An Equisingular Specialization of the Compactified Jacobian and its Applications

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 Jₖ̄ 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 Jₖ̄ 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.