Relative Jacobians of elliptic fibrations with reducible fibers

Abstract

We prove that if $X$ and $S$ are smooth varieties and $f:X\to S$ is an elliptic fibration with singular fibers curves of types I${}_N$ with $N\ge 1$, II, III and IV, then the relative Jacobian $\hat{f}:{\bar{M}}_{X/S}\to S$ of $f$, defined as the relative moduli space of semistable pure dimension one sheaves of rank 1 and degree 0 on the fibers of $f$, is an elliptic fibration such that all its fibers are irreducible. This extends known results when fibers are integral or of type I${}_2$.

Latex file with two figures