Secant loci of scrolls over curves

Given a curve $C$ and a linear system $\mathrm{\ell }$ on $C$, the secant locus $V_e^{e-f}(\mathrm{\ell })$ parametrises effective divisors of degree $e$ which impose at most $e-f$ conditions on $\mathrm{\ell }$. For $E\to C$ a vector bundle of rank $r$, we define determinantal subschemes $H_e^{e-f}(\mathrm{\ell })\subseteq {\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}}^e(\mathbb{P}E)$ and $Q_e^{e-f}(V)\subseteq {\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}^{0,e}(E^{\ast })$ which generalise $V_e^{e-f}(\mathrm{\ell })$, giving several examples. We describe the Zariski tangent spaces of $Q_e^{e-f}(V)$, and give examples showing that smoothness of $Q_e^{e-f}(V)$ is not necessarily controlled by injectivity of a Petri map. We generalise the Abel–Jacobi map and the notion of linear series to the context of Quot schemes. We give some sufficient conditions for nonemptiness of generalised secant loci, and a criterion in the complete case when $f=1$ in terms of the Segre invariant $s_1(E)$. This leads to a geometric characterisation of semistability similar to that in arxiv:1812.00706. Using these ideas, we also give a partial answer to a question of Lange on very ampleness of ${\mathcal{O}}_{\mathbb{P}E}(1)$, and show that for any curve, $Q_e^{e-1}(V)$ is either empty or of the expected dimension for sufficiently general $E$ and $V$. When $Q_e^{e-1}(V)$ has and attains expected dimension zero, we use formulas of Oprea–Pandharipande and Stark to enumerate $Q_e^{e-1}(V)$. We mention several possible avenues of further investigation.