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

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].

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

Journal of Algebra and Its Applications ◽

10.1142/s021949881450056x ◽

2014 ◽

Vol 13 (08) ◽

pp. 1450056 ◽

Cited By ~ 3

Author(s):

Joachim Jelisiejew

Keyword(s):

Hilbert Scheme ◽

Hilbert Function ◽

Hilbert Schemes ◽

Gorenstein Algebras ◽

Punctual Hilbert Scheme ◽

Finite Dimensional ◽

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.

MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

Modern Physics Letters A ◽

10.1142/s0217732398002904 ◽

1998 ◽

Vol 13 (34) ◽

pp. 2731-2742 ◽

Cited By ~ 1

Author(s):

YUTAKA MATSUO

Keyword(s):

Hilbert Scheme ◽

Matrix Theory ◽

Fock Space ◽

Homology Class ◽

Orthogonal Basis ◽

Inner Product ◽

Virasoro Symmetry ◽

The Matrix ◽

Boson Fock Space

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relates the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These bases can be related to the eigenfunctions of Calogero–Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.

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

Forum of Mathematics Sigma ◽

10.1017/fms.2020.35 ◽

2020 ◽

Vol 8 ◽

Author(s):

Burt Totaro

Keyword(s):

Natural Number ◽

Hilbert Scheme ◽

Obstruction Theory ◽

Projective Surface ◽

Hilbert Schemes ◽

Complex Projective Surface ◽

Smooth Complex ◽

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.

On a cell decomposition of the Hilbert scheme of points in the plane

Inventiones mathematicae ◽

10.1007/bf01389372 ◽

1988 ◽

Vol 91 (2) ◽

pp. 365-370 ◽

Cited By ~ 14

Author(s):

Geir Ellingsrud ◽

Stein Arild Str�mme

Keyword(s):

Hilbert Scheme ◽

Cell Decomposition ◽

A Cell

Pathologies on the Hilbert scheme of points

Inventiones mathematicae ◽

10.1007/s00222-019-00939-5 ◽

2019 ◽

Vol 220 (2) ◽

pp. 581-610 ◽

Cited By ~ 2

Author(s):

Joachim Jelisiejew

Keyword(s):

Hilbert Scheme ◽

On the derived category of the Hilbert scheme of points on an Enriques surface

Selecta Mathematica ◽

10.1007/s00029-015-0178-x ◽

2015 ◽

Vol 21 (4) ◽

pp. 1339-1360 ◽

Cited By ~ 5

Author(s):

Andreas Krug ◽

Pawel Sosna

Keyword(s):

Hilbert Scheme ◽

Derived Category ◽

Enriques Surface ◽

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

Compositio Mathematica ◽

10.1112/s0010437x12000607 ◽

2013 ◽

Vol 149 (3) ◽

pp. 481-494 ◽

Cited By ~ 21

Author(s):

François Charles ◽

Eyal Markman

Keyword(s):

Hilbert Scheme ◽

K3 Surfaces ◽

Cohomology Algebra ◽

Hilbert Schemes ◽

Projective Varieties ◽

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.