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

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 $\psi \colon S \dashrightarrow \mathbb{P}^n$ 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.