Secant loci of scrolls over curves

Abstract

Given a curve C and a linear system ℓ on C, the secant locus V e−f e (ℓ) parametrises effective divisors of degree e which impose at most e − f conditions on ℓ. For E → C a vector bundle of rank r, we define determinantal subschemes H e−f e (ℓ) ⊆ Hilb e (PE) and Q e−f e (V ) ⊆ Quot 0,e (E * ) which generalise V e−f e (ℓ), giving several examples. We describe the Zariski tangent spaces of Q e−f e (V ), and give examples showing that smoothness of Q e−f e (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 [Hit19]. Using these ideas, we also give a partial answer to a question of Lange on very ampleness of O PE (1), and show that for any curve, Q e−1 e (V ) is either empty or of the expected dimension for sufficiently general E and V . When Q e−1 e (V ) has and attains expected dimension zero, we use formulas of Oprea-Pandharipande and Stark to enumerate Q e−1 e (V ). We mention several possible avenues of further investigation. e (ℓ), let E → C be a vector bundle of rank r and degree d. The scheme Quot 0,e (E * ) parametrises torsion quotients of E * of length e; equivalently, subsheaves F * ⊂ E * of rank r and degree −d − e. Let M be a line bundle, and suppose that V ⊆ H 0 (C, E * ⊗ M ) is a subspace of dimension n + 1. We define