Torelli theorem for stable curves

TL;DR

This paper investigates the Torelli morphism for stable curves, characterizing its fibers, injectivity, and bounds, revealing that the map is injective outside separating nodes for genus up to 4.

Contribution

It provides new characterizations of the fibers of the Torelli morphism and establishes precise bounds on their sizes and dimensions, extending classical Torelli results to stable curves.

Findings

The Torelli map is injective away from separating nodes for genus ≤ 4.

Sharp upper bounds are given for the size of finite fibers.

The dimension of infinite fibers is bounded.

Abstract

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.