Betti numbers for the hilbert function strata of the punctual hilbert scheme in two variables

We consider the question of irreducibility of the Hilbert scheme of points ℋilbdℙn and its Gorenstein locus. This locus is known to be reducible for d ≥ 14. For d ≤ 11 the irreducibility of this locus was proved in the series of papers [G. Casnati and R. Notari, On the Gorenstein locus of some punctual Hilbert schemes, J. Pure Appl. Algebra 213(11) (2009) 2055–2074; On the irreducibility and the singularities of Gorenstein locus of the Punctual Hilbert scheme of degree 10, J. Pure Appl. Algebra 215(6) (2011) 1243–1254; Irreducibility of the Gorenstein locus of the Punctual Hilbert Scheme of degree 11, preprint (2012)] and Iarrobino conjectured that the irreducibility holds for d ≤ 13. In this paper, we prove that the subschemes corresponding to the Gorenstein algebras having Hilbert function (1, 5, 5, 1) are smoothable, i.e. lie in the closure of the locus of smooth subschemes. This result completes the proof of irreducibility of the Gorenstein locus of ℋilb12ℙn, see Theorem 2.

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Hilbert schemes and Betti numbers over Clements–Lindström rings

Compositio Mathematica ◽

10.1112/s0010437x1100741x ◽

2012 ◽

Vol 148 (5) ◽

pp. 1337-1364 ◽

Cited By ~ 1

Author(s):

Satoshi Murai ◽

Irena Peeva

Keyword(s):

Hilbert Scheme ◽

Hilbert Function ◽

Betti Numbers ◽

Hilbert Schemes

AbstractWe show that the Hilbert scheme, that parameterizes all ideals with the same Hilbert function over a Clements–Lindström ring W, is connected. More precisely, we prove that every graded ideal is connected by a sequence of deformations to the lex-plus-powers ideal with the same Hilbert function. This is an analogue of Hartshorne’s theorem that Grothendieck’s Hilbert scheme is connected. We also prove a conjecture by Gasharov, Hibi, and Peeva that the lex ideal attains maximal Betti numbers among all graded ideals in W with a fixed Hilbert function.

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Bundles on with vanishing lower cohomologies

Canadian Journal of Mathematics ◽

10.4153/s0008414x20000292 ◽

2020 ◽

pp. 1-20

Author(s):

Mengyuan Zhang

Keyword(s):

Hilbert Function ◽

Betti Numbers ◽

Projective Spaces ◽

Hilbert Functions ◽

Free Resolutions ◽

Rational Varieties ◽

Minimal Free Resolutions ◽

Graded Lattice

Abstract We study bundles on projective spaces that have vanishing lower cohomologies using their short minimal free resolutions. We partition the moduli $\mathcal{M}$ according to the Hilbert function H and classify all possible Hilbert functions H of such bundles. For each H, we describe a stratification of $\mathcal{M}_H$ by quotients of rational varieties. We show that the closed strata form a graded lattice given by the Betti numbers.

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