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

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

2010-05-18

arxiv.org/abs/1005.3201v2

Mathematics

Algebraic Geometry

Scientific paper

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.

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