Secant loci of scrolls over curves

Given a curve $C$ and a linear system $\ell$ on $C$, the secant locus $V_e^{e-f}( \ell )$ parametrises effective divisors of degree $e$ which impose at most $e-f$ conditions on $\ell$. For $E \to C$ a vector bundle of rank $r$, we define determinantal subschemes $H_e^{e-f} ( \ell ) \subseteq \mathrm{Hilb}^e ( \mathbb{P} E )$ and $Q_e^{e-f} (V) \subseteq \mathrm{Quot}^{0, e} ( E^* )$ which generalise $V_e^{e-f} ( \ell )$, 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.