The Betti numbers of the Hilbert scheme of points on a smooth projective surface

1990 ◽  
Vol 286 (1-3) ◽  
pp. 193-207 ◽  
Author(s):  
Lothar Göttsche
Keyword(s):  
Hilbert Scheme ◽  
Betti Numbers ◽  
Projective Surface ◽  
Smooth Projective Surface ◽  
Hilbert Scheme Of Points

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 Stability of tautological bundles on the Hilbert scheme of two points on a surface

2014 ◽  
Vol 214 ◽  
pp. 79-94 ◽  
Author(s):  
Malte Wandel
Keyword(s):  
Vector Bundle ◽  
Hilbert Scheme ◽  
Projective Surface ◽  
Stable Vector Bundle ◽  
Free Sheaf ◽  
Smooth Projective Surface ◽  
Rank 2 ◽  
Torsion Free

AbstractLet (X, H) be a polarized smooth projective surface satisfyingH1(Χ, OΧ) = 0, and letƑbe either a rank 1 torsion-free sheaf or a rank 2μH-stable vector bundle onΧ. Assume thatc1(Ƒ) ≠ 0. This article shows that the rank 2—respectively, rank 4—tautological sheafƑ[2]associated withƑon the Hilbert squareΧ[2]isμ-stable with respect to a certain polarization.

scholarly journals Stability of tautological bundles on the Hilbert scheme of two points on a surface

2014 ◽  
Vol 214 ◽  
pp. 79-94
Author(s):  
Malte Wandel
Keyword(s):  
Vector Bundle ◽  
Hilbert Scheme ◽  
Projective Surface ◽  
Stable Vector Bundle ◽  
Free Sheaf ◽  
Smooth Projective Surface ◽  
Rank 2 ◽  
Torsion Free

AbstractLet (X, H) be a polarized smooth projective surface satisfyingH1(Χ, OΧ) = 0, and letƑbe either a rank 1 torsion-free sheaf or a rank 2μH-stable vector bundle onΧ. Assume thatc1(Ƒ) ≠ 0. This article shows that the rank 2—respectively, rank 4—tautological sheafƑ[2]associated withƑon the Hilbert squareΧ[2]isμ-stable with respect to a certain polarization.

scholarly journals The degree of the tangent and secant variety to a projective surface

2020 ◽  
Vol 20 (2) ◽  
pp. 233-248
Author(s):  
Andrea Cattaneo
Keyword(s):  
Projective Space ◽  
Hilbert Scheme ◽  
Projective Surface ◽  
Secant Variety ◽  
Smooth Projective Surface

AbstractWe present a way of computing the degree of the secant (resp. tangent) variety of a smooth projective surface, under the assumption that the divisor giving the embedding in the projective space is 3-very ample. This method exploits the link between these varieties and the Hilbert scheme 0-dimensional subschemes of length 2 of the surface.

scholarly journals LEHN’S FORMULA IN CHOW AND CONJECTURES OF BEAUVILLE AND VOISIN

2020 ◽  
pp. 1-39
Author(s):  
Davesh Maulik ◽  
Andrei Neguţ
Keyword(s):  
Hilbert Scheme ◽  
Cohomology Ring ◽  
K3 Surface ◽  
Projective Surface ◽  
Weak Version ◽  
Chow Ring ◽  
Hilbert Scheme Of Points ◽  
Chern Characters ◽  
Hyperkähler Manifold ◽  
Voisin Conjecture

The Beauville–Voisin conjecture for a hyperkähler manifold $X$ states that the subring of the Chow ring $A^{\ast }(X)$ generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of  $X$ . We prove a weak version of this conjecture when $X$ is the Hilbert scheme of points on a K3 surface for the subring generated by divisor classes and tautological classes. This in particular implies the weak splitting conjecture of Beauville for these geometries. In the process, we extend Lehn’s formula and the Li–Qin–Wang $W_{1+\infty }$ algebra action from cohomology to Chow groups for the Hilbert scheme of an arbitrary smooth projective surface  $S$ .

scholarly journals MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

1998 ◽  
Vol 13 (34) ◽  
pp. 2731-2742 ◽  
Author(s):  
YUTAKA MATSUO
Keyword(s):  
Hilbert Scheme ◽  
Matrix Theory ◽  
Fock Space ◽  
Homology Class ◽  
Orthogonal Basis ◽  
Inner Product ◽  
Virasoro Symmetry ◽  
Hilbert Scheme Of Points ◽  
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.

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

1988 ◽  
Vol 91 (2) ◽  
pp. 365-370 ◽  
Author(s):  
Geir Ellingsrud ◽  
Stein Arild Str�mme
Keyword(s):  
Hilbert Scheme ◽  
Cell Decomposition ◽  
Hilbert Scheme Of Points ◽  
A Cell

scholarly journals Pathologies on the Hilbert scheme of points

2019 ◽  
Vol 220 (2) ◽  
pp. 581-610 ◽  
Author(s):  
Joachim Jelisiejew
Keyword(s):  
Hilbert Scheme ◽  
Hilbert Scheme Of Points

scholarly journals Fibrations with moving cuspidal singularities

1991 ◽  
Vol 122 ◽  
pp. 161-179 ◽  
Author(s):  
Yoshifumi Takeda
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Surface ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Smooth Projective Surface ◽  
Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

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