Determinant Line Bundles on Moduli Spaces of Pure Sheaves on Rational Surfaces and Strange Duality

Abstract

Let $M^H_X (u)$ be the moduli space of semi-stable pure sheaves of class $u$ on a smooth complex projective surface $X$. We specify $u = (0, L, \xi(u) = 0)$, i.e. sheaves in $u$ are of dimension 1. There is a natural morphism $\pi$ from the moduli space $M^H_X (u)$ to the linear system $|L|$. We study a series of determinant line bundles $\lambda_{c_n^r}$ on $M^H_X (u)$ via $\pi$. Denote $g_L$ the arithmetic genus of curves in $|L|$. For any $X$ and $g_L \le 0$, we compute the generating function $Z^r(t) = \sum_n h^0(M^H_X (u), \lambda_{c_n^r})t^n$. For $X$ being $\mathbb{P}^2$ or $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1} (−e))$ with $e = 0, 1$, we compute $Z^1(t)$ for $g_L \gt 0$ 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 Göttsche’s computation. And in addition, we get an interesting corollary (Corollary 4.2.13) in the theory of compactified Jacobian of integral curves.