Torelli theorem for stable curves

Understand

We study the Torelli morphism from the moduli space of stable curves to the moduli space of principally polarized stable semi-abelic pairs.

• We give two characterizations of its fibers, describe its injectivity locus, and give a sharp upper bound on the cardinality of the finite fibers.
• We also bound the dimension of the infinite fibers.
• In particular, we obtain that the compactified Torelli map is injective away from curve with separating nodes if and only if the genus is at most 4.