2023/02/08 by George H. Hitching, Hitching, George H. · 1 citation
Mathematics · #14H60 (Primary) 14N07 #14M12 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2302.04328
openalex publication_date 2023/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a curve C and a linear system ℓ on C, the secant locus Vee-f( 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 Hee-f ( ℓ ) ⊆ Hilbe ( ℙ E ) and Qee-f (V) ⊆ Quot0, e ( E^* ) which generalise Vee-f ( ℓ ), giving several examples. We describe the Zariski tangent spaces of Qee-f (V), and give examples showing that smoothness of Qee-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 s1 (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 Oℙ E (1), and show that for any curve, Qee-1 (V) is either empty or of the expected dimension for sufficiently general E and V. When Qee-1 (V) has and attains expected dimension zero, we use formulas of Oprea--Pandharipande and Stark to enumerate Qee-1 (V). We mention several possible avenues of further investigation.