Divisors of secant planes to curves

TL;DR

This paper computes the classes of divisors related to secant conditions on linear series of very general curves, improving bounds on the effective cone of symmetric products and exploring their moduli space implications.

Contribution

It introduces a method to compute divisor classes of secant conditions on symmetric products of curves, enhancing understanding of the effective cone and moduli space geometry.

Findings

Derived explicit divisor classes for secant conditions

Provided improved bounds for the slope of the effective cone

Analyzed divisor classes within the moduli space of stable pointed curves

Abstract

Inside the symmetric product of a very general curve, we consider the codimension-one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension. We compute the classes of such divisors, and thus obtain improved bounds for the slope of the cone of effective divisor classes on symmetric products of a very general curve. By letting the moduli of the curve vary, we study more generally the classes of the related divisors inside the moduli space of stable pointed curves.