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

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. $${M_X^H(u)}$$ with u = (0, L, χ(u) = 0) and L an effective line bundle on X, together with a series of determinant line bundles associated to $${r[\mathcal{O}_X]-n[\mathcal{O}_{pt}]}$$ in the Grothendieck group of X. Let g L denote the arithmetic genus of curves in the linear system |L|. For g L ≤ 2, we give a upper bound of the dimensions of sections of these line bundles by restricting them to a generic projective line in |L|. Our result gives, together with Gottsche’s computation, a first step of a check for the strange duality for some cases for X a rational surface.