On the Gorenstein locus of the punctual Hilbert scheme of degree 11. (English) Zbl 1287.13013
The Hilbert scheme \({\text{Hilb}}_d (\mathbb P_k^n)\) parametrizing length \(d\) closed subschemes \(X \subset \mathbb P_k^n\) is irreducible for small values of \(d\) (because it is the closure of the open locus \({\mathcal R}\) corresponding to \(d\) distinct points), but for \(d \gg 0\) it was shown to be reducible by A. Iarrobino [Invent. Math. 15, 72–77 (1972; Zbl 0227.14006)]. The first explicit example \({\text{Hilb}}_8 (\mathbb P_k^4)\) was given by A. Iarrobino and J. Emsalem [Compos. Math. 36, 145–188 (1978; Zbl 0393.14002)]; more recently Cartwright, Erman, Velasco and Viray showed that \({\text{Hilb}}_8 (\mathbb P_k^n)\) has exactly two irreducible components for \(n \geq 4\) [D. A. Cartwright et al., Algebra Number Theory 3, No. 7, 763–795 (2009; Zbl 1187.14005)].
Naturally the open Gorenstein locus \({\text{Hilb}}^G_d (\mathbb P_k^n) \subset {\text{Hilb}}_d (\mathbb P_k^n)\) also contains \(\mathcal R\) and is irreducible for small \(d\), but Iarrobino and V. Kanev showed \({\text{Hilb}}^G_{14} (\mathbb P_k^6)\) is reducible and conjectured irreducibility for \(d < 14\) [A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci. With an appendix ‘The Gotzmann theorems and the Hilbert scheme’ by Anthony Iarrobino and Steven L. Kleiman. Lecture Notes in Mathematics. 1721. Berlin: Springer. (1999; Zbl 0942.14026)]. The authors of the paper under review earlier showed that \({\text{Hilb}}^G_{d} (\mathbb P_k^n)\) is irreducible for \(n \leq 3\) or \(d \leq 10\) [J. Pure Appl. Algebra 213, No. 11, 2055–2074 (2009; Zbl 1169.14003) and J. Pure Appl. Algebra 215, No. 6, 1243–1254 (2011; Zbl 1215.14009)] and here they extend the result to \(d=11\).
The problem reduces to studying zero dimensional local Gorenstein \(k\)-algebras \(A\) of length \(d = 11\), which via Macaulay’s theory of inverse systems can be written \(k[[x_1, \dots x_n]]/\text{Ann}(F)\) for suitable \(F \in k[y_1, \dots y_n]\) via the action of \(k[[x_1, \dots x_n]]\) on \(k[y_1, \dots y_n]\) given by \(x_i = \partial/\partial y_i\). Using a filtration of \(\mathrm{gr}(A)\) appearing in work of T. Iarrobino [Mem. Am. Math. Soc. 514, 115 p. (1994; Zbl 0793.13010)], the authors write the equation for \(F\) in a special form with an integer invariant \(f_3\). In general they show that if \(f_3 \leq 3\), then the scheme corresponding to the algebra \(A\) is in \({\overline {\mathcal R}}\) and \(f_3 > 3\) leads to particular Hilbert functions (especially \(H_A = (1,4,4,1,1)\)) which they handle with special arguments. They also give an almost complete description of the singular locus of \({\text{Hilb}}^G_{11} (\mathbb P_k^n)\).
Naturally the open Gorenstein locus \({\text{Hilb}}^G_d (\mathbb P_k^n) \subset {\text{Hilb}}_d (\mathbb P_k^n)\) also contains \(\mathcal R\) and is irreducible for small \(d\), but Iarrobino and V. Kanev showed \({\text{Hilb}}^G_{14} (\mathbb P_k^6)\) is reducible and conjectured irreducibility for \(d < 14\) [A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci. With an appendix ‘The Gotzmann theorems and the Hilbert scheme’ by Anthony Iarrobino and Steven L. Kleiman. Lecture Notes in Mathematics. 1721. Berlin: Springer. (1999; Zbl 0942.14026)]. The authors of the paper under review earlier showed that \({\text{Hilb}}^G_{d} (\mathbb P_k^n)\) is irreducible for \(n \leq 3\) or \(d \leq 10\) [J. Pure Appl. Algebra 213, No. 11, 2055–2074 (2009; Zbl 1169.14003) and J. Pure Appl. Algebra 215, No. 6, 1243–1254 (2011; Zbl 1215.14009)] and here they extend the result to \(d=11\).
The problem reduces to studying zero dimensional local Gorenstein \(k\)-algebras \(A\) of length \(d = 11\), which via Macaulay’s theory of inverse systems can be written \(k[[x_1, \dots x_n]]/\text{Ann}(F)\) for suitable \(F \in k[y_1, \dots y_n]\) via the action of \(k[[x_1, \dots x_n]]\) on \(k[y_1, \dots y_n]\) given by \(x_i = \partial/\partial y_i\). Using a filtration of \(\mathrm{gr}(A)\) appearing in work of T. Iarrobino [Mem. Am. Math. Soc. 514, 115 p. (1994; Zbl 0793.13010)], the authors write the equation for \(F\) in a special form with an integer invariant \(f_3\). In general they show that if \(f_3 \leq 3\), then the scheme corresponding to the algebra \(A\) is in \({\overline {\mathcal R}}\) and \(f_3 > 3\) leads to particular Hilbert functions (especially \(H_A = (1,4,4,1,1)\)) which they handle with special arguments. They also give an almost complete description of the singular locus of \({\text{Hilb}}^G_{11} (\mathbb P_k^n)\).
Reviewer: Scott Nollet (Fort Worth)
MSC:
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 14C05 | Parametrization (Chow and Hilbert schemes) |
| 14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |
Citations:
Zbl 0227.14006; Zbl 0393.14002; Zbl 1187.14005; Zbl 0942.14026; Zbl 1169.14003; Zbl 1215.14009; Zbl 0793.13010Software:
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