Simpson Jacobians of reducible curves

Abstract

For any projective curve $X$ let ${\bar{M}}^d(X)$ be the Simpson moduli space of pure dimension one rank 1 degree $d$ sheaves that are semistable with respect to a fixed polarization $H$ on $X$. When $X$ is a reduced curve the connected component of ${\bar{M}}^d(X)$ that contains semistable line bundles can be considered as the compactified Jacobian of $X$. In this paper we give explicitly the structure of this compactified Simpson Jacobian for the following projective curves: tree-like curves and all reduced and reducible curves that can appear as Kodaira singular fibers of an elliptic fibration, that is, the fibers of types $III$, $IV$ and $I_N$ with $N\ge 2$

Latex file with 13 figures. Final version to appear in J. Reine. Angew. Math