. simple singularities; A-D-E singularities; curve singularities Singularities in algebraic geometry, Plane and space curves, Deformation of singularities Fully simple singularities of plane and space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R = \mathbb{Z}[x,y]\) denote the ring of polynomials in \(x\) and \(y\) with integer coefficients and fix a grading by some abelian group \(A\). The main result of this paper establishes the smoothness and irreducibility of the multigraded Hilbert scheme parametrizing those ideals of \(R\) with a given Hilbert function.
The paper's generality is remarkable on two fronts: one is that the polynomial ring only assumes integer coefficients, which implies that the theorem extends to schemes over any fixed scheme via change of base; the other is that the the group \(A\) is permitted to be any abelian group, even one with non-zero torsion. As is pointed out in the paper, the major restriction is the use of two variables, which is essential. Indeed it is known that in general multigraded Hilbert schemes may fail to even be connected and can be highly non-singular.
The proof is broken down into several steps. The fundamental idea is to connect any point on the Hilbert scheme to a distinguished point and show that the tangent space to any point along the connecting path has constant dimension. The paper is well written with thorough proofs. The authors do an excellent job of bringing the reader up to the current state of knowledge on relevant issues concerning the multigraded Hilbert scheme. Hilbert schemes; multigraded rings; combinatorial commutative algebra DOI: 10.1016/j.aim.2009.10.003 Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Smooth and irreducible multigraded 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 finds a closed formula for any multiplicative characteristic class \(\phi\) evaluated on the tangent bundle of the Hilbert scheme of points on a non-compact surface \(X\). \textit{S. Boissiére} and \textit{M. Nieper-Wisskirchen} have earlier [J. Algebra 315, No. 2, 924--953 (2007; Zbl 1126.14005); J. Algebraic Geom. 14, No. 14, 761--787 (2005; Zbl 1120.14002)] found a universal formula expressing such a \(\phi\) as sums of certain creation operators induced from the surface \(X\), and a series of coefficients. The formula is referred to as universal since the coefficients do only depend on the characteristic class \(\phi\), and not on the actual surface. In this article, by specializing to a particular surface and by computing the equivariant cohomology of the associated Hilbert scheme, the author obtain closed expressions for the coefficients in the universal formula. As a corollary he gets a closed formula for the Chern character of the tangent bundle of the Hilbert scheme. Hilbert schemes; characteristic classes; universal formulas. M A Nieper-Wisskirchen, Characteristic classes of the Hilbert schemes of points on non-compact simply-connected surfaces, JP J. Geom. Topol. 8 (2008) 7 Parametrization (Chow and Hilbert schemes) Characteristic classes of the Hilbert schemes of points on non-compact simply-connected surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be an irreducible complex smooth projective surface and \(\mathrm{Hilb}^d(X)\) be the Hilbert scheme of zero-dimensional subschemes of \(X\) of length \(d\). Let \(\mathcal{Z}\subset X\times\mathrm{Hilb}^d(X)\) be the universal subscheme and let \(p,q\) denote the projections of \(\mathcal{Z}\) to \(X\) and \(\mathrm{Hilb}^d(X)\) respectively. Now to any vector bundle \(E\) on \(X\) is naturally associated the vector bundle \(\mathcal{F}_d(E)=q_*p^*E\) on \(\mathrm{Hilb}^d(X)\).
In this paper the authors prove that if \(E\) and \(G\) are two vector bundles on \(X\) such that \(E\) is semistable and \(\mathcal{F}_d(E)\) is isomorphic to \(\mathcal{F}_d(G)\), then also \(E\) and \(G\) are isomorphic. Hilbert scheme; algebraic surface; semistability; direct image Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Parametrization (Chow and Hilbert schemes) On vector bundles over surfaces 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 \(X={\mathbb C}^{2}//{\Gamma}\) be the minimal resolution of \({\mathbb C}^{2}/{\Gamma},\) where \({\Gamma}\) is a cyclic finite subgroup of \(SL_{2}({\mathbb C}).\) In the paper the authors study the equivariant cohomology ring of Hilbert schemes of points on \({\mathbb C}^{2}/{\Gamma}.\) The study uses the vertex algebra techniques. For \({\Gamma}\)-trivial the results of the paper specialize to those in the afffine plane case [cf. \textit{E. Vasserot}, C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 1, 7--12 (2001; Zbl 0991.14001) and \textit{W.-P. Li, Z. Qin} and \textit{W. Wang}, Int. Math. Res. Not. 2004, No. 40, 2085--2104 (2004; Zbl 1086.14005)].
The authors generalizing the results of Vasserot [loc. cit.] introduce a ring structure on \({\mathbb H}_{n}=H_{T}^{2n}(({\mathbb C}^{2}//{\Gamma})^{[n]}).\) This ring structure encodes the equivariant cohomology ring structure of \(H_{T}^{*}(({\mathbb C}^{2}//{\Gamma})^{[n]}),\) where \(T=C^{*}.\) The authors construct the equivariant analog of the Heisenberg algebra [cf. \textit{H. Nakajima}, Lectures on Hilbert schemes of points on surfaces. University Lecture Series. 18. Providence, RI: American Mathematical Society (1999; Zbl 0949.14001), Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)] which acts irreducibly on \({\mathbb H}={\bigoplus}_{n=0}^{\infty}{\mathbb H}_{n} \) and an explicit map \({\mathbb H}_{n}\rightarrow R({\Gamma}_{n}).\) Here \(R({\Gamma}_{n})\) denotes the class algebra of \({\Gamma}_{n}\) where \({\Gamma}_{n}\) is the wreath product of \({\Gamma}\) and \(S_{n}.\)
This map is shown to be an isomorphism. Let \({\mathcal G}^{*}_{\Gamma}(n)\) be the graded ring associated to a natural filtration of \(R({\Gamma}_{n})\) [cf. \textit{W. Wang}, Adv. Math. 187, No. 2, 417--446 (2004; Zbl 1112.19001)]. The authors construct an explicit graded ring isomorphism from \(H^{*}(({\mathbb C}^{2}//{\Gamma})^{[n]})\) to \({\mathcal G}^{*}_{\Gamma}(n),\) which in fact establishes \textit{Y. Ruan}'s conjecture [Contemp. Math. 312, 187--233 (2002; Zbl 1060.14080)] for the crepant resolution \({\pi}_{n}: ({\mathbb C}^{2}//{\Gamma})^{[n]} \rightarrow {\mathbb C}^{2n}/{\Gamma}_{n}.\) A family of moduli spaces of sheaves on \({\mathbb C}^{2}//{\Gamma}\), which are isomorphic to \(({\mathbb C}^{2}//{\Gamma})^{[n]}\) and parametrized by certain integral lattice in \(H^{2}({\mathbb C}^{2}//{\Gamma})\) is introduced. The authors study the \(T\)-equivariant Chern characters of certain \(T\)-equivariant tautological bundles over the moduli spaces.
This leads to description of Chern character operators in terms of some familiar operators acting on the fermionic Fock space associated to the integral lattice in \(H^{2}_{T}({\mathbb C}^{2}//{\Gamma})\) and also to the description of generating functions of the equivariant intersection numbers of these Chern characters. Hilbert scheme; vertex algebra; Heisenberg algebra; equivariant cohomology; Chern character Z. Qin and W. Wang. Hilbert schemes of points on the minimal resolution and soliton equations. In \textit{Lie algebras, vertex operator algebras and their applications}, volume 442 of \textit{Contemp. Math.}, pages 435--462. Amer. Math. Soc., Providence, RI, 2007. Parametrization (Chow and Hilbert schemes), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Vertex operators; vertex operator algebras and related structures Hilbert schemes of points on the minimal resolution and soliton equations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 vanishing of \(h^1 (\mathcal I_Z (a_1, a_2))\) for a zero-dimensional scheme \(Z \subset\mathbb P_1 \times\mathbb P_1\), when \(\deg(Z) \leq 4a_2 + 1\) and \(a_1 \geq a_2\). Hilbert function; postulation; smooth quadric surface; zero-dimensional scheme Projective techniques in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On the Hilbert function of zero-dimensional schemes of the smooth quadric surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert scheme of irreducible surfaces of degree \(m\) in projective \({\mathbb P}^{m+1}\) is irreducible, except for \(m=4\) when the Hilbert scheme has two components. The article provides a new proof of this result using generic coverings of the projective plane. Hilbert scheme; irreducible surfaces of minimal degree; coverings of the plane Parametrization (Chow and Hilbert schemes) On the irreducibility of Hilbert scheme of surfaces of minimal degree | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(X\) denote the subvariety of the grassmannian of nets of quadrics consisting of nets spanned by the minors of a \(3\times 2\) matrix of linear forms. Let \(I\) be the subvariety of \(X\) of nets with a fixed component. Denote by \(H\) the Hilbert scheme of twisted cubic curves. There is a natural map \(h:H\to X\) defined by assigning to a (possibly degenerate) twisted cubic the quadratic part of its homogeneous ideal. We prove the following:
Theorem. \(h:H\to X\) is the blowup of \(X\) along \(I\). grassmannian; nets of quadrics; Hilbert scheme of twisted cubic curves I. Vainsencher,A note on the Hilbert scheme of twisted cubics, Bol. S.B.M.18, \#1 (1987), 81-89. Parametrization (Chow and Hilbert schemes), Plane and space curves, Grassmannians, Schubert varieties, flag manifolds A note on the Hilbert scheme of twisted cubics | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 connection between the geometry of Hilbert schemes and integrable hierarchies. The main goal of the paper is to establish a direct link between equivariant cohomology rings of Hilbert schemes \(X^{[n]}\) of \(n\)-points on a quasi-projective surface \(X\) and integrable hierarchies as well as the correspondence with stationary Gromov-Witten theory. The authors study the case of \(X={\mathbb C}^2\) -- the affine plane. The action of \(T={\mathbb C}^*\) given by the formula \(t(w,z)=(tw,t^{-1}z)\) induces an action on the Hilbert scheme \(X^{[n]}\) with finitely many fixed points parametrized by partitions of \(n\) [cf. \textit{G. Ellingsrud} and \textit{S. A. Strømme}, Invent. Math. 87, 343--352 (1987; Zbl 0625.14002)].
\textit{E. Vasserot} [C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 1, 7--12 (2001; Zbl 0991.14001)] has shown that the construction of the Heisenberg algebra given in [\textit{H. Nakajima}, Lectures on Hilbert schemes of points on surfaces (1999; Zbl 0949.14001)] can be extended to the \(T\)-equivariant cohomology of Hilbert schemes. All information of the equivariant cohomology ring \(H^{*}_T(X^{[n]})\) is then encoded in \({\mathbb H}_n=H^{2n}_T(X^{[n]})\) and \({\mathbb H}_{X}=\bigoplus_{n\geq 0}{\mathbb H}_n\) becomes the bosonic Fock space of a Heisenberg algebra. The ring \({\mathbb H}_n\) was identified in [Zbl 0991.14001] with the class algebra of the symmetric group \(S_n\).
This allows the authors to establish the correspondence between the \(k\)-th equivariant Chern characters of the tautological rank \(n\) vector bundle and the \(k\)-th power sum of Jucys-Murphy elements. The authors introduce the moduli spaces \({\mathcal M}(m,n)\) , where \(m\in {\mathbb Z}, n\geq 0\). The equivariant cohomology ring of \({\mathcal M}(m,n)\) corresponds to a ring \({\mathbb H}_n^{(m)}\) and \(\bigoplus_{n,m}{\mathbb H}_n^{(m)}\) can be identified with the fermionic Fock space via the boson-fermion correspondence. The introduction of the spaces \({\mathcal M}(m,n)\) allows one to reduce the study of equivariant intersection theory on the Hilbert schemes to the study of intersection numbers of equivariant Chern characters in \({\mathcal M}(m,n)\). The intersection numbers of equivariant Chern characters are studied via the generating functions.
The authors consider three types of generating functions: the \(N\)-point function, the multipoint trace function and the \(\tau\)-function. The authors show that the first function is related to the \(N\)-point disconnected series of stationary Gromov-Witten invariants of \({\mathbb P}^1\), the second is related to the characters on the fermionic Fock space and the third is actually the \(\tau\)-function for the Toda hierarchy. W.-P. Li, Z. Qin, and W. Wang, ''Hilbert schemes, integrable hierarchies, and Gromov-Witten theory,'' Int. Math. Res. Not. 40 (2004), 2085--2104. Parametrization (Chow and Hilbert schemes), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Virasoro and related algebras Hilbert schemes, integrable hierarchies, and Gromov-Witten theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors formulate a conjecture relating the Poincaré polynomial \(\overline{\mathcal P}(q,a,t)\) of the triply graded (unreduced) Khovanov-Rozansky HOMFLY homologies of an algebraic link to an other series involving the weight polynomials of certain Hilbert schemes of points supported on the corresponding complex plane curve singularity. Denoting the latter series by \(\overline{\mathcal P}_{\mathrm{alg}}(q,a,t)\), the conjectured identity \(\overline{\mathcal P}(q,a,t) = \overline{\mathcal P}_{\mathrm{alg}}(q,a,t)\) setting \(t = -1\) specializes to an earlier conjecture of \textit{A. Oblomkov} and \textit{V. Shende} [Duke Math. J. 161, No. 7, 1277--1303 (2012; Zbl 1256.14025)] relating the HOMFLY polynomial of an algebraic link to a series with coefficients being the Euler characteristics of the above Hilbert scheme spaces associated to the corresponding plane curve singularity.
Several results are proved to support the proposed conjecture. The first is a statement about the structure of \(\overline{\mathcal P}_{\mathrm{alg}}(q,a,t)\) in terms of the number of branches, the arithmetic genus and the multiplicity of the singularity. A symmetry property with respect to the variable change \(t \mapsto 1/qt\) is also proved.
It is also discussed that the lowest \(a\)-degree part \(\overline{\mathcal P}_{\mathrm{alg}}^{\min}\) of \(\overline{\mathcal P}_{\mathrm{alg}}(q,a,t)\) can be computed from the compactified Jacobian of the singularity.
Next, some explicit formulas are proved for \(\overline{\mathcal P}_{\mathrm{alg}}\) in the case of algebraic torus knots, through a stratification of the Hilbert scheme type moduli space corresponding to these singularities. The limiting case \((n,\infty)\) of the \((n,k)\) torus knot results matches the formula for the stable superpolynomial of torus knots conjectured in [\textit{N. M. Dunfield} et al., Exp. Math. 15, No. 2, 129--159 (2006; Zbl 1118.57012)].
In the \((n, mn+1)\) torus knot case, a further conjectural formula (based on a conjecturally identical stratification) is proposed using Cherednik algebras expressing the lowest \(a\)-degree part \(\overline{\mathcal P}_{\mathrm{alg}}^{\min}\) (and, as explained further in [\textit{E. Gorsky} et al., Duke Math. J. 163, No. 14, 2709--2794 (2014; Zbl 1318.57010)], even the entire \(\overline{\mathcal P}_{\mathrm{alg}}\)) in terms of an expression involving partitions of \(n\).
The Appendix by E. Gorsky relates the Hilbert scheme stratification of torus knot type plane curve singularities to the combinatorics of diagonal harmonics and DAHA representations, justifying some conjectures involving \((q,t)\)-Catalan numbers from \textit{E. A. Gorsky} [Contemp. Math. 566, 213--232 (2012; Zbl 1294.57007)]. plane curve; Hilbert scheme; Khovanov homology A. Oblomkov, R. Rasmussen, and V. Shende, The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link. \textit{Geom. Topol. }22 (2018), no. 2, 645--691. With an appendix by E. Gorsky.MR 3748677 Zbl 1388.14087 Singularities of curves, local rings, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Invariants of knots and \(3\)-manifolds The Hilbert scheme of a plane curve singularity and the \textsc{HOMFLY} homology of its link | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We construct \(G\)-Hilbert schemes for finite group schemes \(G\). We find a construction of \(G\)-Hilbert schemes as relative \(G\)-Hilbert schemes over the quotient that does not need the Hilbert scheme of \(n\) points, works under more natural assumptions and gives additional information about the morphism from the \(G\)-Hilbert scheme to the quotient. \(G\)-Hilbert scheme; McKay correspondence Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Hopf algebras and their applications, Ordinary representations and characters Construction of \(G\)-Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities From author's abstract: We present an elementary construction of the multigraded Hilbert scheme of \(d\) points of \(\mathbb{A}^n_k=\text{Spec}(k[x_1,\dots,x_n])\), where \(k\) is an arbitrary commutative and unitary ring. This Hilbert scheme represents the functor from \(k\)-schemes to sets that associates to each \(k\)-scheme \(T\) the set of closed subschemes \(Z\subset T\times_k \mathbb{A}^n_k\) such that the direct image (via the first projection) of the structure sheaf of \(Z\) is locally free of rank \(d\) on \(T\). It is a special case of the general multigraded Hilbert scheme constructed by \textit{M. Haiman} and \textit{B. Sturmfels} [J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)]. Our construction proceeds by gluing together affine subschemes representing subfunctors that assign to \(T\) the subset of \(Z\) such that the direct image of the structure sheaf on \(T\) is free with a particular set of \(d\) monomials as basis. The coordinate rings of the subschemes representing the subfunctors are concretely described, yielding explicit local charts on the Hilbert scheme. Hilbert scheme of points; affine space; syzygies Huibregtse M.E.: An elementary construction of the multigraded Hilbert scheme of points. Pac. J. Math. 223, 269--315 (2006) Parametrization (Chow and Hilbert schemes), Syzygies, resolutions, complexes and commutative rings, Computational aspects and applications of commutative rings An elementary construction of the multigraded 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 smooth complex quasi-projective variety and \(\text{Hilb}^n (X)\) its Hilbert scheme of zero dimensional subschemes of length \(n\). The author expresses the virtual Hodge polynomials of \(\text{Hilb}^n (X)\) -- defined by cohomology with compact support -- in terms of those of \(X\) and the Hilbert scheme of subschemes of length \(n\) supported at a point of \(X\). -- The proof proceeds by comparison with the \(n\)-fold symmetric product of \(X\) and related spaces and uses a lemma on point Hilbert schemes from \(L\). Göttsche's 1991 Bonn thesis [see \textit{L. Göttsche}, ``Hilbertschemata nulldimensionaler Unterschemata glatter Varietäten'', Bonner Math. Schr. 243 (1991; Zbl 0846.14002)]. The key properties of the virtual Hodge polynomial used in the proof are its additivity over stratifications and multiplicativity for fibrations. The results extend those found for the Poincaré and Hodge polynomials of surfaces in a paper by \textit{L. Göttsche} [Math. Ann. 286, No. 1-3, 193-207 (1990; Zbl 0679.14007)]. Hilbert scheme; virtual Hodge polynomials; symmetric product J. Cheah, ''On the Cohomology of Hilbert Schemes of Points,'' J. Algebr. Geom. 5, 479--511 (1996). Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the cohomology of Hilbert schemes of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S\) be a smooth algebraic surface over an algebraically closed field, \(\sigma : \widetilde S\rightarrow S\) the blow-up of \(S\) at a point \(x_0\) and \(l_0 = {\sigma}^{-1}(x_0)\) the exceptional divisor. Let \(H\) (resp., \(\widetilde H\)) denote the Hilbert scheme of 0-dimensional subschemes of \(S\) (resp., \(\widetilde S\)) of length \(d\). \(\sigma \) induces a birational map \(f : \widetilde H\dashrightarrow H\).
The author shows that there exists a closed subscheme \(R\) of \(\widetilde H\) (defined by a suitable Fitting ideal sheaf) with support \(\{ [Z]\in \widetilde H \mid \text{length}(Z\cap l_0)\geq 2\} \) such that, if \(\widehat H\) is the blow-up of \(\widetilde H\) along \(R\), \(f\) can be extended to a \textit{morphism} \(\widehat f : \widehat H\rightarrow H\).
If \(d = 2\) then \(R\) is smooth (in fact, \(R\simeq S^2l_0\simeq {\mathbb P}^2\)) hence \(\widetilde H\) is smooth, and \(\widehat f\) is the composition of two blow-ups with smooth centres. The first one is the blow-up \(\tau : H^{\prime}\rightarrow H\) of \(H\) along \(Q = \{ [Z]\in H\mid x_0\in Z\} \simeq \widetilde S\) and the second one is the blow-up of \(H^{\prime}\) along the unique section \(m_0\) with self-intersection -3 of the rational ruled surface \({\tau}^{-1}(l_0)\simeq {\mathbb F}_3\).
For \(d\geq 3\), however, \(\widehat H\) is singular. birational map; blow-up Parametrization (Chow and Hilbert schemes), Rational and birational maps, Surfaces and higher-dimensional varieties On birational transformations of Hilbert schemes of an algebraic surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We review the deformed instanton equations making connection with Hilbert schemes and integrable systems. A single \(U(1)\) instanton is shown to be anti-self-dual with respect to the Burns metric. deformed instanton equations; anti-self-dual; Burns metric Braden, H. W.; Nekrasov, N. A.: Instantons, Hilbert schemes and integrability. Dynamical symmetries of integrable quantum field theories (2000) Groups and algebras in quantum theory and relations with integrable systems, Noncommutative geometry in quantum theory, Yang-Mills and other gauge theories in quantum field theory, Many-body theory; quantum Hall effect, Parametrization (Chow and Hilbert schemes) Instantons, Hilbert schemes and integrability | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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{D. Chen} et al. [Commun. Algebra 39, 3021--3043 (2011; Zbl 1238.14012] studied the Hilbert scheme component \(H_n \subset \text{Hilb} (\mathbb P^n)\) whose general point corresponds to the union of a pair of codimension two linear subspaces meeting transversely, showing that \(H_n\) is smooth, isomorphic to the blow up of the symmetric square of \(\mathbb G (n-2,n)\) along the diagonal, meets only one other component of the Hilbert scheme, and is a Mori dream space. Extending this work, the author offers a very complete study of the component \(\mathcal H_{c,d}^n \subset \text{Hilb} (\mathbb P^n)\) whose general point parametrizes a union of two linear subspaces of codimensions \(2 \leq c \leq d\) which meet transversely. He shows that \(\mathcal H_{c,d}^n\) is smooth, constructing it as (a) an iterated blow up over the symmetric square of \(\mathbb G (n-c,n)\) if \(c=d\) or (b) an iterated blow up over \(\mathbb G (n-c,n) \times \mathbb G (n-d,n)\) if \(d > c\). To show these isomorphisms, he constructs a Gröbner basis for the ideals of schemes parametrized by \(\mathcal H_{c,d}^n\), which leads to a classification of those schemes and the conclusion that \(\mathcal H_{c,d}^n\) has a unique Borel fixed point. In particular, there are exactly \(2^c\) schemes parametrized by \(\mathcal H_{c,d}^n\) up to projective equivalence. He goes on to determine the effective and Nef cones of \(\mathcal H_{c,d}^n\), concluding that \(\mathcal H_{c,d}^n\) is a Mori dream space. Furthermore \(\mathcal H_{c,d}^n\) is Fano if and only if \(c=3\) and \(n=5\) or \(c \neq 3\) and \(n=2c-1\) or \(2c\). Hilbert schemes; Grassmann varieties; Borel fixed points; Mori dream spaces; effective cones; Fano varieties Parametrization (Chow and Hilbert schemes), Rational and birational maps, Minimal model program (Mori theory, extremal rays), Syzygies, resolutions, complexes and commutative rings, Grassmannians, Schubert varieties, flag manifolds, Deformations and infinitesimal methods in commutative ring theory The Hilbert scheme of a pair of linear spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f\in \mathbb{C}[x_{1},x_{2},x_{3}]\) an \(ADE\) simple singularities and \(f^{T}\in \mathbb{C}[x_{1},x_{2},x_{3}]\) the corresponding Berglund-Hübsch dual. To construct a matrix model for the Fan-Jarvis-Ruan-Witten (FJRW) invariant of \(f^{T}\), similar to Kontesevich's model as in [\textit{M. Kontsevich}, Commun. Math. Phys. 147, No. 1, 1--23 (1992; Zbl 0756.35081)], one needs to find explicit identification of the generating function of FJRW invariants of \(f^{T}\) with a tau-function of a specific Kac-Wakimoto hierarchy. The identification involves rescaling parameters of the Kac-Wakimoto hierarchy and the precise value for the rescaling constants. Such a computation seems to be straightforward and it is not done in the paper. Instead, the authors explain in the introduction how the technical details leads to a problem in singularity theory.
On the other hand, in [\textit{H. Fan} et al., Ann. Math. (2) 178, No. 1, 1--106 (2013; Zbl 1310.32032)] the authors showed that generating functions of \(FJRW\) invariants of \(f^{T}\) coincides with the total descendant potentials of \(f\) (or, t.d.p. of \(f\), for short). In [\textit{E. Frenkel} et al., Funct. Anal. Other Math. 3, No. 1, 47--63 (2010; Zbl 1203.37108)] and [\textit{A. B. Givental} and \textit{T. E. Milanov}, Prog. Math. 232, 173--235 (2005; Zbl 1075.37025)] have proved that if \(f\) is a \(ADE\) singularity then t.d.p of \(f\) is a tau-function of the principal Kac-Wakimoto hierarchy of the same type \(ADE\). However, the problem is not solved yet, because the identification with the Milnor ring of the singularity and the Cartan subalgebra of the simple Lie algebra is not clearly explicit yet.
The contribution of the paper under review comes in showing that. The proofs are very well developed and they were done by approaching the ADE-singularity case by case. The paper offers a very pleasant reading. simple singularities; period map; mirror symmetry; topological K-theory Structure of families (Picard-Lefschetz, monodromy, etc.), Deformations of complex singularities; vanishing cycles, Equivariant \(K\)-theory Integral structure for simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(H(d,g)_{\mathrm{sc}}\) be the Hilbert scheme of smooth connected space curves. In the literature, many questions have been considered including smoothness of components, the existence of non-reduced components, the dimension of components, etc. In this paper the authors consider maximal irreducible closed subsets \(W\) of \(H(d,g)_{\mathrm{sc}}\) whose general element corresponds to a curve \(C\) lying on a surface \(S\) of degree \(s\), especially when \(s=4\) and \(s=5\). The authors ask when \(W\) is a non-reduced, or generically smooth, component. They also determine \(\dim W\), and they obtain connections with the Picard group of \(S\). In an appendix, the first author finds new classes of non-reduced components of \(H(d,g)_{\mathrm{sc}}\), making progress on a conjecture about non-reduced components for maximal families \(W \subset H(d,g)_{\mathrm{sc}}\) of linearly normal curves on a smooth cubic surface; he significantly extends the known range where the conjecture holds. space curves; quartic surfaces; cubic surfaces; Hilbert scheme; Hilbert-flag scheme Kleppe, J.O., Ottem, John C.: Components of the Hilbert scheme of space curves on low-degree smooth surfaces. Int. J. Math. \textbf{26}(2) (2015) Parametrization (Chow and Hilbert schemes), Divisors, linear systems, invertible sheaves, Picard schemes, higher Jacobians, \(K3\) surfaces and Enriques surfaces, Plane and space curves Components of the Hilbert scheme of space curves on low-degree smooth 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 authors consider \(\mathcal A\)-simple analytic germs from \(\mathbb{C}\) to \(\mathbb{C}^ 3\), and give a complete classification of such germs. The classification is partly derived by using the method of complete transversals. This method was first used by \textit{A. Dimca} and the first author for classification under \(\mathcal K\)-equivalence [Math. Scand. 56, 15-28 (1985; Zbl 0579.32014)]. In a forthcoming paper of Bruce and du Plessis the method has been extended to subgroups of \(\mathcal K\) including the groups \({\mathcal A}_ k\) of coordinate transformations with \(k\)-jet equal the identity. This makes the method available also for classification under \(\mathcal A\)-equivalence. simple analytic germs; classification; complete transversals Gibson, C. G.; Hobbs, C. A., Simple singularities of space curves, \textit{Math. Proc. Camb. Phil. Soc.}, 113, 297-310, (1993) Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Plane and space curves, Singularities of curves, local rings Simple singularities of space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We construct five families of 2D moduli spaces of parabolic Higgs bundles (respectively, local systems) by taking the equivariant Hilbert scheme of a certain finite group acting on the cotangent bundle of an elliptic curve (respectively, twisted cotangent bundle). We show that the Hilbert scheme of \(m\) points of these surfaces is again a moduli space of parabolic Higgs bundles (respectively, local systems), confirming a conjecture of \textit{P. Boalch} [Publ. Math., Inst. Hautes Étud. Sci. 116, 1--68 (2012; Zbl 1270.34204)] in these cases and extending a result of \textit{A. Gorsky} et al. [Commun. Math. Phys. 222, No. 2, 299--318 (2001; Zbl 0985.81107)]. Using the McKay correspondence, we establish the autoduality conjecture for the derived categories of the moduli spaces of Higgs bundles under consideration. Gröchenig, M., Hilbert schemes as moduli of Higgs bundles and local systems, Int. math. res. not. IMRN, 2014, 23, 6523-6575, (2014) Relationships between algebraic curves and integrable systems, Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, McKay correspondence Hilbert schemes as moduli of Higgs bundles and local 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 The author studies the moduli spaces of semistable objects with respect to the stability conditions of \textit{T. Bridgeland} [Ann. Math. 166, No. 2, 317--345 (2007; Zbl 1137.18008)]. For the derived category \(\text{D}^{\text{b}} (\mathbb P^3)\) of coherent sheaves on \(\mathbb P^3\), a stability condition consists of a pair \((Z,\mathcal P)\), where \(Z: K(\text{D}^{\text{b}} (\mathbb P^3)) \to \mathbb C\) is a homomorphism called the \textit{central charge} and \(\mathcal P(\phi) \subset \text{D}^{\text{b}} (\mathbb P^3)\) are full additive subcategories indexed by \(\phi \in \mathbb R\) satisfying further hypotheses. \textit{A. Bayer} et al. constructed a family of stability conditions \((Z_{\alpha,\beta, s}, {\mathcal A}^{\alpha,\beta})\) (along with associated stability functions \(\lambda_{\alpha,\beta,s}\) satisfying the Harder-Narasimhan property) parametrized by \((\alpha, \beta, s) \in \mathbb R_{>0} \times \mathbb R \times \mathbb R_{>0}\) [J. Algebraic Geom. 23, No. 1, 117--163 (2014; Zbl 1306.14005)], thus describing a subset of the space \(\text{Stab} (\mathbb P^3)\) of all locally finite stability conditions.
Letting \(C \subset \mathbb P^3\) be the twisted cubic curve, the author analyzes a path \(\gamma\) in the space of triples \((\alpha,\beta,s)\) constructed by \textit{B. Schmidt} [``Bridgeland stability on threefolds -- some wall crossings'', Preprint, \url{arXiv:1509.04608}], describing the moduli spaces \(M_{\lambda_{\alpha,\beta,s}} (v)\) of \(\lambda_{\alpha,\beta,s}\)-semistable objects with Chern character \(v = \text{ch} (\mathcal I_C)\) as \(\gamma\) crosses three walls while passing through four chambers. The first chamber yields no semistable objects, but after crossing the first wall one encounters a smooth projective irreducible moduli space \({\mathbf M}_1\) of dimension 12, which was earlier described by Schmidt [loc. cit.] using quiver representations. The second wall crossing is a \textit{simple} wall crossing: here the author identifies a \(5\)-dimension smooth subspace \(H \subset {\mathbf M}_1\) which is modified as \(\gamma\) passes over the wall, the result being that the third moduli space \({\mathbf M}_2\) has two irreducible components \({\mathbf B} \) and \({\mathbf P}\), where \({\mathbf B}\) is the blow up of \({\mathbf M}_1\) along \(H\), \({\mathbf P}\) is a \(\mathbb P^9\)-bundle over \(\mathbb P^3 \times (\mathbb P^3)^*\) and the two components intersect transversely along the exceptional divisor of \({\mathbf B}\): the transversality is proven by computing the second order Kuranishi map for complexes. Similarly the author identifies a \(5\)-dimension smooth subspace \(K \subset {\mathbf M}_2\) that is modified as \(\gamma\) crosses the third wall, which turns out to be entirely contained in \({\mathbf P}\). Crossing the third wall, the moduli space \({\mathbf M}_3\) seen in the fourth chamber is isomorphic to \({\mathbf M}_2\) blown up along \(K\).
The moduli space \({\mathbf M}_3\) is precisely the Hilbert scheme of twisted cubics studied by \textit{R. Piene} and \textit{M. Schlessinger} [Am. J. Math. 107, 761--774 (1985; Zbl 0589.14009)] and the work here gives an independent proof of their results. This also recovers some results from work of \textit{G. Ellingsrud} et al. [Lect. Notes Math. 1266, 84--96 (1987; Zbl 0659.14027)]. Bridgeland stability; derived categories; wall crossing; Hilbert scheme of twisted cubics Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Plane and space curves Hilbert scheme of twisted cubics as a simple wall-crossing | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 his fundamental paper [``Techniques de construction et théorèmes d'existence en géométrie algébrique. IV: Les schemes de
Hilbert'', Sémin. Bourbaki, exp. 221 (1961; Zbl 0236.14003)], \textit{A. Grothendieck} introduced the so called Hilbert scheme, which parametrizes all projective subschemes of the projective space with fixed Hilbert polynomial. Problems naturally arising in the study of the Hilbert scheme are irreducibility and number of components, dimension and smoothness. For instance, one knows that if \(X\subset \mathbb{P}^r\) is a local complete intersection projective subscheme and \(h^1(X,\mathcal N_{X,\mathbb{P}^r})=0\) (\(\mathcal N_{X,\mathbb{P}^r}=\) normal bundle of \(X\) in \(\mathbb{P}^r\)), then \(X\) is unobstructed, i.e. the corresponding point \([X]\) in the Hilbert scheme is smooth, and in such case the local dimension at \([X]\) is \(h^0(X,\mathcal N_{X,\mathbb{P}^r})\). But, in general, a necessary and sufficient condition for a subscheme to be unobstructed is not known [see also \textit{D. Mumford}, Am. J. Math. 84, 642--648 (1962; Zbl 0114.13106)] and \textit{E. Sernesi} [``Topics on families of projective schemes'', Queen's Pap. Pure Appl. Math. 73 (1986)].
Continuing previous works by \textit{J. O. Kleppe} [``The Hilbert-flag scheme, its properties and its connection with the Hilbert scheme. Applications to curves in \(3-\)space'', Preprint (part of thesis), Univ. of Oslo, March (1981)], \textit{G. Bolondi} [Arch. Math. 53, No. 3, 300--305 (1989; Zbl 0658.14005)], and \textit{M. Martin-Deschamps} and \textit{D. Perrin} [``Sur la classification des courbes gauches'', Astérisque 184--185 (1990; Zbl 0717.14017)], in the paper under review the author exhibits sufficient conditions and necessary conditions for unobstructedness of space curves \(C\subset \mathbb{P}^3\) which satisfy \(_{0}{\text{Ext}}^2_R(M,M)=0\) (e.g. of diameter\((M)\leq 2\)), and computes the dimension of the Hilbert scheme \(H(d,g)\) at \([C]\) under the sufficient conditions. Here \(C\subset \mathbb{P}^3\) denotes an equidimensional, locally Cohen-Macaulay subscheme of dimension one, \(d\) and \(g\) the degree and the arithmetic genus of \(C\subset \mathbb{P}^3\), \(M=\bigoplus_{v}H^1(\mathbb{P}^3, \mathcal I_C(v))\) denotes the Hartshorne-Rao module of \(C\), \(R=k[x_0,x_1,x_2,x_3]\) the polynomial ring over an algebraically closed field \(k\) of characteristic zero, and diameter\((M):=\max\{v\,| \,H^1(\mathbb{P}^3, \mathcal I_C(v))\neq 0\}-\min\{v\,| \,H^1(\mathbb{P}^3, \mathcal I_C(v))\neq 0\}+1\) (when \(H^1(\mathbb{P}^3, \mathcal I_C(v))=0\) for all \(v\), i.e. when \(C\) is arithmetically Cohen-Macaulay, then by \textit{G. Ellingsrud} [Ann. Sci. Éc. Norm. Supér. (4) 8, 423--431 (1975; Zbl 0325.14002)] one already knows that \(C\) is unobstructed).
In the diameter one case, the necessary and sufficient conditions coincide, and the unobstructedness of \(C\) turns out to be equivalent to the vanishing of certain graded Betti numbers of the free minimal resolution of the ideal \(I=\bigoplus_{v}H^0(\mathbb{P}^3, \mathcal I_C(v))\subset R\) of \(C\). The author also gives a description of the number of irreducible components of \(H(d,g)\) which contain an obstructed diameter one curve, and shows that in the diameter one case every irreducible component is reduced. Hilbert scheme; space curve; Buchsbaum curve; unobstructedness; cup-product; graded Betti numbers; ghost terms; linkage; normal module; postulation Hilbert scheme Dan, A.: Non-reduced components of the Noether-Lefschetz locus. Preprint arXiv:1407.8491v2 Parametrization (Chow and Hilbert schemes), Plane and space curves, Linkage, complete intersections and determinantal ideals, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series The Hilbert scheme of space curves of small diameter | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 classify stably simple reducible curve singularities in complex spaces of any dimension. This extends the same classification of irreducible curve singularities obtained by V. I. Arnold. The proof is essentially based on the method of complete transversals by \textit{J. W. Bruce, N. P. Kirk} and \textit{A. A. du Plessis} [Nonlinearity 10, 253--275 (1997; Zbl 0929.58019)]. Kolgushkin, P.A., Sadykov, R.R.: Simple singularities of multigerms of curves. Rev. Mat. Complut. \textbf{XIV}(2), 311-334 (2001) Singularities in algebraic geometry, Plane and space curves Simple singularities of multigerms 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 We define and analyze various generalizations of the punctual Hilbert scheme of the plane, associated to complex or real Lie algebras. Out of these, we construct new geometric structures on surfaces whose moduli spaces share multiple properties with Hitchin components, and which are conjecturally homeomorphic to them. For simple complex Lie algebras, this generalizes the higher complex structure. For real Lie algebras, this should give an alternative description of the Hitchin-Kostant-Rallis section. higher Teichmüller theory; Hitchin components; geometric structures; punctual Hilbert schemes Teichmüller theory for Riemann surfaces, General geometric structures on manifolds (almost complex, almost product structures, etc.), Parametrization (Chow and Hilbert schemes) Generalized punctual Hilbert schemes and \(\mathfrak{g}\)-complex structures | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 0742.00065. This paper is a continuation of the second named author's paper in Math. Z. 191, 489-506 (1986; Zbl 0589.17012)].
The authors compute the generalized Cartan matrix (GCM) associated to the simple-elliptic surface singularities \(\tilde E_ 6\), \(\tilde E_ 7\), and \(\tilde E_ g\) (see the paper of the second author cited above for the definition of the associated GCM), and show that in the case of \(\tilde E_ 7\) and \(\tilde E_ 8\) singularities the associated GCM is not a topological invariant. They also show that in the case of \(\tilde E_ 6\)-singularities, the stable derivation algebra [in the sense of \textit{E. Schenkman}; see his paper in Am. J. Math. 73, 453-474 (1951; Zbl 0054.018)] of the solvable Lie algebra \(L(V)\) is not a topological invariant, even though the nilradical of \(L(V)\) is indeed a topological invariant in this case. Recall that \(L(V)\) is, by definition, the Lie algebra of all the derivations of the moduli algebra \(A(V)\) associated to the singularity \(V\). Lie algebra of derivations of the moduli algebra; generalized Cartan matrix; GCM; simple-elliptic surface singularities Craig Seeley and Stephen S.-T. Yau, Algebraic methods in the study of simple-elliptic singularities, Algebraic geometry (Chicago, IL, 1989) Lecture Notes in Math., vol. 1479, Springer, Berlin, 1991, pp. 216 -- 237. Singularities of surfaces or higher-dimensional varieties, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Singularities in algebraic geometry Algebraic methods in the study of simple-elliptic 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 Families of curves in projective space have long been in studied in algebraic geometry. In the modern language, let \(H_{d,g} (\mathbb P^r)\) denote the Hilbert scheme parametrizing curves \(C \subset \mathbb P^r\) of degree \(d\) and arithmetic genus \(g\) and let \({\mathcal I}_{d,g,r} \subset H_{d,g} (\mathbb P^r)\) be the union of irreducible components whose general member is a smooth irreducible non-degenerate curve. Almost one hundred years ago \textit{F. Severi} claimed that \({\mathcal I}_{d,g,r}\) is irreducible if \(d \geq g+r\) [Vorlesungen über algebraische Geometrie. Leipzig u. Berlin: B. G. Teubner (1921; JFM 48.0687.01)]. While \textit{L. Ein} showed that the claim is true for \(r=3\) [Ann. Scient. Ec. Norm. Sup. 19, 469--478 (1986; Zbl 0606.14003)] and \(r=4\) [\textit{L. Ein}, Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], [\textit{E. Mezzetti} and \textit{G. Sacchiero}, Lect. Notes Math. 1389, 183--194 (1989; Zbl 0716.14012)] and [\textit{C. Keem}, Proc. Am. Math. Soc. 122, No. 2, 349--354 (1994; Zbl 0860.14003)] produced counterexamples for \(r \geq 6\): these examples have been produced by exhibiting families of curves whose general member is an \(m\)-fold cover of \(\mathbb P^1\) for \(m \geq 3\). In the paper under review, the authors produce new irreducible components of \({\mathcal I}_{d,g,r}\) whose general member \(C \subset \mathbb P^r\) has the property that the morphism defined by the linear system \(\mathbb P H^0(C, \omega_C (-1))\) has degree two (possibly after removing base points). Hilbert scheme of smooth connected curves; Brill-Noether theory; double covers Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) Reducibility of the Hilbert scheme of smooth curves and families of double covers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Denote by \(H_{d,g,r}\) the Hilbert scheme parametrizing projective smooth irreducible complex curves of degree \(d\) and genus \(g\) in \({\mathbb P^r}\). A natural question concerning \(H_{d,g,r}\), which goes back to [\textit{F. Severi}, Vorlesungen über algebraische Geometrie, Teubner (1921; JFM 48.0687.01)], is whether it is irreducible under the assumption \(d\geq g+r\).
In more recent years \textit{J. Harris} [see \textit{L. Ein}, Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, Part 1, Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], and \textit{C. Keem} [Proc. Am. Math. Soc., 122, No.~2, 349--354 (1994; Zbl 0860.14003)] proved by examples that \(H_{d,g,r}\) may be reducible for \(d\geq g+r\) and \(r\geq 6\). Moreover \textit{L. Ein} [loc. cit.; Ann. Sci. Éc. Norm. Supér., IV. Sér., 19, No.~4, 469--478 (1986; Zbl 0606.14003)], and \textit{C. Keem} and \textit{S. Kim} [J. Algebra, 145, No.~1, 240--248 (1992; Zbl 0783.14002)] proved the irreducibility of \(H_{d,g,r}\) when \(d\geq g+r\), for \(r=3\) and \(r=4\). In their paper, C. Keem and S. Kim also proved the irreduciblity of \(H_{g+2,g,3}\) if \(g\geq 5\), and of \(H_{g+1,g,3}\) if \(g\geq 11\).
Continuing the quoted works of L. Ein, C. Keem and S. Kim, and using Brill-Noether Theory as developed by [\textit{E. Arbarello, M. Cornalba, P. Griffiths} and \textit{J. Harris}, ``Geometry of Algebraic Curves'', Grundlehren der Mathematischen Wissenschaften, 267 (1985; Zbl 0559.14017)], in the paper under review the author further refines the irreducibility range of \(H_{d,g,r}\) for \(3\leq r\leq 4\), proving that \(H_{g,g,3}\) is irreducible if \(g\geq 13\), and that \(H_{g+i,g,4}\) is irreducible for \(2\leq i\leq 3\) if \(g\geq 23-6i\). projective space curve; Brill-Noether Theory; line bundle; normal bundle; linear series \textsc{H. Iliev}, On the irreducibility of the Hilbert scheme of space curves, Proc. Amer. Math. Soc. \textbf{134} (2006), 2823-2832. Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) On the irreducibility of the Hilbert scheme of space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \({\mathcal H}(d,g,r)\) denote the Hilbert scheme of smooth curves of degree \(d\) and genus \(g\) in the complex projective space \(\mathbb{P}^r\). The authors are primarily interested in the Hilbert scheme of curves of low gonality and its geometric properties. In particular, they examine the so-called Brill-Noether-Petri properties for the Hilbert scheme \({\mathcal H}(2g-8,g,g-5)\) in section 2. In section 3, they look for the components of Hilbert schemes which violates the Brill-Noether-Petri properties. In fact they exhibit extra components of Hilbert schemes with an open dense subset considering of \(k\)-gonal curves. Finally, they look at the Hilbert scheme consisting of nearly extremal curves; they give an example of a Hilbert scheme with two components consisting of smooth curves which achieve nearly maximal genus with respect to the degree in projective space \(\mathbb{P}^r\). Brill-Noether-Petri properties Special divisors on curves (gonality, Brill-Noether theory), Parametrization (Chow and Hilbert schemes) On the Hilbert scheme of trigonal curves and nearly extremal 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 In a recent paper [the first and the second author, in: Symmetries, integrable systems and representations. Proceedings of the conference on infinite analysis: frontier of integrability, Tokyo, Japan, July 25--29, 2011 and the conference on symmetries, integrable systems and representations, Lyon, France, December 13--16, 2011. London: Springer. 15--33 (2013; Zbl 1307.14007)], the first two authors proved that the generating series of the Poincare polynomials of the quasihomogeneous Hilbert schemes of points in the plane has a simple decomposition in an infinite product. In this paper, we give a very short geometrical proof of that formula. Qian, C J, A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems, 4708-4715, (2005) Parametrization (Chow and Hilbert schemes), (Co)homology theory in algebraic geometry A simple proof of the formula for the Betti numbers of the quasihomogeneous 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 description of the Gorenstein locus of zero-dimensional schemes of degree 10 in projective \(N\)-space. Specifically, let \(H\) denote this open subscheme of the Hilbert scheme. The authors show that \(H\) is irreducible and give a complete description of its singular locus.
The general question of ``what is the structure of the Hilbert scheme'' is very difficult and has attracted the interest of many mathematicians. Aside from Hartshorne's proof that it is always connected, not too much is known in complete generality. However, the case where the common dimension of the parametrized schemes is zero is particularly interesting, and the present paper builds on a growing pile of results in various specific cases. The paper extends previous work by the same authors which establishes the irreducibility of the same set when the degree is less than 10. Hilbert scheme; Gorenstein scheme; zero-dimensional schemes Casnati, G; Notari, R, On the irreducibility and the singularities of the gorenstain locus of the punctual Hilbert scheme of degree 10, J. Pure Appl. Algebra, 215, 1243-1254, (2011) Parametrization (Chow and Hilbert schemes), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the irreducibility and the singularities of the Gorenstein locus of the punctual Hilbert scheme of degree 10 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities R. Vakil defined singularity type as an equivalence class of pointed schemes under the relation generated by \((X,x)\sim(Y,y)\) if there is a smooth morphism \((X,x)\rightarrow(Y,y)\) [\textit{R. Vakil}, Invent. Math. 164, No. 3, 569--590 (2006; Zbl 1095.14006)]. He showed that every singularity type over \(\mathbb Z\) appears on various moduli spaces of well-behaved objects: one says that Murphy's law holds for these moduli spaces.
The author investigates the Hilbert scheme of points, which is missing in Vakil's list. The main theorem states that Murphy's law holds up to retraction for \(\mathrm{Hilb}_{\mathrm{pts}}(\mathbb A^{16}_{\mathbb Z})\). The proof proceeds by a series of reductions from objects with more structure. The main role is played by a generalized Białynicki-Birula decomposition [\textit{J. Jelisiejew} and \textit{Ł. Sienkiewicz}, J. Math. Pures Appl. (9) 131, 290--325 (2019; Zbl 1446.14030)]. In order to construct the local retractions, the author uses TNT frames.
Using concrete singularity types, the author shows that \(\mathrm{Hilb}_{\mathrm{pts}}(\mathbb A^{16}_{\mathbb Z})\) and \(\mathrm{Hilb}_{\mathrm{pts}}(\mathbb A^{16}_{\mathbb C})\) are non-reduced, answering questions raised by J. Fogarty [\textit{J. Fogarty}, Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)]. He also shows that not all finite schemes over finite fields lift to characteristic zero, answering a question by R. Hartshorne [\textit{R. Hartshorne}, Deformation theory. Berlin: Springer (2010; Zbl 1186.14004)].
As a corollary of the main theorem, Murphy's law holds up to retraction for \(\mathrm{Hilb}_{\mathrm{pts}}(\mathbb P^{16}_{\mathbb Z})\). Since the forgetful functor from embedded to abstract deformations of a finite scheme is smooth, the above described pathologies appear for abstract deformations of finite schemes.
The author highlights that the choice of ambient dimension \(n=16\) was made for sake of transparency in the proof, but he suggests that the result may hold for \(n=6\) or even \(n=4\). Hilbert scheme of points; Vakil's Murphy's Law Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Local deformation theory, Artin approximation, etc., Deformations and infinitesimal methods in commutative ring theory, Deformations of singularities Pathologies on 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 In the paper under review, the authors study the Hilbert scheme parametrizing schemes and families of subschemes of a fixed projective space \(\mathbb{P}^n\) with a fixed Hilbert polynomial \(p(t)\). \textit{G. Gotzmann} [Math. Z. 158, 61--70 (1978; Zbl 0352.13009)] gave an optimal upper bound of the Castelnuovo-Mumford regularity of schemes parametrized by a given Hilbert scheme. However, in many cases this bound turns out to be far higher than the regularity of the most meaningful schemes from a geometric point of view. Hence, it is natural and interesting to study loci of Hilbert schemes with bounded regularity.
The authors tackle the problem of determining equations for these loci, starting from the classical construction of the Hilbert scheme as subscheme of a suitable Grassmannian. By semicontinuity, imposing an upper bound of the regularity corresponds to an open condition, so these loci turn out to be locally closed subschemes of the Grassmannian. The authors prove that such subschemes can be defined by equations of degree at most \(\deg p(t)+2\) in an open subscheme of the Grassmannian defined by the non-vanishing of several linear forms. The tools and techniques used in this paper come from another paper of the third author with \textit{J. Brachat}, \textit{P. Lella} and \textit{B. Mourrain} [``Extensors and the Hilbert scheme'', to appear on Ann. Sc. Norm. Super. Pisa, Cl. Sci. (2014); \url{doi:10.2422/2036-2145.201407\_003}]. Hilbert scheme; Castelnuovo-Mumford regularity; Borel-fixed ideal; marked basis Ceria, M.: JMBConst.lib. A library for Singular 4-0-2 which constructs J-Marked Schemes. Available at: http://www.singular.uni-kl.de/svn/trunk/Singular/LIB/JMSConst.lib (2012) Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) The locus of points of the Hilbert scheme with bounded regularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 expository paper is the content of the author's talk at the international conference on ``Zero-dimensional schemes'' in Ravello, 1992. He gives an overview of several important topics in the subject of zero-schemes. He discusses the (projectivized) tangent cone to an affine curve with an ordinary singularity at the origin, various uniformity conditions, the ideal generation and minimal resolution conjectures, the problem of finding hypersurfaces with prescribed singularities at a finite set of points, ``fat points'', and embeddings of blow-ups of \(\mathbb{P}^ 2\). He also gives several examples, describes results obtained by himself and others on these topics, and discusses open problems. ideal generation conjectures; fat points; zero-schemes; tangent cone; minimal resolution conjectures Schemes and morphisms, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Singularities of curves, local rings Zero-dimensional schemes: Singular curves and rational surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be an integral projective curve over an algebraically closed field. Let \(J(X)\) be the generalized Jacobian of degree zero invertible sheaves on \(X\) and let \({\mathcal M}(X)\) be the moduli space of degree zero and rank one torsion free sheaves on \(X\). \({\mathcal M}(X)\) is a projective scheme containing \(J(X)\) as an open subscheme. Altman, Iarrobino, and Kleiman showed that if every singular point of \(X\) has embedding dimension 2 then \({\mathcal M}(X)\) is an integral projective variety which is a local complete intersection. The paper treats the case when \(X\) admits at most simple singularities. compactified Jacobian; generalized Jacobian; simple singularities P. R. Cook, Compactified Jacobians and curves with simple singularities, preprint, University of Liverpool. Jacobians, Prym varieties, Singularities of curves, local rings Compactified Jacobians and curves with simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(C\) be a curve in \(\mathbb P^3\) over an algebraically closed field \(k\) and let \(R = k[x_0,x_1,x_2,x_3]\). Denote by \(I(C)\) its homogeneous ideal, and by \(\mathcal I_C\) its ideal sheaf. The \textit{Rao module} (sometimes \textit{Hartshorne-Rao module} or \textit{deficiency module}) of \(C\) is the graded \(R\)-module \(M = H_*^1(\mathcal I_C) := \bigoplus_{t \in \mathbb Z} H^1(\mathbb P^3, \mathcal I_C (t))\). The curve \(C\) is said to be \textit{Buchsbaum} if \(M\) is annihilated by the irrelevant ideal of \(R\). This is true in particular when \(M\) has only one non-zero component, and such curves are the central object of study of this paper. Assume from now on that \(C\) is such a curve. We consider all the components, \(V\), of the Hilbert scheme \(H(d,g)\) that contain \(C\). The author determines \(V\) from the point of view of describing the graded Betti numbers of the generic curve of \(V\) in terms of the graded Betti numbers of \(C\). This deep and careful study of the behavior of the Betti numbers of curves in these families, combined with earlier work of the author, gives us a good understanding of the Hilbert scheme of \(C\), and its singular locus. Hilbert scheme; space curve; Buchsbaum curve; graded Betti numbers; ghost term; linkage Kleppe, J.O., The Hilbert scheme of Buchsbaum space curves, Ann. inst. Fourier, 62, 6, 2099-2130, (2012) Parametrization (Chow and Hilbert schemes), Plane and space curves, Linkage, Syzygies, resolutions, complexes and commutative rings, Linkage, complete intersections and determinantal ideals The Hilbert scheme of Buchsbaum space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities To any 0-dimensional, reduced, degree \(s\), projective scheme \(X\) we associate a set \(S_X\) of sequences of \(s\) natural numbers; these sequences turn out to be a suitable permutation of a nondecreasing sequence \((d_1,\dots,d_s)\) which is in \(1-1\) correspondence with the first difference \(\Gamma_X\) of the Hilbert function of \(X\). The structure of \(S_X\) allows us to read geometric properties of \(X\). In its turn, the set \(\mathcal S_X\) of all sequences allowed for at least one scheme \(X\) with \(\Gamma_X=\Gamma\) can be partitioned into equivalence classes, giving information on all the schemes \(X\) with \(\Gamma_X=\Gamma\). We start to afford the problem of producing all the sequences equivalent to a given one.
Cf. also the review [in: Zero-dimensional schemes and applications. Proceedings of the workshop, Naples, Italy, February 9-12, 2000. Kingston: Queen's University. Queen's Pap. Pure Appl. Math. 123, 221--230 (2002; Zbl 1008.14010)]. Beccari, G.; Massaza, C.: A new approach to the Hilbert function of a 0-dimensional projective scheme, J. pure appl. Algebra 165, No. 3, 235-253 (2001) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Graded rings, Relevant commutative algebra, Computational aspects in algebraic geometry A new approach to the Hilbert function of a 0-dimensional projective 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 A classical way to study a finite set of points in a projective space \(\mathbb{P}^r\) over an algebraically closed field is to look at relations between its Castelnuovo function (i.e. the first difference function of the Hilbert function of its homogeneous coordinate ring) of geometric properties of the point set. In this extended abstract (containing no proofs), the authors refine that method as follows.
Given a sequence \(X= (P_1,\dots, P_s)\) of points in \(\mathbb{P}^r\), let \(S= (d_1,\dots, d_s)\in \mathbb{N}^s\) be defined by \(d_1=0\) and \(d_k=\) least degree of a hypersurface separating \(P_k\) from \(P_1,\dots, P_{k-1}\) for \(k>1\). Then the multiplicity sequence \(\gamma_S(n)= \#\{i\mid d_i=n\}\) equals the Castelnuovo function of \(X\) and does not depend on the order of the points. Hence it makes sense to study which sequences \(S\) are realizable and to try to classify all point sets \(X\) with given Castelnuovo function \(\Delta HF_X\) according to their sequences \(S\).
Here the authors announce some steps in this direction by examining the effect of neighbour transposition in the point sequence on the degree sequence. They discover that a set \(X\) gives rise to only one non-decreasing sequence \(S\) if and only if \(X\) is in uniform position. Moreover, maximal growth of the Castelnuovo function is shown to correspond to sequences of the form \(S= (0,1,\dots, n+h, S')\) with non-decreasing sequence \(S'\). All results are amply illustrated by examples. 0-dimensional scheme; Hilbert function; geometric properties of the point set; Castelnuovo function Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Cycles and subschemes A new approach to the Hilbert function of a 0-dimensional projective scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper ist devoted to the classification of simple surface singularities, i.e., singularities of germs of complex surfaces such that there are only finitely many isomorphism classes of singularities in their versal deformation.
Starting with rational double points and rational triple points which are known to be simple (the classical ADE list due to Arnold in the first case and Frühbis-Krüger, Neumer in the second), the author conjectures the following:
``Simple normal surface singularities are exactly those rational singularities whose resolution graphs can be obtained from the graphs of rational double points and rational triple points by making any number of vertex weights more negative.''
Singularities of such graphs are taut in the sense of Laufer, i.e., their analytic type is determined by the graph. The conjecture would imply that for normal surfaces there do not exist rigid singularities (an open question so far).
In the article, the following is shown:
(a) If a rational singularity is simple, it satisfies the condition of the conjecture.
(b) Conversely, the conjecture on simpleness is satisfied for the classes of quotient singularities, rational quadruple points and sandwiched singularities. simple singularities; rational singularities; tautness; normal surface singularities; rational double point; rational quadruple point; sandwiched singularities Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, Deformations of complex singularities; vanishing cycles, Deformations of singularities, Singularities in algebraic geometry, Infinitesimal methods in algebraic geometry Simple 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 study ``straight equisingular deformations'', a linear subfunctor of all equisingular deformations and describe their seminuniversal deformation by an ideal containing the fixed Tjurina ideal. Moreover, we show that the base space of the seminuniversal straight equisingular deformation appears as the fibre of a morphism from the \(\mu\)-constant stratum onto a punctual Hilbert scheme parametrizing certain zero-dimensional schemes concentrated in the singular point. Although equisingular deformations of plane curve singularities are very well understood, we believe that this aspect may give a new insight in their inner structure. Singularities in algebraic geometry, Deformations of singularities, Local complex singularities, Equisingularity (topological and analytic) Straight equisingular deformations and 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 Let \(X\) be a smooth quasi-projective surface. It is well known by work of \textit{J. Fogarty} that the Hilbert scheme \(\mathrm{Hilb}^n (X)\) parametrizing closed subschemes of length \(n\) is a smooth irreducible variety of dimension \(2n\) [Amer. J. Math. 90, 511--521 (1968; Zbl 0176.18401)]. The corresponding universal family \(Z^n \subset \mathrm{Hilb}^n (X) \times X\) is smooth for \(n=2\) (it is isomorphic to the blow-up of \(X \times X\) along the diagonal), but singular for \(n>2\). \textit{J. Fogarty} showed that \(Z^n\) is irreducible, normal, Cohen-Macaulay and satisfies Serre's condition \(R_3\) [Am. J. Math. 95, 660--687 (1973; Zbl 0299.14020)]
In this note the author gives more detailed information about the local structure of \(Z^n\), proving that its singularities are rational but not \(\mathbb Q\)-Gorenstein. He also proves that at a closed point \(\zeta = (\xi,p) \in Z^n\), the Samuel multiplicity \(\mu\) is given by \(\binom{b_2+1}{2}\), where \(b_2\) is the dimension of the socle of the local ring \({\mathcal O}_{\xi,p}\). He also proves a sharp upper bound on \(b_2\), namely \(\displaystyle b_2 \leq \lfloor \frac{\sqrt{1+8n}-1}{2} \rfloor\). This implies that the Samuel multiplicity satisfies \(\mu \leq n\), corroborating a result of \textit{M. Haiman} [J. Amer. Math. Soc. 14, No. 4, 841--1006 (2001; Zbl 1009.14001)]. Hilbert scheme of points on a surface; universal family; rational singularities; Samuel multiplicity Parametrization (Chow and Hilbert schemes) On the universal family of Hilbert schemes of points on a surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S = k[x_1, \dots , x_n]\) be a polynomial ring graded by \(\deg(x_i) = 1\) for all \(i\) over an algebraically closed field \(k\) of characteristic zero. Let \(P = (x^{a_1}_ 1 ,\dots , x^{a^n}_n )\), with \(a_1\leq a_2\leq \dots\leq a_n\leq \infty\) (where \(x_i^{\infty} = 0\)) and set \(W = S/P\). Then \(W\) is called a \textit{Clements-Lindström} ring. We refer to \(x^{a_1}_ 1 ,\dots , x^{a^n}_n\) as the \(P\)-powers. The \textit{Clements-Lindström} Theorem states that Macaulay's Theorem holds over \(W\), that is, for every graded ideal in \(W\) there exists a lex ideal with the same Hilbert function.
Let Hilbert scheme \(H_{W}(h)\) be the scheme that parametrizes all graded ideals in \(W\) with a fixed Hilbert function \(h\). Equivalently, this Hilbert scheme parametrizes all graded ideals in \(S\) containing the \(P\)-powers and with a fixed Hilbert function. Let us define a \(P\)-deformation, that is, a deformation that connects ideals containing the \(P\)-powers. With this notation in Section 3, \(\S 3.7\), the authors prove Theorem 1.3, that the Hilbert scheme \(H_{W}(h)\) is connected; and that every graded ideal in the polynomial ring \(S\) that contains the \(P\)-powers, is connected by a sequence of \(P\)-deformations to the lex-plus- powers ideal with the same Hilbert function.
Macaulay's Theorem was generalized to Betti numbers by Bigatti, Hulett, and Pardue as follows: (see Theorem 1.4.) Every lex ideal in \(S\) attains maximal Betti numbers among all graded ideals with the same Hilbert function.
Aramova, Herzog, and Hibi proved that every lex ideal in an exterior algebra attains maximal Betti numbers among all graded ideals with the same Hilbert function. It was conjectured by Gasharov, Hibi, and Peeva that Theorem 1.4 holds over the \textit{Clements-Lindström} ring \(W\). In Section 4, the main theorem is Theorem 4.6.5, i.e., the proof of the conjecture: (see Theorem 1.5.) Every lex ideal in \(W\) attains maximal Betti numbers among all graded ideals with the same Hilbert function.
Note that Theorem 1.4 is about finite resolutions, while Theorem 1.5 is about infinite ones. In order to prove Theorem 1.5 the authors construct special changes of coordinates and use them to build a construction that starting with a monomial ideal yields a lex-closer ideal with bigger Betti numbers. Hilbert scheme; deformations; Betti numbers S. Murai and I. Peeva, Hilbert schemes and Betti numbers over a Clements-Lindström ring, submitted. Formal methods and deformations in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Syzygies, resolutions, complexes and commutative rings Hilbert schemes and Betti numbers over Clements-Lindström rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper studies tropicalizations of Hilbert schemes given as subschemes of the Grassmanian. The connection to the tropical Grassmanian enables the authors to draw nice consequences about degree bounds for tropical bases. The cases of hypersurfaces, and of pairs of points in the plane are considered in detail. tropicalization; Hilbert scheme; Grassmanian; tropical bases D. Alessandrini, M. Nesci, On the tropicalization of the Hilbert scheme, arxiv:0912.0082. , Fine and coarse moduli spaces On the tropicalization 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 Let \(M\) be a smooth algebraic variety over a field \(k\), algebraically closed and of characteristic zero. Let \(X\subset M\) be a hypersurface, i.e., a closed subscheme locally defined by one equation. Then the singular locus \(Y\) of \(X\) has a natural scheme structure defined by the jacobian ideal.
The author defines a class \(\mu_{\mathcal L}(Y)\) in the Chow group of \(Y\), depending on the data \((M,X)\): he then shows that the class only depends on \(Y\) and on the line bundle \({\mathcal L}:={\mathcal O}_M(X)|_Y\), and gives explicit methods for computing it in the case where \(Y\) is itself smooth.
In \S 2 it is proven that \(\deg(\mu_{\mathcal L}(Y))\) is Parusiński's generalized Milnor number; the class is then used to study local properties near \([X]\) of the dual variety of \((M,{\mathcal O}_M(X))\), i.e., the variety of sections of \({\mathcal O}_M(X)\) defining singular hypersurfaces, recovering and partially extending previous results of Ein, Holme, Landman, Parusiński and Zak.
In \S 3 it is shown that the \(\mu\) class imposes strong restrictions on which schemes \(Y\) can appear as singular subschemes of a hypersurface. hypersurface singularity; Milnor number; jacobian ideal; Chow group Paolo Aluffi, ``On the singular schemes of hypersurfaces'', Duke Math. J.80 (1995) no. 2, p. 325-351 Hypersurfaces and algebraic geometry, Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes) Singular schemes of hypersurfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be an adjoint semisimple algebraic group over an algebraically closed field. \textit{C. De Concini} and \textit{C. Procesi} [in: Invariant theory, Proc. 1st 1982 Sess. C.I.M.E., Montecatini/Italy, Lect. Notes Math. 996, 1-44 (1983; Zbl 0581.14041)] proved in characteristic zero the following fundamental result:
Theorem. \(G\) admits a canonical smooth completion \(\overline{G}\) such that:
(i) the action of \(G\times G\) by left and right multiplication on \(G\) extends to \(\overline{G}\),
(ii) the boundary \(\overline{G}\setminus G\) is the union of smooth irreducible divisors intersecting transversely along the unique closed \(G\times G\)-orbit, and
(iii) the partial intersections of these boundary divisors are exactly the orbit closures.
Subsequently this result was extended in positive characteristic by \textit{E. S. Strickland} [see Math. Ann. 277, 165-171 (1987; Zbl 0595.14037)]. In both papers the method was via representation theory. In the paper under review the author obtains algebro-geometric realizations of \(\overline{G}\) as follows:
Let \(P\) be a parabolic subgroup of \(G\) and set \(X:=G/P\). The group \(G\times G\) acts on the Hilbert scheme \(\text{Hilb}(X\times X)\) via the natural action of \(G\times G\). When the action of \(G\) on \(X\) is faithful, then the \(G\times G\)-orbit of the diagonal \(\Delta_X\) (regarded as a point of \(\text{Hilb}(X\times X)\)) is isomorphic to \(G\).
Then the author proves that \(\overline{G}\) is isomorphic to the closure of this orbit. completion of algebraic group; action of algebraic group; Hilbert scheme Michel Brion, Group completions via Hilbert schemes, J. Algebraic Geom. 12 (2003), 605-626. Group actions on varieties or schemes (quotients), Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations Group completions via 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 shows the algebraicity of the Hilbert functor \(\text{Hilb}^d_X\) of \(d\) points on a separated algebraic space or stack \(X\). By taking a presentation \(U\to X\), one observes that the push-forward map determines a presentation of the Hilbert functor \(\text{Hilb}^d_X\). One then reduces the question concerning the algebraicity of \(\text{Hilb}^d_X\) to the algebraicity of the Hilbert functor of points for affine \(U\), where it is known.
An important shift of perspective is obtained by considering the Hilbert \textit{stack} of points on \(X\). The classical Hilbert functor of \(d\) points on \(X\) parametrizes closed subschemes \(Z\subseteq X\) that are flat and finite of rank \(d\) over the base. The small, and elegant, generalization is to replace the closed immersion \(Z\subseteq X\) with a morphism \(Z \to X\). The algebraicity for the Hilbert stack of points on an algebraic stack \(X\) is essentially deduced in the same way as for the Hilbert functor. Hilbert scheme; Hilbert stack; Weil restriction; Hom stack; non-separated D. Rydh, ''Representability of Hilbert schemes and Hilbert stacks of points,'' Comm. Algebra, vol. 39, iss. 7, pp. 2632-2646, 2011. Parametrization (Chow and Hilbert schemes), Generalizations (algebraic spaces, stacks) Representability of Hilbert schemes and Hilbert stacks 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 We consider the Prym map \(\mathcal P : \mathcal R_g\to \mathcal A_{g-1}\) from the moduli space of unramified double covers of projective irreducible and non-singular curves of genus \(g\geq 6\) to the moduli space of principally polarized abelian varieties of dimension \(g-1\). If \(\pi: \tilde{C}\to C\) is such a double cover with \(C\) non-hyperelliptic, we consider the natural embedding \(\tilde{C} \subset P\) (defined up to translation) of \(\tilde{C}\) into the Prym variety \(P\) of \(\pi\) and we study the local structure of the Hilbert scheme \(Hilb^P\) of \(P\) at the point \([\tilde{C}]\). We show that this structure is
related to the local geometry of the Prym map, or more precisely with the validity
of the infinitesimal version of Torelli's theorem for Pryms at \([\pi]\) (see Section 3 for the definitions).
The results we prove are the following.
Proposition. If the infinitesimal Torelli theorem for Pryms holds at \([\pi]\) then \(Hilb^P\) is non-singular of dimension \(g-1\) at \([\tilde{C}]\) (i.e. \(\tilde{C}\) is unobstructed) and the only
deformations of \(\tilde{C}\) in \(P\) are translations.
Theorem. Assume that the following conditions are satisfied:
(a) the infinitesimal Torelli theorem for Pryms fails at \([\pi]\);
(b) \([\pi]\) is an isolated point of the fibre \(\mathcal P^{-1}(\mathcal P ([\pi]))\).
Then \(\tilde{C}\) is obstructed. Moreover the only local deformations of \(\tilde{C}\) in \(P\) are translations and the only irreducible component of \(Hilb^P\) containing \([\tilde{C}]\) is everywhere non-reduced.
Conversely, if \(\tilde{C}\) is obstructed then the infinitesimal Torelli theorem for Pryms fails at \([\pi]\).
Using these results we give some examples in which \(\tilde{C}\subset P\) is obstructed and
some in which we have unobstructedness but the infinitesimal Torelli theorem for
Pryms fails. Lange H., Sernesi E.: On the Hilbert scheme of a Prym variety. Ann. di Matem. 183, 375--386 (2004) Jacobians, Prym varieties, Parametrization (Chow and Hilbert schemes), Torelli problem On the Hilbert scheme of a Prym variety | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\text{Hilb}^{P}({X})\) be the Hilbert scheme of smooth connected non-degenerate curves \(C \subset X\) with Hilbert polynomial \(P(m)=dm+1-g\) where \(X\) is either \({\mathbb P}^3\), \({\mathbb P}^4\) or a smooth quadric threefold \(Q\) in \({\mathbb P}^4\). Let \(U\) be any irreducible component of \(\text{Hilb}^{P}({X})\).
In this paper, the author gives a good lower bound for the dimension of \(U\) for \(X= {\mathbb P}^3\) in the range \(g^2 \geq d^3\) (Theorem 1.3). In proving this result, the author needs to calculate the Euler characteristic \( \chi(N_{C/S})\) of the normal sheaf where \(S\) is a smooth surface containing \(C\) and the variation of \( \chi(N_{C/S})\) which we need when \(S\) is singular (the latter is not easy). If \(X=Q\) the author shows that if \(g^2 > d^3/2\) (resp. \(g^2 < 4d^3/1125\)) up to lower degree terms, then \(\dim U \) is always greater than the expected value \(3d\) (resp. there exists a component with \(\dim U = 3d\)). To get the existence result he uses smoothing techniques introduced by \textit{E. Sernesi} [Invent. Math. 75, 25--57 (1984; Zbl 0541.14024)].
Finally recall that a curve \(C \subset {\mathbb P}^r\) is called rigid if every deformation of \(C\) is induced by an automorphism of \({\mathbb P}^r\). For \(X={\mathbb P}^4\) the author shows that there are no rigid curve in a range asymptotically given by \(g^2 > 9d^3\), thus contributing to a conjecture of \textit{J. Harris} and \textit{I. Morrison} [Moduli of curves. New York, NY: Springer (1998; Zbl 0913.14005)], stating that only normal rational curves are rigid.
Reviewer's remark: In Theorem 2.1 there is a missing number 1 in Peskine-Gruson's result on the maximum genus. This makes Proposition 2.2 slightly inaccurate; one should there replace \(g\) by \(g-1\), or exclude e.g. complete intersections in the conclusion. To correct the inaccuracy, one may in Theorem 1.3 redefine \(\mu\) by replacing \(g\) by \(g-1\) in the definition. Hilbert scheme; deformation; space curve; rigid curve; smoothing Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Plane and space curves On the dimension of the Hilbert scheme 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 \textit{R. Hartshorne} [Publ. Math., Inst. Hautes Étud. Sci. 29, 5-48 (1966; Zbl 0171.41502)] showed that the full Hilbert scheme for projective subschemes with a fixed Hilbert polynomial is connected. Often one studies certain subsets of the Hilbert scheme which parametrize subschemes satisfying a certain property. For example, one can consider the Hilbert scheme \(H(d,g)\) of locally Cohen-Macaulay curves in the projective space \(\mathbb{P}^3\) of fixed degree \(d\) and genus \(g\). In this case, it is not known whether \(H(d,g)\) is connected. In the present paper the author shows that \(H(3,g)\) is connected. This is achieved by giving a classification of the locally Cohen-Macaulay space cubics of genus \(g\), determining the irreducible components of \(H(3,g)\), and giving certain specializations to show connectedness. Curiously, there are curves which lie in the closure of each irreducible component of \(H(3,g)\). locally Cohen-Macaulay space curve; Rao module; Koszul module; Hilbert scheme; connectedness [MDP1]\textsc{M. Martin-Deschamps--D. Perrin},\textit{Sur la classification des courbes gauches}, Astérisque, Vol.\textbf{184-185}, 1990. Parametrization (Chow and Hilbert schemes), Plane and space curves, Topological properties in algebraic geometry The Hilbert schemes of degree three 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 projective surface with \(p_g=q=0\). We show how to use derived categorical methods to study the geometry of certain special iterated Hilbert schemes associated to \(S\) by showing that they contain a smooth connected component isomorphic to \(S\). Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, \(K3\) surfaces and Enriques surfaces, Surfaces of general type Smooth components on special iterated 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 Over an algebraically closed field \(K\) of characteristic 0, simple singularities are classically known as hypersurfaces \(R=K[[X_1,\dots ,X_{d+1}]]/(f)\), where \(f\) is (up to contact equivalence) one of the equations ADE [cf. \textit{V. I. Arnol'd, S. M. Gusejn-Zade} and \textit{A. N. Varchenko}, Singularities of differentiable maps. Volume I: The classification of critical points, caustics and wave fronts. Monographs in Mathematics, Vol. 82. Boston-Basel-Stuttgart: Birkhäuser (1985; Zbl 0554.58001)]. Among many others, those are equivalent conditions for a (formal) hypersurface:
(i) \(R\) is simple.
(ii) \(R\) is of finite CM-representation type (i.e. the number of isomorphism classes of indecomposable maximal Cohen-Macaulay modules is finite).
(iii) \(R\) is of finite deformation type.
This classification was extended by \textit{G.-M. Greuel} and \textit{H. Kröning} [Math. Z. 203, No. 2, 339--354 (1990; Zbl 0715.14001)] to arbitrary characteristic, where the equations \(f\) look slightly different in some cases.
The article under review gives a new characterization of simple singularities. In this context, \(R\) is \textit{simple} iff it is a hypersurface of finite CM-representation type.
The authors discuss the following condition: Let \(R\) be a local noetherian ring and \((\mathbf{mod}_R)\) the category of finitely generated \(R\)-modules. Let \({\mathcal G}(R)\) denote the full subcategory of modules \( M\in (\mathbf{mod}_R)\) such that there exists an exact sequence
\[
\dots \to F_n\to \dots \to F_1 \to \dots \to F_0 \to M\to 0
\]
with \(F_i\) f.g. free and such that the dual
\[
\dots \to F_n^*\to \dots \to F_1^* \to F_0^*
\]
of the complex \((F_i)_{i\in\mathbb N}\) is exact (\(^*\) denotes the algebraic dual \(\text{Hom} (- ,R)\)). Such modules are said to be of \textit{Gorenstein dimension} 0 (in the sense of Auslander-Bridger) or \textit{totally reflexive} (in the sense of Avramov-Martsinkovsky).
The authors show: Theorem A. Let \(R\) be complete. If the set of isomorphism classes of indecomposable modules in \({\mathcal G} (R)\) different from the class of \(R\) is finite and not empty, then \(R\) is simple.
There is, in fact, a close relation to the notion of MCM-representations. Thus it is reasonable to speak of \textit{finite Gorenstein representation type} in the above case.
A theorem of \textit{J. Herzog} [Math. Ann. 233, 21--34 (1978; Zbl 0358.13009)] states: If \(R\) is Gorenstein and of finite MCM-representation type then \(R\) is an abstract hypersurface, i.e. \(\hat{R} \cong A/xA\) with \(A\) regular, \(x\in m_A\).
On the other hand, if \(R\) is Gorenstein, \({\mathcal G} (R)= \mathcal{MCM}(R)\). Thus theorem A is a corollary of the following which does not make any assumption on the local noetherian ring \(R\).
Theorem B. Let \(R\) be a local noetherian ring. If the set of isomorphism classes of indecomposable modules in \({\mathcal G}(R)\) is finite, then \(R\) is Gorenstein or every module in \({\mathcal G}(R)\) is free.
In this general context, the known theory of CM-approximation is not appropriate any more, but it gives an idea how to prove the above result for the Gorenstein case. The authors develop an approximation theory of modules in \((\mathbf{mod}_R)\) with respect to \({\mathcal G}(R)\). More generally, it turns out necessary to do this for any \textit{reflexive subcategory} \(\mathcal B\) of \((\mathbf{mod}_R)\) i.e. for a subcategory \(\mathcal B\) with the following properties: Denote
\[
{\mathcal B}^\perp := \{ L\in (\mathbf{mod}_R) \mid \text{Ext}_R^i(B,L)=0 \;\;\forall B\in {\mathcal B}, i>0 \} .
\]
Then
\(\bullet\) \(R\in {\mathcal B} \cap {\mathcal B}^\perp \)
\(\bullet\) \(\mathcal B\) is closed under direct sums and direct summands.
\(\bullet\) \(\mathcal B\) is closed under syzygies.
\(\bullet\) \(\mathcal B\) is closed under algebraic duality.
\({\mathcal G} (R)\) is known to be the largest reflexive subcategory of \((\mathbf{mod}_R)\), and for any reflexive subcategory \(\mathcal B\) there are inclusions
\[
{\mathcal F} (R) \subseteq {\mathcal B} \subseteq {\mathcal G} (R),
\]
where \({\mathcal F} (R)\) denotes the full subcategory of free modules in \((\mathbf{mod}_R)\).
Using a result of Takahashi, conditions on the existence of covers are obtained: Assuming the Krull-Remak-Schmidt property for \((\mathbf{mod}_R)\), a module \(M\) has a \(\mathcal B\)-precover iff it has a \(\mathcal B\)-approximation.
There is the following relation to MCM-approximation:
\({\mathcal G}(R) \subseteq \mathcal{MCM}(R)\) iff \(R\) is Cohen-Macaulay.
Central part in the proof of Theorem B is Theorem C, which reduces Theorem B to showing that the residue field \(k\) of \(R\) has a reflexive hull.
Corollary. For a local ring \(R\) such that \({\mathcal G}(R)\) contains not only free modules, the following conditions are equivalent:
\(\bullet\) \(R\) is Gorenstein.
\(\bullet\) The residue field \(R/m\) has a \({\mathcal G}(R)\)-approximation.
\(\bullet\) All modules in \( (\mathbf{mod}_R)\) have a minimal \({\mathcal G} (R)\)-approximation. approximations; Cohen-Macaulay representation type; covers; Gorenstein dimension; precovers; simple singularity; totally reflexive modules Christensen, L. W.; Piepmeyer, G.; Striuli, J.; Takahashi, R., Finite Gorenstein representation type implies simple singularity, Adv. Math., 218, 1012-1026, (2008) Singularities in algebraic geometry, Relative homological algebra, projective classes (category-theoretic aspects), Cohen-Macaulay modules Finite Gorenstein representation type implies simple singularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R = \mathbb{C}[x_{1},\dots,x_{N}]\) and let \(F =\{ f_{1},\dots,f_{t}\} \subset R\) be a set of generators for an ideal \(I\). Let \(Y =\{ y_{1},\dots,y_{\ell}\} \subset {\mathbb{C}}^{N}\) be a subset of the set of isolated solutions of the zero locus of \(F\). Let \(\mathfrak{m}_{y_{i}}\) denote the maximal ideal of \(y_i\) and let \(\mathcal{P}_{y_{i}}\) denote the \(\mathfrak{m}_{y_{i}}\)-primary component of \(I\). Let \(J = \cap _{i=1}^{l}\mathcal{P}_{y_{i}}\) and let \(\mathcal{Z}\) denote the corresponding zero dimensional subscheme supported on \(Y\). This article presents a numerical algorithm for computing the Hilbert function and the regularity of \(\mathcal{Z}\). In addition, the algorithm produces a monomial basis for \(R\slash J\). The input for the algorithm is the polynomial system \(F\) and a numerical approximation of each element in \(Y\). Griffin, Zachary A.; Hauenstein, Jonathan D.; Peterson, Chris; Sommese, Andrew J.: Numerical computation of the Hilbert function of a zero-scheme. Springer Proceedings in mathematics \& statistics (2011) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Numerical computation of the Hilbert function and regularity of a zero dimensional 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 \(\Gamma \subset \mathrm{SL}_2(\mathbb{C})\) be a nontrivial finite subgroup and the surface \(S = \widehat{\mathbb{C}^2/\Gamma}\) be the minimal resolution of \(\mathbb{C}/\Gamma\). Associated to \(\Gamma\) is a Heisenberg algebra of affine type, \(\mathfrak{h}_\Gamma\), and the Hilbert schemes of points \(\mathrm{Hilb}^n(S)\). \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)] construct a representation of the Heisenberg algebra (actually a slightly different version from the one considered in this paper) on the cohomology of the Hilbert schemes. Algebraically, \textit{I. Frenkel}, \textit{N. Jing} and \textit{W. Wang} [Int. Math. Res. Not. 2000, No. 4, 195--222 (2000; Zbl 1011.17020)] construct the basic representation of \(\mathfrak{h}_\Gamma\) on the Grothendieck group of the category of \(\mathbb{C}[\Gamma^n \rtimes S_n]\)-modules.
In this paper, the authors define a 2-category \(\mathcal{H}_\Gamma\) and their first main result (3.4) states that \(\mathcal{H}_\Gamma\) categorifies the Heisenberg algebra \(\mathfrak{h}_\Gamma\).
The second main result of the paper (4.4) is a categorical action of \(\mathcal{H}_\Gamma\) on a 2-category \(\bigoplus_{n\geq 0} D(A_n^\Gamma -\mathrm{gmod})\). Here \(D(A_n^\Gamma -\mathrm{gmod})\) denotes the bounded derived category of finite-dimensional, graded \(A_n^\Gamma\)-modules, where
\[
A_n^\Gamma = [(\mathrm{Sym}^*((\mathbb{C}^2)^\vee) \rtimes \Gamma) \otimes \ldots \otimes (\mathrm{Sym}^*((\mathbb{C}^2)^\vee) \rtimes \Gamma) ] \rtimes S_n.
\]
As explained in Section 8, \(D(A_n^\Gamma -\mathrm{gmod})\) is known to be equivalent to \(D\mathrm{Coh}(\mathrm{Hilb}^n(S))\) and thus the second main theorem categorifies a representation similar to that of Grojnowski [Zbl 0879.17011] and Nakajima [Zbl 0915.14001].
In Section 9, another 2-representation of \(\mathcal{H}_\Gamma\) is introduced that is related to the first by Koszul duality. In section 9.6, it is shown that this 2-representation categorifies an action similar to that constructed by Frenkel, Jing and Wang [Zbl 1011.17020].
For the most part geometry appears only in Section 8. The main definitions are algebraic and a number of the proofs are based on graphical calculus. Section 10 contains a description of various connections to other categorical actions and some open problems.
As the case \(\Gamma = \mathbb{Z}/2\) differs slightly, the necessary modifications are addressed separately in a short appendix. categorification; Heisenberg algebra; McKay correspondence; Hilbert scheme S. Cautis & A. Licata, ``Heisenberg categorification and Hilbert schemes'', Duke Math. J.161 (2012) no. 13, p. 2469-2547 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Quantum groups and related algebraic methods applied to problems in quantum theory, Frobenius induction, Burnside and representation rings, Lie algebras and Lie superalgebras, Parametrization (Chow and Hilbert schemes) Heisenberg categorification 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 \(d\geq 2\) and \(g\) be integers and let \(H_{d,g}\) denote the Hilbert scheme of locally Cohen-Macaulay space curves of degree \(d\) and arithmetic genus \(g\). \(H_{d,g}\) is an open part of the Hilbert scheme parametrizing closed subschemes of \({\mathbb P}^3\) with Hilbert polynomial \(dt+1-g\). It is related to the classification of smooth, irreducible space curves via the process of linkage. \(H_{d,g}\) is non-empty precisely when \(g=(d-1)(d-2)/2\) (plane curves) or when \(g\leq (d-2)(d-3)/2\). The main conjecture concerning \(H_{d,g}\) asserts its connectedness. It has been proved, by several authors, for \(g\geq (d-3)(d-4)/2-1\) by deforming curves to curves of degree \(d\) containing a plane curve of degree \(d-1\) as a subscheme (so called ``extremal curves''). On the other hand, \textit{S. Nollet} [Ann. Sci. Éc. Norm. Supér., IV Ser. 30, No. 3, 367--384 (1997; Zbl 0892.14004)] described the irreducible components of the schemes \(H_{3,g}\) and proved that all these schemes are connected.
In these paper, the authors do the same thing for the schemes \(H_{4,g}\). The idea is the following one: a non-reduced component of a (locally Cohen-Macaulay) space curve of degree 4 must be a double line or conic, or a triple line, or a quadruple line. Double structures on smooth space curves were described in an important paper of \textit{D. Ferrand} [C. R. Acad. Sci. Paris 281, 345--347 (1975; Zbl 0315.14019)] and triple structures on space lines by \textit{R. Hartshorne} [Math. Ann. 238, 229--280 (1978; Zbl 0411.14002)]. Quadruple structures on space lines can be described using the general results of \textit{C. Bănică} and \textit{O. Forster} [in: Algebraic geometry, Proc. Lefschetz Centen. Conf., Mexico City/Mex. 1984, Part I, Contemp. Math. 58, 47--64 (1986; Zbl 0605.14026)]. According to the terminology of Bănică and Forster, there exist quasiprimitive 4-lines and thick 4-lines (i.e., 4-lines of embedding dimension 3 at every point).
The main contribution of the authors consists of two degeneration results : 1) quasiprimitive 4-lines can specialize to thick 4-lines, and 2) certain types of quasiprimitive 4-lines are specializations of disjoint unions of two double lines. In order to prove the connectedness of \(H_{4,g}\) they show that, for \(g\leq -3\), thick 4-lines form an irreducible component \(G_4\) of \(H_{4,g}\), disjoint from the component \(E\) of extremal curves, that any other component can be connected to either \(E\) or \(G_4\) and that there exists a component meeting both \(E\) and \(G_4\). space curve; locally Cohen-Macaulay curve; multiple structure on a smooth curve; connectedness Nollet S., Schlesinger E.: Hilbert schemes of degree four curves. Compos. Math. 139(2), 169--196 (2003) Plane and space curves, Parametrization (Chow and Hilbert schemes) Hilbert schemes of degree four curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We use basic algebraic topology and Ellingsrud-Strømme results on the Betti numbers of punctual Hilbert schemes of surfaces to compute a generating function for the Euler characteristic numbers of the Douady spaces of ''\(n\)-points'' associated with a complex surface. The projective case was first proved by \textit{L. Göttsche} [Math. Ann. 286, No. 1-3, 193--207 (1990; Zbl 0679.14007)]. Mark Andrea A. de Cataldo, Hilbert schemes of a surface and Euler characteristics, Arch. Math. (Basel) 75 (2000), no. 1, 59 -- 64. Parametrization (Chow and Hilbert schemes), Complex-analytic moduli problems Hilbert schemes of a surface and Euler characteristics | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, we study properties of the Hilbert schemes of ideals of finite algebras over an algebraically closed field. We prove a duality theorem for the Hilbert schemes of a finite Gorenstein algebra. We also study some properties of finite algebras obtained from informations on their Hilbert schemes. We give examples of finite algebras \(A\) such that the sequences \(\{\chi(\mathrm{Hilb}^r(A))\}_r\) are unimodal. They are examples of a generalization of a combinatorial conjecture by Stanton. finite algebras; Hilbert schemes Structure of finite commutative rings, Parametrization (Chow and Hilbert schemes) On the Hilbert schemes of finite algebras over an algebraically closed field | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(W\) be a finite dimensional linear representation of a reductive algebraic group \(G\) over an algebraically closed field \(k\) of characteristic 0. Let \(\mathcal{H}\) denote the invariant Hilbert scheme \(\mathrm{Hilb}^G_{h_W}(W)\) parametrizing \(G\)-stable closed subschemes \(Z\) of \(W\) with \(h_W\) being the Hilbert function of the general fiber of the (categorical) quotient morphism \(\nu : W \rightarrow W/\!\!/G = \mathrm{Spec}(k[W]^G)\). The article under review addresses the following question: in which cases is the Hilbert-Chow morphism from \(\gamma : \mathcal{H} \rightarrow W/\!\!/G\), possibly restricted to the main component, a desingularization of \(W/\!\!/G\)?
This question was studied before only for finite groups \(G\), in which case the \(G\)-Hilbert scheme of \textit{Y. Ito} and \textit{I. Nakamura} [Proc. Japan Acad., Ser. A 72, No. 7, 135--138 (1996; Zbl 0881.14002)] coincides with the main component of~\(\mathcal{H}\). They gave a positive answer for finite groups of \(\mathrm{SL}(2)\), then \textit{T. Bridgeland} et al. [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] for finite subgroups of \(\mathrm{SL}(3)\), and \textit{M. Lehn} and \textit{C. Sorger} [in: Geometric methods in representation theory. II. Selected papers based on the presentations at the summer school, Grenoble, France, June 16 -- July 4, 2008. Paris: Société Mathématique de France. 429--435 (2012; Zbl 1312.14007)] for a single 4-dimensional symplectic group.
The article under review gives first results in the case of infinite group~\(G\). The author considers four classical groups (SL, O, Sp, GL) with chosen series of natural representations. The main theorem states that in cases which are small enough (that is, they satisfy certain bounds on parameters of chosen representations), \(\mathcal{H}\) is a desingularization of \(W/\!\!/G\). The proof is based on a reduction principle, which allows to obtain information on all cases from the description of certain small ones. Two of four series of representations are analysed in the article, the details for remaining two can be found in the author's PhD thesis. algebraic group; quotient; desingularization; Hilbert scheme R. Terpereau, \textit{Invariant Hilbert schemes and desingularizations of quotients by classical groups}, Transform. Groups \textbf{19} (2014), no. 1, 247-281. Algebraic cycles, Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Invariant Hilbert schemes and desingularizations of quotients by classical groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We provide an overview of the \textit{VersalDeformations} package for \textit{Macaulay2} which computes versal deformations of isolated singularities and local multigraded Hilbert schemes. Ilten N.\ O., Versal deformations and local Hilbert schemes, J. Software Algebra Geom. 3 (2012), 12-16. Formal methods and deformations in algebraic geometry, Parametrization (Chow and Hilbert schemes), Software, source code, etc. for problems pertaining to algebraic geometry Versal deformations and local 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 This is a survey article on moduli of affine schemes equipped with an action of a reductive group. The emphasis is on examples and applications to the classification of spherical varieties. M. Brion, \textit{Invariant Hilbert schemes}, in: \textit{Handbook of Moduli}, Vol. I, Adv. Lect. Math. (ALM) \textbf{24}, Int. Press, Somerville, MA, 2013, pp. 64-117. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Compactifications; symmetric and spherical varieties Invariant 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 In this paper, the classification of real simple singularities of surfaces is investigated with respect to the actions of complex conjugations on exceptional loci. It is given a procedure for determining the action of complex conjugation on a non-singular fiber of a semi- universal deformation of a real simple singularity of surfaces. Dynkin diagram; complex conjugations on exceptional loci; semi-universal deformation of a real simple singularity of surfaces Singularities of surfaces or higher-dimensional varieties, Deformations of singularities, Real algebraic and real-analytic geometry, Real-analytic manifolds, real-analytic spaces, Deformations of complex singularities; vanishing cycles, Singularities in algebraic geometry, Complex singularities Real simple singularities of surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We extend the usual Hilbert property for varieties over fields to arithmetic schemes over integral domains by demanding the set of near-integral points (as defined by Vojta) to be non-thin. We then generalize results of \textit{L. Bary-Soroker} et al. [Ann. Inst. Fourier 64, No. 5, 1893--1901 (2014; Zbl 1359.12001); ``Ramified covers of abelian varieties over torsion fields'', Preprint, \url{arXiv:2206.01582}] and \textit{P. Corvaja} and \textit{U. Zannier} [Math. Z. 286, No. 1--2, 579--602 (2017; Zbl 1391.14044)] by proving several structure results related to products and finite étale covers of arithmetic schemes with the Hilbert property. integral points; Hilbert property; Hilbert's irreducibility theorem; arithmetic schemes Rational points, Arithmetic varieties and schemes; Arakelov theory; heights The Hilbert property for arithmetic 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 give a survey about the development and the results concerning Grothendieck's conjectures relating simple singularities of surfaces and the geometry of finite dimensional complex Lie algebras. Let \(\mathfrak{g}\) be a simple complex Lie algebra of type \(A_n, D_n, E_6, E_7, E_8\) and \(G\) the corresponding simply connected simple Lie group. Let \(\mathfrak{h}\) be a Cartan subalgebra of \(\mathfrak{g}\) and \(\mathfrak{B}\) a Borel subalgebra of \(\mathfrak{g}\). Let \(B\) be the corresponding Borel subgroup of \(G\). There is a canonical map \(\gamma: \mathfrak{g}\to \mathfrak{h}/W\), \(W\) the Weyl group, defined by \(\gamma(x)=(\gamma_1(x), \ldots, \gamma_r(x))\) where \(\gamma_1, \ldots,\gamma_r\) are the homogeneous \(G\)--invariant polynomials generating \(\mathbb{C}[\mathfrak{g}]^G\). Grothendiek conjectured that \(\gamma\) has a simultaneous resolution. He also conjectured that for a subregular nilpotent element \(y\) and a transversal slice \(X\) at \(y\) to the orbit \(Gy\) the germ of the surface \(X\cap \gamma^{-1}(\gamma(0))\) at \(y\) is a simple surface singularity of the same type as the Lie algebra \(\mathfrak{g}\). simple singularities; resolution of singularities; Lie algebras; subregular nilpotent elements Lê, D. T.; Tosun, M., Simple singularities and simple Lie algebras, \textit{TWMS J. Pure Appl. Math.}, 2, 1, 97-111, (2011) Singularities in algebraic geometry, Complex surface and hypersurface singularities, Coadjoint orbits; nilpotent varieties, Deformation of singularities Simple singularities and simple 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 In his book ``Vorlesungen über algebraische Geometrie'' (Leipzig 1921; F 48.068701), \textit{F. Severi} has asserted with an incomplete proof that the subscheme \({\mathcal I}_{d, g,r}'\) which is the union of the irreducible components of the Hilbert scheme \({\mathcal H}_{d, g,r}\) whose general points correspond to smooth, irreducible, and nondegenerate curves of degree \(d\) and genus \(g\) in \(\mathbb{P}^r\) is irreducible if \(d\geq g+r\). Also \textit{J. Harris} [``Curves in projective space'', Sem. Math. Sup. 85 (Montreal 1982; Zbl 0511.14014)] has conjectured that \({\mathcal I}_{d,g,r}\) is irreducible if the Brill-Noether number \(\rho (d,g, r): =g-(r+1) (g-d+r)\) is positive.
In this paper we demonstrate various reducible examples of the subscheme \({\mathcal I}_{d,g,r}'\) with positive Brill-Noether number. Indeed an example of a reducible \({\mathcal I}_{d,g,r}'\) with positive \(\rho (d,g,r)\), namely the example \({\mathcal I}_{2g-8,g,g-8}'\) (or other variations of it), has been known to some people (including the author), but it seems to have first appeared in the literature in a paper by \textit{D. Eisenbud} and \textit{J. Harris} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No. 1, 33-53 (1989; Zbl 0691.14006)]. The purpose of the paper under review is to add a wider class of examples to the list of such reducible examples by using general \(k\)-gonal curves. We also show that \({\mathcal I}_{d,g,r}'\) is irreducible for the range of \(d\geq 2g-7\) and \(g-d+r \leq 0\). Throughout we will be working over the field of complex numbers. reducible Hilbert scheme; F 48.068701; irreducible components of the Hilbert scheme; positive Brill-Noether number Keem, C, Reducible Hilbert scheme of smooth curves with positive brill-Noether number, Proc. Amer. Math. Soc., 122, 349-354, (1994) Parametrization (Chow and Hilbert schemes), Curves in algebraic geometry Reducible Hilbert scheme of smooth curves with positive 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 For an arbitrary scheme \(W\), the \(m\)-th jet scheme \(W_m\) parametrizes morphisms \(\text{Spec} \mathbb{C} [t]/(t^{m+1})\to W\). Main result:
If \(X\) is a smooth variety, \(Y\subset X\) a closed subscheme, and \(q>0\) a rational number, then:
(1) The pair \((X,q\cdot Y)\) is log canonical if and only if \(\dim Y_m\leq (m+1) (\dim X-q)\), for all \(m\).
(2) The pair \((X,q\cdot Y)\) is Kawamata log terminal if and only if \(\dim Y_m< (m+1)(\dim X-q)\), for all \(m\).
The main technique we use in the proof of this result is motivic integration, a technique due to Kontsevich, Batyrev, and Denef and Loeser. As a consequence of the above result, we obtain a formula for the log canonical threshold:
Corollary. If \(X\) is a smooth variety and \(Y\subset X\) is a closed subscheme, then the log canonical threshold of the pair \((X,Y)\) is given by \(c(X,Y)= \dim X-\sup_{m\geq 0} {\dim Y_m\over m+1}\).
We apply this corollary to give simpler proofs of some results on the log canonical threshold proved by \textit{J.-P. Demailly} and \textit{J. Kollár} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 34, No. 4, 525-556 (2001; Zbl 0994.32021)] using analytic techniques. jet schemes; log canonical threshold; motivic integration; Kawamata log terminal M. Mustaţǎ, Singularities of pairs via jet schemes, \textit{J. Amer. Math. Soc.} 15 no. 3 (2002) 599-615 (electronic). Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Minimal model program (Mori theory, extremal rays) Singularities of pairs via jet 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 surface (over \({\mathbb{C}})\) and \(S^{[n]}\) the component of the Hilbert scheme of S which parametrizes subschemes of finite length n on S. Recently \textit{G. Ellingsrud} and \textit{S. A. Strømme} have computed the Betti numbers of \(S^{[n]}\) for \(S={\mathbb{P}}_ 2\) [Invent. Math. 87, 343-352 (1987; Zbl 0625.14002)].
In this paper the Betti numbers of \(S^{[n]}\) are computed for an arbitrary smooth projective surface S. For \(X=S\) or \(X=S^{[n]}\) let \(b_ i(X)\) be the i-th Betti number and \(p(X,z)=\sum_{i}b_ i(X)z^ i \) the Poincaré polynomial. Then the main result is:
\[
\sum^{\infty}_{n=0}p(S^{[n]},z)t^ n=\prod^{\infty}_{k=1}\frac{(1+z^{2k-1}t^ k)^{b_ 1(S)}(1+z^{2k+1}t^ k)^{b_ 1(S)}}{(1-z^{2k-2}t^ k)^{b_ 0(S)}(1-z^{2k}t^ k)^{b_ 2(S)}(1-z^{2k+2}t^ k)^{b_ 0(S)}}.
\]
The proof uses reduction modulo q and the Weil conjectures.
We also make some remarks about the Hodge numbers. Hilbert scheme; Betti numbers; Poincaré polynomial; Weil conjectures; Hodge numbers L. G''ottsche. The Betti numbers of the Hilbert scheme of points on a smooth projective surface. Math. Ann., 286(1-3):193--207, 1990. Topological properties in algebraic geometry, Parametrization (Chow and Hilbert schemes) The Betti numbers of the Hilbert scheme of points on a smooth projective surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors obtain an explicit description of the irreducible component of the Hilbert scheme parametrizing complete intersections of two quadrics in \(\mathbb{P}^n\) for \(n\geq 3\), as a suitable double blow up of the Grassmann variety of pencils of quadrics in \(\mathbb{P}^n\). As an application they compute the number 52.832.040 of elliptic quartics intersecting 16 lines and the number 47.867.287.590.090 of Del Pezzo surfaces intersecting 26 lines. Hilbert scheme; complete intersections; Grassmann variety of pencils of quadrics Avritzer, D., Vainsencher, I.: The Hilbert Scheme component of the intersection of 2 quadrics, Comm. Alg., 27 (1999), 2995--3008 Parametrization (Chow and Hilbert schemes), Complete intersections, Families, moduli of curves (algebraic), Pencils, nets, webs in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry The Hilbert scheme component of the intersection of two quadrics | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 proved the connectedness of the real locus of several families of space curves with assigned cohomology. As explained by the author, the main idea is to show that, under some assumptions, if a certain property is known to be true for a nonempty Zariski open subset of an integral variety \(V\), then it is also true for all points of \(V\) outside finitely many subvarieties of codimension \(\geq t\geq 2\). connectedness of the real locus; families of space curves Plane and space curves, Topology of real algebraic varieties, Parametrization (Chow and Hilbert schemes) On the real part of the Hilbert scheme of space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given an algorithm for resolution of singularities that satisfies certain conditions (``a good algorithm''), natural notions of simultaneous algorithmic resolution, and of equi-resolution, for families of embedded schemes (parametrized by a reduced scheme \(T\)) are defined. It is proved that these notions are equivalent. Something similar is done for families of sheaves of ideals, where the goal is algorithmic simultaneous principalization. A consequence is that given a family of embedded schemes over a reduced \(T\), this parameter scheme can be naturally expressed as a disjoint union of locally closed sets \(T_j\), such that the induced family on each part \(T_j\) is equi-resolvable. In particular, this can be applied to the Hilbert scheme of a smooth projective variety; in fact, our result shows that, in characteristic zero, the underlying topological space of any Hilbert scheme parametrizing embedded schemes can be naturally stratified in equi-resolvable families. resolution of singularities; algorithmic resolution; simultaneous resolution; Hilbert schemes Encinas, S., Nobile, A. and Villamayor, O.: On algorithmic equi-resolution and stratification of Hilbert schemes. Proc. London Math. Soc. 86 (2003), no. 3, 607-648. Global theory and resolution of singularities (algebro-geometric aspects) On algorithmic equi-resolution and stratification of Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(C\) be a locally planar curve. Its versal deformation admits a stratification by the genera of the fibres. The strata are singular; we show that their multiplicities at the central point are determined by the Euler numbers of the Hilbert schemes of points on \(C\). These Euler numbers have made two prior appearances. First, in certain simple cases, they control the contribution of \(C\) to the Pandharipande-Thomas curve counting invariants of three-folds. In this context, our result identifies the strata multiplicities as the local contributions to the Gopakumar-Vafa BPS invariants. Second, when \(C\) is smooth away from a unique singular point, a conjecture of Oblomkov and the present author identifies the Euler numbers of the Hilbert schemes with the `\(U(\infty )\)' invariant of the link of the singularity. We make contact with combinatorial ideas of Jaeger, and suggest an approach to the conjecture. V. Shende, Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation , to appear in Compos. Math., preprint, [math.AG] 1009.0914v2 Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 the notion of a quaternionic \(r\)-Kronecker structure of rank \(k\) on a manifold \(M\). This is a certain type of bundle map \(\alpha:E\otimes \mathbb{C}^r\to T^{\mathbb{C}}M\), where \(E\) is a quaternionic vector bundle of rank \(k\). In the case where \(\alpha\) is an isomorphism and \(r=2\), this reduces to an almost hypercomplex structure. They also introduce a complex analog, as well as twistor spaces of (integrable and regular) Kronecker structures. These are complex manifolds admitting a natural holomorphic submersion to \(\mathbb{C}\mathrm{P}^{r-1}\) from which the original manifold may be recovered as a suitable space of sections, analogous to the so-called twistor lines in the hypercomplex case. Conversely, it is proven that for any complex manifold \(Z\) with a surjective holomorphic submersion \(\pi:Z\to \mathbb{C}\mathrm{P}^{r-1}\), certain spaces of sections carry natural \(r\)-Kronecker structures.
After discussing connections to hypercomplex geometry, the authors apply these concepts to certain open subsets \(M_{d,g}\) of the Hilbert scheme of curves of fixed genus \(g\) and degree \(d\) in \(\mathbb{C}\mathrm{P}^3\), showing that the subspace \(M_{d,g}^\sigma\) of real curves admits a quaternionic \(4\)-Kronecker structure of rank \(2d\) (though it may be empty for some values of \(d,g\)). They also consider curves in \(\mathbb{C}\mathrm{P}^n\) for \(n>3\), where the assumptions needed to ensure smoothness impose stronger constraints on \(d\) and \(g\). In the final section, the authors return to the case \(n=3\), and study \(M_{1,0}^\sigma=S^4\) in more detail. projective curves; quaternionic structures; twistor methods Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Twistor methods in differential geometry, Parametrization (Chow and Hilbert schemes), Plane and space curves Differential geometry of Hilbert schemes of curves in a projective space | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities One triumph of the functorial approach in algebraic geometry is \textit{Grothendieck's} construction of the Hilbert scheme \(\mathbf{Hilb}_{p(t)}^n\), which parametrizes closed subschemes \(X \subset \mathbb P^n_k\) having Hilbert polynomial \(p(t)\) [\textit{A. Grothendieck}, in: Sem. Bourbaki 13(1960/61), No.221, 28 p. (1961; Zbl 0236.14003)]. \textit{G. Gotzmann} found the minimal value of \(r\) for which \({\mathcal O}_X\) is \(r\)-regular in terms of \(p(t)\) [Math. Z. 158, 61--70 (1978; Zbl 0352.13009)], which allows one to embed the Hilbert scheme into the Grassmann variety \({\mathbb G}^{N(r)}_{p(r)}\) (here \(N(r)=\binom{n+r}{n}\)) by associating each closed subscheme \(X\) to the vector space quotient \(H^0(\mathbb P^n, {\mathcal O} (r)) \to H^0(X, {\mathcal O}_X (r))\). \textit{A. Iarrobino} and \textit{V. Kanev} produced explicit equations of degree \(N(r+1)-p(r+1)-1\) [Power sums, Gorenstein algebras, and determinantal loci. With an appendix `The Gotzmann theorems and the Hilbert scheme' by Anthony Iarrobino and Steven L. Kleiman. Berlin: Springer (1999; Zbl 0942.14026)]. More recently \textit{M. Haiman and B. Sturmfels} [J. Algebr. Geom. 13, 725--769 (2004; Zbl 0172.14007)] produced equations of degree \(n+1\) conjectured by Bayer in his thesis.
In the paper under review, the authors exploit the action of \(\text{PGL}(n+1)\) to find significantly lower degree equations when \(\text{char} \; k=0\). They use the group action and monomials corresponding to a certain Borel-fixed ideal to define the \textit{Borel open cover} for the Grassmann functor and a corresponding Borel cover for the Hilbert functor. Using the theory of \(J\)-marked sets developed by \textit{P. Lella} and \textit{M. Roggero} [``On the functoriality of marked families'', Preprint, \url{arXiv:1307.7657}], they reprove the existence of the Hilbert scheme and derive equations for the universal family in \(\mathbb P^n \times {\mathbb G}^{N(r)}_{p(r)}\) of bi-degree \((r,1)\) by exploiting properties of extensors. Consequently they find explicit equations for the Hilbert scheme in the ring of Plücker coordinates of degree at most \(\dim X + 2\). They illustrate their method by describing their equations for Hilbert schemes of two points in \(\mathbb P^n\) with \(n=2,3,4\) (each is generated by quadrics in their respective Grassmann variety), including an interesting table comparing to the equations of Iarrobino, Kleiman, Haiman and Sturmfels [loc. cit.]. Hilbert schemes; Plücker coordinates; Borel fixed ideals; extensors Brachat, J., Lella, P., Mourrain, B., Roggero, M.: Extensors and the Hilbert scheme. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (2015). 10.2422/2036-2145.201407_003 Parametrization (Chow and Hilbert schemes), Exterior algebra, Grassmann algebras Extensors 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 \textit{A. N. Tyurin} considered in [``Quantization, classical and quantum field theory and theta functions''. CRM Monograph Series 21 (2003; Zbl 1083.14037)] a special case of topologically trivial sheaves on curves with rational irreducible components, and a trivalent dual graph.
The authors gives a construction of the moduli space \({\mathcal M}_0(X,r)\) of topologically trivial bundles of rank \(r\geq 1\) on a curve \(X\), and its compactification \(\overline{\mathcal M}_0(X,r)\), the moduli space of (semi)stable topologically trivial bundles of rank \(r\geq 1\).
The cases \(r=1\) of the generalized Jacobian of \(X\) and \(\overline{\mathcal M}_0(X,2,0)\) of rank two bundles with trivial determinant are considered in detail. Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles Topologically trivial sheaves on curves with simplest 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 biregular geometry of punctual Hilbert schemes in dimensions 2 and 3, i.e. of schemes parametrizing fixed-length zero-dimensionaI subschemes supported at a given point on a smooth surface or a smooth three-dimensional variety, is studied. A precise biregular description of these schemes has only been known for the trivial cases of lengths 3 and 4 in dimension 2. The next case of length 5 in dimension 2 and the two first nontrivial cases of lengths 3 and 4 in dimension 3 are considered. A detailed description of the biregular properties of punctual Hilbert schemes and of their natural desingularizations by varieties of complete punctual flags is given. Stein expansion; Briançon classification; punctual Hilbert schemes Parametrization (Chow and Hilbert schemes) Punctual Hilbert schemes of small length in dimensions 2 and 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 Hilbert scheme \(X^{[n]}\) of points in a smooth projective surface \(X\) is a desingularization of the \(n\)-th symmetric product of \(X\). An element \(\xi \in X^{[n]}\) is a length-\(n\) \(0\)-dimensional closed subscheme of \(X\).
In the paper under review the authors study curves contained in the Hilbert scheme \(X^{[n]}\) when \(X\) is a simply-connected smooth projective surface for the case \(n \geq 2\). For a fixed point \(x \in X\), let \(M_2(x)=\{\xi\in X^{[2]} \mid {\text{supp}}(\xi)=\{x\}\}\). It is known that \(M_2(x)\cong \mathbb{P}^1\). Let
\[
\beta_n =\{\xi+x_1+\ldots+x_{n-2}\in X^{[n]} \mid \xi\in M_2(x)\} \tag{1}
\]
for fixed distinct points \(x, x_1,\ldots,x_{n-2}\). Using well-known results about Hilbert schemes, the authors characterize all the curves \(\gamma\) in the Hilbert scheme \(X^{[n]}\) homologous to \(\beta_n\). It turns out that the moduli space \(\mathcal{M}(\beta_n)\) of all these curves has dimension \((2n-2)\) and its top stratum consists of all the curves \(\gamma\) of the form (1). Moreover they determine the normal bundles of the curves \(\gamma\) of the form (1). Now let \(X=\mathbb{P}^2\). For a fixed line \(\ell \subset X\) and distinct points \(x_1, \ldots, x_{n-1} \in X\) such that \(x_i \notin \ell\), where \(i=1,\ldots, (n-1)\), let \(\beta_{\ell}\) be the curve in \(X^{[n]}\) of the form
\[
\beta_{\ell} =\{x+x_1+\ldots+x_{n-1}\in X^{[n-1]} \mid x \in \ell\} \tag{2}
\]
The authors prove that the effective cone of the Hilbert scheme \(X^{[n]}\) is spanned by \(\beta_n\) and \(\beta_{\ell}-(n-1)\beta_n\), and determine the nef cone of \(X^{[n]}\). In addition, they show that a curve \(\gamma\) in \(X^{[n]}\) is homologous to \(\beta_{\ell}-(n-1)\beta_n\) if and only if there exists a line \(C\) in \(X=\mathbb{P}^2\) such that \(\gamma\) is a line in \(C^{[n]}\subset X^{[n]}\) (Theorem 5.1). So all these curves \(\gamma\) are parametrized by the Grassmannian bundle over the dual space \((\mathbb{P}^2)^{*}\).
Finally, the authors remark that results of the paper may be used to compute the \(1\)-point Gromov-Witten invariants in the Hilbert scheme \(X^{[n]}\), and to study the quantum cohomology of \(X^{[n]}\). simply-connected smooth projective surface; algebraic curve; normal bundle; homology group Li W.-P., Qin Z.B., Zhang Q.: Curves in the Hilbert schemes of points on surfaces. Contemp. Math. 322, 89--96 (2003) Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Curves in the 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 There are several classical theorems concerning the Hilbert scheme of Grothendieck which parametrize the ideals of a graded polynomial ring with a given Hilbert polynomial. Among these are: Macaulay's theorem, which characterize all possible Hilbert functions; Hartshorne's theorem that the Hilbert scheme is connected; and finally the result of Bigatti, Hulett, and Pardue that among all ideals with a given Hilbert function the Lexicographic ideal attains the maximal Betti numbers.
In this paper the authors prove each of these facts over a different ring, namely the coordinate ring of a Veronese embedding of projective space. By consequence the authors provide numerical versions of Macaulay's theorem as well as Gotzmann's persistence theorem over Veronese rings. As another consequence they also show that the graded Betti numbers of any ideal in a Veronese ring can be obtained by consecutive cancellations (reducing the Betti numbers in the same column and consecutive rows each by one). The paper contains numerous useful examples and as well is an excellent source of background information on Hilbert schemes, Hilbert functions, and Betti numbers. Hilbert schemes; Hilbert functions; Veronese embeddings; Macaulay's Theorem; Gotzmann's persistence; maximal Betti numbers; consecutive cancellations Gasharov, V; Murai, S; Peeva, I, Hilbert schemes and maximal Betti numbers over Veronese rings, Math. Z., 267, 155-172, (2011) Parametrization (Chow and Hilbert schemes), Syzygies, resolutions, complexes and commutative rings, Graded rings Hilbert schemes and maximal Betti numbers over Veronese rings | 0 |
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