Secant loci of scrolls over curves
George H. Hitching

TL;DR
This paper generalizes secant loci of scrolls over curves using determinantal schemes on Hilbert and Quot schemes, providing conditions for nonemptiness, smoothness, and enumerations, and exploring their geometric properties.
Contribution
It introduces new determinantal subschemes generalizing secant loci, describes their tangent spaces, and extends classical concepts like Abel–Jacobi maps to Quot schemes, offering new criteria and enumerations.
Findings
Conditions for nonemptiness of generalized secant loci
Criteria for smoothness and expected dimension
Enumeration formulas for zero-dimensional cases
Abstract
Given a curve and a linear system on , the secant locus parametrises effective divisors of degree which impose at most conditions on . For a vector bundle of rank , we define determinantal subschemes and which generalise , giving several examples. We describe the Zariski tangent spaces of , and give examples showing that smoothness of 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 in terms of the Segre invariant $s_1…
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Taxonomy
TopicsTensor decomposition and applications · Advanced Algebra and Geometry · Algebraic Geometry and Number Theory
