scholarly journals Local finite-dimensional Gorenstein k-algebras having Hilbert function (1,5,5,1) are smoothable

2014 ◽  
Vol 13 (08) ◽  
pp. 1450056 ◽  
Author(s):  
Joachim Jelisiejew
Keyword(s):  
Hilbert Scheme ◽  
Hilbert Function ◽  
Hilbert Schemes ◽  
Gorenstein Algebras ◽  
Punctual Hilbert Scheme ◽  
Finite Dimensional ◽  
Hilbert Scheme Of Points

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.

scholarly journals The integral cohomology of the Hilbert scheme of points on a surface

2020 ◽  
Vol 8 ◽  
Author(s):  
Burt Totaro
Keyword(s):  
Natural Number ◽  
Hilbert Scheme ◽  
Obstruction Theory ◽  
Projective Surface ◽  
Hilbert Schemes ◽  
Complex Projective Surface ◽  
Smooth Complex ◽  
Hilbert Scheme Of Points ◽  
Complex Projective ◽  
Torsion Free

Abstract We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme $X^{[n]}$ has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces.

scholarly journals The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces

2013 ◽  
Vol 149 (3) ◽  
pp. 481-494 ◽  
Author(s):  
François Charles ◽  
Eyal Markman
Keyword(s):  
Hilbert Scheme ◽  
K3 Surfaces ◽  
Cohomology Algebra ◽  
Hilbert Schemes ◽  
Projective Varieties ◽  
Hilbert Scheme Of Points ◽  
Complex Projective ◽  
Symplectic Varieties

AbstractWe prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky’s theory of hyperholomorphic sheaves and a study of the cohomology algebra of Hilbert schemes of K3 surfaces.

scholarly journals Stability of some vector bundles on Hilbert schemes of points on K3 surfaces

2021 ◽  
Author(s):  
Fabian Reede ◽  
Ziyu Zhang
Keyword(s):  
Moduli Space ◽  
Hilbert Scheme ◽  
Vector Bundles ◽  
K3 Surfaces ◽  
Hilbert Schemes ◽  
Connected Component ◽  
Stable Vector Bundles ◽  
Universal Family ◽  
Hilbert Scheme Of Points ◽  
Flat Family

AbstractLet X be a projective K3 surfaces. In two examples where there exists a fine moduli space M of stable vector bundles on X, isomorphic to a Hilbert scheme of points, we prove that the universal family $${\mathcal {E}}$$ E on $$X\times M$$ X × M can be understood as a complete flat family of stable vector bundles on M parametrized by X, which identifies X with a smooth connected component of some moduli space of stable sheaves on M.

scholarly journals Generalized punctual Hilbert schemes and 𝔤-complex structures

2021 ◽  
Author(s):  
Alexander Thomas
Keyword(s):  
Lie Algebras ◽  
Moduli Spaces ◽  
Hilbert Scheme ◽  
Complex Structure ◽  
Complex Structures ◽  
Hilbert Schemes ◽  
Simple Complex ◽  
Geometric Structures ◽  
Punctual Hilbert Scheme ◽  
Alternative Description

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.

scholarly journals Incidence stratifications on Hilbert schemes of smooth surfaces, and an application to Poisson structures

2016 ◽  
Vol 27 (01) ◽  
pp. 1650006 ◽  
Author(s):  
Ziv Ran
Keyword(s):  
Poisson Structure ◽  
Hilbert Scheme ◽  
Smooth Surface ◽  
Smooth Curve ◽  
Smooth Surfaces ◽  
Poisson Structures ◽  
Hilbert Schemes ◽  
Hilbert Scheme Of Points

Given a smooth curve on a smooth surface, the Hilbert scheme of points on the surface is stratified according to the length of the intersection with the curve. The strata are highly singular. We show that this stratification admits a natural log-resolution, namely the stratified blowup. As a consequence, the induced Poisson structure on the Hilbert scheme of a Poisson surface has unobstructed deformations.

scholarly journals Hilbert schemes and Betti numbers over Clements–Lindström rings

2012 ◽  
Vol 148 (5) ◽  
pp. 1337-1364 ◽  
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.

scholarly journals Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points

2017 ◽  
Vol 4 (1) ◽  
pp. 16-36 ◽  
Author(s):  
Roger Bielawski
Keyword(s):  
Moduli Space ◽  
Hilbert Scheme ◽  
Double Cover ◽  
Hilbert Schemes ◽  
Hilbert Scheme Of Points ◽  
Adjoint Orbits ◽  
Hyperkähler Metrics

Abstract We show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of GL(n, ℂ) is isomorphic to the deformation of the D2-singularity if n = 2, the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if n = 3, and to the Atiyah-Hitchin manifold itself if n = 4. For higher n, such slices to the sum of two orbits, each having only two distinct eigenvalues, are either empty or biholomorphic to open subsets of the Hilbert scheme of points on one of the above surfaces. In particular, these open subsets of Hilbert schemes of points carry complete hyperkähler metrics. In the case of the double cover of the Atiyah-Hitchin manifold this metric turns out to be the natural L2-metric on a hyperkähler submanifold of the monopole moduli space.

A NOTE ON THE HILBERT SCHEME OF POINTS ON A CUSP CURVE

2012 ◽  
Vol 22 (03) ◽  
pp. 1250025
Author(s):  
HWAYOUNG LEE
Keyword(s):  
Hilbert Scheme ◽  
Punctual Hilbert Scheme ◽  
Hilbert Scheme Of Points

We study the Hilbert scheme of points on a cusp curve, mostly the punctual Hilbert scheme [Formula: see text] which parameterizes m points supported at a cusp. We show that the reduced punctual scheme [Formula: see text] is isomorphic to ℙ1 and the Hilbert scheme has one singularity along [Formula: see text].

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