247 research outputs found
The Standard Conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces
We 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.Comment: 15 pages, minor change
Borel Degenerations of Arithmetically Cohen-Macaulay curves in P^3
We investigate Borel ideals on the Hilbert scheme components of
arithmetically Cohen-Macaulay (ACM) codimension two schemes in P^n. We give a
basic necessary criterion for a Borel ideal to be on such a component. Then
considering ACM curves in P^3 on a quadric we compute in several examples all
the Borel ideals on their Hilbert scheme component. Based on this we conjecture
which Borel ideals are on such a component, and for a range of Borel ideals we
prove that they are on the component.Comment: 20 pages, shorter and more effective versio
Universal Polynomials for Tautological Integrals on Hilbert Schemes
We show that tautological integrals on Hilbert schemes of points can be
written in terms of universal polynomials in Chern numbers. The results hold in
all dimensions, though they strengthen known results even for surfaces by
allowing integrals over arbitrary "geometric" subsets (and their
Chern-Schwartz-MacPherson classes).
We apply this to enumerative questions, proving a generalised G\"ottsche
Conjecture for all singularity types and in all dimensions. So if L is a
sufficiently ample line bundle on a smooth variety X, in a general subsystem
P^d of |L| of appropriate dimension the number of hypersurfaces with given
singularity types is a polynomial in the Chern numbers of (X,L). When X is a
surface, we get similar results for the locus of curves with fixed "BPS
spectrum" in the sense of stable pairs theory.Comment: 44 pages, minor changes and correction
- …