What question did this study set out to answer?

The study aims to establish the equivalence of Betti and mixed Hodge numbers of the compactified Jacobian of nodal curves with those of a product space.

May 24, 2026

An Equisingular Specialization of the Compactified Jacobian and its Applications

Key Points

The study aims to establish the equivalence of Betti and mixed Hodge numbers of the compactified Jacobian of nodal curves with those of a product space.
Constructed a local family of projective varieties containing both the compactified Jacobian and the product space as fibers.
Utilized the Kunneth type formula to derive explicit formulas for Betti and Hodge numbers from the product space.
The Betti numbers of the compactified Jacobian $ar{J_k}$ match those of $J_0 \times R^k$.
The mixed Hodge numbers of $ar{J_k}$ are equivalent to those of $J_0 \times R^k$.

Abstract

ABSTRACT For any positive integer k, let Xₖ be a projective irreducible nodal curve with k nodes. We show that the Betti numbers and the mixed Hodge numbers of the compactified Jacobian Jₖ of an irreducible nodal curve Xₖ with k nodes are the same as the Betti numbers and the mixed Hodge numbers of J₀ Rᵏ, where J₀ is the Jacobian of the normalization of the irreducible nodal curve and R denotes the rational nodal curve with one node. We prove it by constructing a trivial topologically locally family of projective varieties that contain both Jₖ and J₀ Rᵏ as fibers. Explicit formulas for the Betti and Hogde numbers are obtained by applying the Kunneth type formula on the Betti and Hodge numbers of the product space J₀ Rᵏ.

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