Quot schemes, Segre invariants, and inflectional loci of scrolls over curves

Abstract

Let E be a vector bundle over a smooth curve C, and $$S = {{\mathbb {P}}}E$$ the associated projective bundle. We describe the inflectional loci of certain projective models $$\psi :S \dashrightarrow {{\mathbb {P}}}^n$$ in terms of Quot schemes of E. This gives a geometric characterisation of the Segre invariant $$s_1 (E)$$, which leads to new geometric criteria for semistability and cohomological stability of bundles over C. We also use these ideas to show that for general enough S and $$\psi $$, the inflectional loci are all of the expected dimension. An auxiliary result, valid for a general subvariety of $${{\mathbb {P}}}^n$$, is that under mild hypotheses, the inflectional loci associated to a projection from a general centre are of the expected dimension.