Codimension of jumping loci

TL;DR

This paper investigates how the Harder-Narasimhan filtration of vector bundles on smooth projective varieties varies across families of curves, revealing conditions for expected codimension behavior and applying findings to rank 2 bundles on projective planes and moduli space singularities.

Contribution

It identifies geometric conditions that determine when the expected codimension of jumping loci in Harder-Narasimhan filtrations holds and applies these results to specific geometric contexts.

Findings

Codimension of jumping loci depends linearly on slope jumps under certain conditions

Characterization of geometric properties influencing Harder-Narasimhan filtration jumps

Applications to rank 2 bundles on  and singularities in moduli spaces

Abstract

Suppose that $\mathcal{E}$ is a vector bundle on a smooth projective variety $X$. Given a family of curves $C$ on $X$, we study how the Harder-Narasimhan filtration of $\mathcal{E}|_{C}$ changes as we vary $C$ in our family. Heuristically we expect that the locus where the slopes in the Harder-Narasimhan filtration jump by $\mu$ should have codimension which depends linearly on $\mu$. We identify the geometric properties which determine whether or not this expected behavior holds. We then apply our results to study rank $2$ bundles on $\mathbb{P}^{2}$ and to study singular loci of moduli spaces of curves.