Embedding of a compactified Jacobian and theta divisors

Abstract

Using a general stable vector bundle, we give an embedding $\alpha _Y$ of the compactified Jacobian $\bar{J}(Y)$ of an integral nodal curve Y into the moduli space $U_Y(r,d)$ of semistable torsion free sheaves of rank r and degree d on Y. We also give an embedding of the normalisation $\tilde{J}(Y)$ of $\bar{J}(Y)$ in the normalisation P(r, d) of $U_Y(r,d)$. We determine a relation between the restriction of the theta line bundle on P(r, d) to $\tilde{J}(Y)$ and the theta line bundle on $\tilde{J}(Y)$. We show that the restriction of the Picard bundle $E_{r,d}$ on $U_Y(r,d)$ to $\bar{J}(Y)$ is stable with respect to any theta divisor $\theta _{\bar{J}(Y)}$ on $\bar{J}(Y)$ if $d > r(2g-1)$ and it is semistable if $d= r(2g-1)$. Finally we prove some results on natural cohomology.