Secant loci of scrolls over curves (Q6607624)

Let \(C\) be a smooth complex projective curve of genus \(g\). For any linear series \(\ell =(L,V)\) on \(C\), let \(V^{e-f}_{e}(\ell)\) denote the set of all degree \(e\) effective divisors on \(C\) giving at most \(e-f\) independent conditions on \(V\). Let \(E\) be a rank \(r\) vector bundle of degree \(d\) on \(C\). The author studies two generalizations of it to higher dimension and higher ranks. Let \(S\) be a smooth projective variety of dimension \(r\ge 1\) and \(\ell = (L,V)\) a linear series on \(S\) of dimension \(m\). Let \(\mathrm{Hilb}^e_{\mathrm{sm}}(S) \subseteq \mathrm{Hilb}^e(S)\) denote the smoothable component of the Hilbert scheme of degree \(e\) zero-dimensional schemes of \(S\). Let \(H^{e-f}_{e}(\ell)\) be the set of all \(Z\in \mathrm{Hilb}^e_{\mathrm{sm}}(S)\) giving at most \(e\) conditions to \(V\). He use the definition for \(E\) taking \(S:= \mathbb{P}E\). Let \(\mathrm{Quot}^{0,e}(E^*)\) be the Quot-scheme parametrizing subsheaves \(F^*\subset E^*\) of rank \(r\) and degree \(-d-e\). Take \(M\in \mathrm{Pic}(C)\). The second generalization is the scheme\N\[\NQ^{e-f}(E,M,V):= \{[F^*\to E^*]\mid h^0(C,F^*\otimes M)\ge n+1-e+f\}.\N\]\NThe author describes the Zariski tangent space of these schemes and show that smoothness is not assured by the injectivity of a Petri map. He generalizes the notion of Abel-Jacoby map and linear series to the Quot-schemes. Lot of examples and several open questions (at the end of the introduction), e.g. relating these loci to the Terracini loci. He gives criteria of nonemptyness for these secant loci and a full criterion of \(V=H^0(L)\) and \(f\) in terms of the Segre invariant \(s_1(E)\). This implies a geometric criteria for semistability similar to the one in [\textit{G. H. Hitching}, Geo. Dedicata 205, 1--19 (2020; Zbl 1442.14109)]. With these tools he gives a partial answer to a question of Lange on the very ampleness of \(\mathcal{O}_{\mathbb{P}E}(1)\) and prove that for all curves \(Q^{e-1}_{e}(V)\) is either empty or of the expected dimension.\N\NFor the entire collection see [Zbl 1545.14003].