MAXIMAL SUBSHEAVES OF TORSION-FREE SHEAVES ON NODAL CURVES

Abstract

Let $Y$ be a reduced irreducible projective curve of arithmetic genus $g\ge 2$ with at most ordinary nodes as singularities. For a subsheaf $F$ of rank $r'$, degree $d'$ of a torsion-free sheaf $E$ of rank $r$, degree $d$, let $s(E,F) = r'd-rd'$. Define $s_{r'}(E) = {\rm min} s(E,F)$, where the minimum is taken over all subsheaves of $E$ of rank $r'$. For a fixed $r', s_{r'}$ defines a stratification of the moduli space $U(r,d)$ of stable torsion-free sheaves of rank $r$, degree $d$ by locally closed subsets $U_{r',s}$. We study the nonemptiness and dimensions of the strata. We show that the general element in each nonempty stratum is a vector bundle and it has only finitely many (respectively unique) maximal subsheaves of rank $r'$ for $s\le r'(r-r')(g-1)$ (respectively $s<r'(r-r')(g-1)$). We prove that the tensor product of two general stable vector bundles on an irreducible nodal curve $Y$ is nonspecial.