Codimension of jumping loci

Suppose that 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 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 μ should have codimension which depends linearly on μ. 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 P² and to study singular loci of moduli spaces of curves.