Estimates of sections of determinant line bundles on Moduli spaces of pure sheaves on algebraic surfaces

TL;DR

This paper provides bounds on the sections of determinant line bundles over moduli spaces of pure sheaves on algebraic surfaces, contributing to the understanding of strange duality in algebraic geometry.

Contribution

It introduces new upper bounds for sections of determinant line bundles on moduli spaces of pure sheaves, aiding in the verification of strange duality for rational surfaces.

Findings

Upper bounds for sections when genus g_L ≤ 2

Results support cases of strange duality conjecture

Extends G"ottsche's computations to new settings

Abstract

Let $X$ be any smooth simply connected projective surface. We consider some moduli space of pure sheaves of dimension one on $X$, i.e. $\mhu$ with $u=(0,L,\chi(u)=0)$ and $L$ an effective line bundle on $X$, together with a series of determinant line bundles associated to $r[\mo_X]-n[\mo_{pt}]$ in Grothendieck group of $X$. Let $g_L$ denote the arithmetic genus of curves in the linear system $\ls$. For $g_L\leq2$, we give a upper bound of the dimensions of sections of these line bundles by restricting them to a generic projective line in $\ls$. Our result gives, together with G\"ottsche's computation, a first step of a check for the strange duality for some cases for $X$ a rational surface.