An Equisingular Specialisation of the Compactified Jacobian and its applications

TL;DR

This paper demonstrates that the Betti and mixed Hodge numbers of the compactified Jacobian of a nodal curve are equivalent to those of a product of simpler Jacobians and rational nodal curves, using a topologically trivial family of varieties.

Contribution

It introduces a new equisingular specialization technique for the compactified Jacobian, linking its topological invariants to those of a product space involving the normalisation and rational nodal curves.

Findings

Betti numbers of  and J_0  R^k are equal.

Mixed Hodge numbers of  and J_0  R^k are equal.

Constructed a topologically locally trivial family of varieties.

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.