Extending Andreotti's proof of Torelli's theorem to nodal curves

Abstract

Let $C$ be a connected nodal curve of genus $g$ over an algebraically closed field $k$ of characteristic zero. Let \[ A(C)=\Big(J(C), {\overline{P^{g-1}_C}}, \Theta(C) \Big) \] be the stable semi-abelic pair (in the sense of Alexeev) associated with its canonical compactified Jacobian $\overline{P^{g-1}_C}$ in degree $g-1$. The canonical triple of $C$ consists of the closure of its canonical image together with two sets of distinguished points encoding the information of the images of the nonseparating nodes and of the branch data of the canonical map. We prove a functorial reconstruction: given connected nodal curves $C$ and $C'$ over $k$, every isomorphism $A(C)\cong A(C')$ of stable semi-abelic pairs canonically induces a projectivity between the corresponding canonical triples of $C$ and $C'$. Passing to isomorphism classes, this recovers the reconstruction statement in the compactified Torelli theorem of Caporaso-Viviani. Our proof extends Andreotti's Gauss-map strategy. The branch locus of the projection from the normalization of the Gauss graph recovers the nonlinear canonical components and branch data by projective biduality; boundary strata of the compactified Jacobian recover the node-images; and incidence arguments recover the linear components. Along the way, we repair technical gaps in the classical treatments of Andreotti's proof of Torelli's theorem for smooth curves.