Geometry of curves with exceptional secant planes: linear series along the general curve

TL;DR

This paper investigates the geometry of linear series on general curves with exceptional secant planes, proving nonexistence in certain cases, and computing their counts in families, with applications to Hilbert schemes and hypergeometric series.

Contribution

It extends Brill-Noether theory to pairs of linear series, providing formulas for counting exceptional secant planes and analyzing their asymptotics.

Findings

General curves have no linear series with exceptional secant planes when the total number is finite.

Derived formulas for counting linear series with exceptional secant planes in families.

Applied results to extremal cases related to Hilbert schemes and hypergeometric series.

Abstract

We study linear series on a general curve of genus $g$, whose images are exceptional with regard to their secant planes. Working in the framework of an extension of Brill-Noether theory to pairs of linear series, we prove that a general curve has no linear series with exceptional secant planes, in a very precise sense, whenever the total number of series is finite. Next, we partially solve the problem of computing the number of linear series with exceptional secant planes in a one-parameter family in terms of tautological classes associated with the family, by evaluating our hypothetical formula along judiciously-chosen test families. As an application, we compute the number of linear series with exceptional secant planes on a general curve equipped with a one-dimensional family of linear series. We pay special attention to the extremal case of $d$-secant $(d-2)$-planes to $(2d-1)$-dimensional series, which appears in the study of Hilbert schemes of points on surfaces. In that case, our formula may be rewritten in terms of hypergeometric series, which allows us both to prove that it is nonzero and to deduce its asymptotics in $d$.