Brill-Noether loci of pencils with prescribed ramification on moduli of curves and on Severi varieties on $K3$ surfaces

TL;DR

This paper investigates Brill-Noether loci with prescribed ramification on moduli spaces of curves and Severi varieties on K3 surfaces, establishing existence, dimension, and dominance results that extend classical theories and connect to algebraic cycles.

Contribution

It proves the existence and expected dimension of Brill-Noether loci with prescribed ramification and analyzes their implications for the geometry of moduli spaces and K3 surfaces.

Findings

Brill-Noether loci have components of expected codimension under certain conditions.

The map from Hurwitz schemes to moduli space is dominant or generically finite based on parameters.

Existence of special curves in Severi varieties with non-generic Brill-Noether behavior.

Abstract

Under the assumption that the adjusted Brill-Noether number $\widetilde{\rho}$ is at least $-g$, we prove that the Brill-Noether loci in $\mathcal{M}_{g,n}$ of pointed curves carrying pencils with prescribed ramification at the marked points have a component of the expected codimension with pointed curves having Brill-Noether varieties of pencils of the minimal dimension. As an application, the map from the Hurwitz scheme to $\mathcal{M}_g$ is dominant if $n+\widetilde{\rho} \geq 0$ and generically finite otherwise, settling a variation of a classical problem of Zariski. In the second part of the paper, we study the analogous loci of curves in Severi varieties on $K3$ surfaces, proving existence of curves with non-general behaviour from the point of view of Brill-Noether theory. This extends previous results of Ciliberto and the first named author to the ramified case. We apply these results to study correspondences and cycles on $K3$ surfaces in relation to Beauville-Voisin points and constant cycle curves.