$k$-canonical divisors through Brill-Noether special points

TL;DR

This paper constructs and analyzes divisors in the moduli space of curves defined by Brill-Noether special points, computing their classes and demonstrating their extremality and rigidity.

Contribution

It introduces a new approach to studying $k$-canonical divisors via Brill-Noether special points, with explicit class calculations and extremality results.

Findings

Computed classes of divisors using incidence geometry and family restrictions.

Proved extremality and rigidity of certain incidence divisors.

Established connections between Brill-Noether theory and $k$-canonical divisors.

Abstract

Inside the projectivized $k$-th Hodge bundle, we construct a collection of divisors obtained by imposing vanishing at a Brill-Noether special point. We compute the classes of the closures of such divisors in two ways, using incidence geometry and restrictions to various families, including pencils of curves on K3 surfaces and pencils of Du Val curves. We also show the extremality and rigidity of the closure of the incidence divisor consisting of smooth pointed curves together with a canonical or 2-canonical divisor passing through the marked point.