 
Summary: COMPACTIFIED JACOBIANS AND TORELLI MAP
VALERY ALEXEEV
Abstract. We compare several constructions of compactified jacobians  us
ing 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 ja
cobian" 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.
There are many papers devoted to compactifying (generalized) jacobians of
curves and families of curves. Some of them are concerned primarily with exis
tence, some provide a finer description. The approaches vary widely: some con
structions use moduli of semistable rank1 sheaves, some use semistable projective
curves, some use combinatorics of cell decompositions; yet others use degenerations
of principally polarized abelian varieties and various notions of stable varieties.
One aim of this survey is to give a definitive account in the case of nodal curves
and to show, pleasingly, that in this case all of the known approaches are equivalent
and produce isomorphic varieties, with the degeneration of PPAVs approach being
the special case of degree g  1. Combining the known results we then describe
these varieties in detail.
