Geometry of intersections of some secant varieties to algebraic curves

TL;DR

This paper investigates the intersection properties of secant divisors on algebraic curves, providing new methods to verify when such intersections are empty and exploring cases with unexpected transversality.

Contribution

It introduces a general method to determine the emptiness of intersections of secant divisor cycles on curves, advancing classical enumerative geometry techniques.

Findings

Identified cases with unexpected transversality properties

Developed a method to verify when intersections are empty

Analyzed specific examples of secant divisor intersections

Abstract

For a smooth projective curve, the cycles of subordinate or, more generally, secant divisors to a given linear series are among some of the most studied objects in classical enumerative geometry. We consider the intersection of two such cycles corresponding to secant divisors of two different linear series on the same curve and investigate the validity of the enumerative formulas counting the number of divisors in the intersection. We study some interesting cases, with unexpected transversality properties, and establish a general method to verify when this intersection is empty.