Universal Series for Hilbert Schemes and Strange Duality

Abstract

We show how the "finite Quot scheme method" applied to Le Potier's strange duality on del Pezzo surfaces leads to conjectures (valid for all smooth complex projective surfaces) relating two sets of universal power series on Hilbert schemes of points on surfaces: those for top chern classes of tautological sheaves, and those for Euler characteristics of line bundles. We have verified these predictions computationally for low order. We then give an analysis of these conjectures in small ranks. We also give a combinatorial proof of a formula predicted by our conjectures: the top chern class of the tautological sheaf on $S^{[n]}$ associated to the structure sheaf of a point is equal to $(-1{)}^n$ times the $n$th Catalan number.

The introduction and exposition have been restructured and (hopefully) improved

References in corpus (1)

• The rank-level duality for non-abelian theta functions

Cited by in corpus (10)

• The virtual K-theory of Quot schemes of surfaces
• Virtual Segre and Verlinde numbers of projective surfaces
• K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface
• Sheaves on surfaces and virtual invariants
• Rank-one sheaves and stable pairs on surfaces
• Blowup formulas for Segre and Verlinde numbers of surfaces and higher rank Donaldson invariants
• The geometry of Hilbert schemes of two points on projective space
• Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi-Yau 4-folds
• Equivariant Segre and Verlinde invariants for Quot schemes
• Representations in Strange Duality: Hilbert Schemes paired with higher rank spaces