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

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

2010-10-09

arxiv.org/abs/1010.1815v1

Mathematics

Algebraic Geometry

Scientific paper

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.

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