Secant loci of scrolls over curves

Abstract

Given a curve C and a linear series on C, the secant locus V e−f e ( ) parametrises effective divisors of degree e which impose at most e − f conditions on . For E → C a vector bundle of rank r, we define etermi nantal subschemes He−f e ( ) ⊆ Hilbe(PE) and Qe−f e (V ) ⊆ Quot0,e(E∗) which generalise V e−f e ( ), giving several examples. We describe the Zariski tangent spaces of Qe−f e (V ), and give examples showing that smoothness of Qe−f e (V ) is not necessarily controlled by injectiveness 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 s1(E). This leads to a geometric characterisation of semistability similar to that in [Quot schemes, Segre invariants, and inflectional loci of scrolls over curves, Geom. Dedicata 205 (2020), 1–19]. Using these ideas, we also give a partial answer to a question of Lange on very ampleness of OPE(1), and show that for any curve, Qe−1 e (V ) is either empty or of the expected dimension for sufficiently general E and V . When Qe−1 e (V ) has and attains expected dimension zero, we use formulas of Oprea–Pandharipande and Stark to enumerate Qe−1 e (V ). We mention several possible avenues of further investigation.