Geometry of a semistable degeneration of moduli of vector bundles on curves

Abstract

We study the Gieseker moduli space $\mathcal{M}_{X_0}$ of stable rank-$2$, degree-$1$ vector bundles on an irreducible curve $X_0$ with one node. Its normalization is a two-step blow-up of the moduli space of generalized parabolic bundles on the normalization $\tilde X_0$, and $\mathcal{M}_{X_0}$ is recovered by gluing along a natural involution on the boundary. From this we compute $\operatorname{Pic}(\mathcal{M}_{X_0})$ and, via a Harder--Narasimhan stratification, the virtual Poincar\'e polynomial of $\mathcal{M}_{X_0}$. Since the normalization morphism is small, the intersection cohomology of $\mathcal{M}_{X_0}$ equals the ordinary cohomology of the smooth normalization, which gives a closed formula in terms of the arithmetic genus.