Compactified Jacobians and Torelli Map

Abstract

We compare several constructions of compactified jacobians — using semistable sheaves, semistable projective curves, degenerations of abelian varieties, and combinatorics of cell decompositions — and show that they are equivalent. We give a detailed description of the “canonical compactified jacobian” in degree $g-1$. Finally, we explain how Kapranov’s compactification of configuration spaces can be understood as a toric analog of the extended Torelli map.