  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.
