Determinant line bundles on Moduli spaces of pure sheaves on rational surfaces and Strange Duality

TL;DR

This paper investigates determinant line bundles on moduli spaces of pure sheaves on rational surfaces, computes generating functions for their sections, and provides evidence for the Strange Duality conjecture in specific cases.

Contribution

It explicitly computes generating functions for sections of determinant line bundles on moduli spaces of pure sheaves, supporting the Strange Duality conjecture in certain rational surface cases.

Findings

Computed $Z^r(t)$ for $g_L leq 0$ on specific surfaces.

Verified predictions of Strange Duality in chosen cases.

Connected results to the theory of compactified Jacobians.

Abstract

Let $\mhu$ be the moduli space of semi-stable pure sheaves of class $u$ on a smooth complex projective surface $X$. We specify $u=(0,L,\chi(u)=0),$ i.e. sheaves in $u$ are of dimension $1$. There is a natural morphism $\pi$ from the moduli space $\mhu$ to the linear system $\ls$. We study a series of determinant line bundles $\lcn$ on $\mhu$ via $\pi.$ Denote $g_L$ the arithmetic genus of curves in $\ls.$ For any $X$ and $g_L\leq0$, we compute the generating function $Z^r(t)=\sum_{n}h^0(\mhu,\lcn)t^n$. For $X$ being $\mathbb{P}^2$ or $\mathbb{P}(\mo_{\pone}\oplus\mo_{\pone}(-e))$ with $e=0,1$, we compute $Z^1(t)$ for $g_L>0$ and $Z^r(t)$ for all $r$ and $g_L=1,2$. Our results provide a numerical check to Strange Duality in these specified situations, together with G\"ottsche's computation. And in addition, we get an interesting corollary in the theory of compactified Jacobian of integral curves.