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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|₂ changes as we vary C in our family. Heuristically we expect that the locus where the slopes in the Harder-Narasimhan filtration jump by should have codimension which depends linearly on. 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 P^2 and to study singular loci of moduli spaces of curves.
Lehmann et al. (Fri,) studied this question.