An Equisingular Specialisation of the Compactified Jacobian and its applications
Sourav Das, A. J. Parameswaran, Subham Sarkar
TL;DR
The paper constructs a topologically locally trivial specialization family that connects the compactified Jacobian $ar{J}_k$ of an irreducible nodal curve with $k$ nodes to the product $J_0 imes R^k$, where $J_0$ is the Jacobian of the normalization and $R$ is the rational nodal curve with one node. This specialization yields an exact match between the Betti numbers and the mixed Hodge numbers of $ar{J}_k$ and those of $J_0 imes R^k$, enabling explicit computations via the Künneth formula. The approach hinges on a careful construction of total spaces, divisors, and twisted identifications, followed by descent to obtain a projective, well-behaved family; Whitney and Thom-Mather theory then ensure topological triviality and the existence of a VHS. The results provide a clear equisingular specialization framework for compactified Jacobians and yield concrete invariants, with potential generalizations to higher-rank moduli spaces. Overall, the work connects the geometry of singular curves to tractable product spaces, facilitating direct computation of topological and Hodge-theoretic invariants.
Abstract
For any positive integer $k$, let $X_k$ be a projective irreducible nodal curve with $k$ nodes. We show that the Betti numbers and the mixed Hodge numbers of the compactified Jacobian $\overline{J_{k}}$ of an irreducible nodal curve $X_k$ with $k$ nodes are the same as the Betti numbers and the mixed Hodge numbers of $J_0\times R^k$, where $J_0$ is the Jacobian of the normalisation of the irreducible nodal curve and $R$ denotes the rational nodal curve with one node. We prove it by constructing a topologically locally trivial family of projective varieties containing $\overline{J_{k}}$ and $J_0\times R^k$ as fibres.