  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.
