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 = PE the associated projective bundle. We describe the inflectional loci of certain projective models ψ: S 99K Pn in terms of Quot schemes of E. This gives a geometric characterisation of the Segre invariant s1(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 ψ, the inflectional loci are all of the expected dimension. An auxiliary result, valid for a general subvariety of Pn, is that under mild hypotheses, the inflectional loci associated to a projection from a general centre are of the expected dimension.