Hurwitz-Brill-Noether theory via K3 surfaces and stability conditions

TL;DR

This paper introduces a new approach to Hurwitz-Brill-Noether theory using Bridgeland stability conditions on elliptic K3 surfaces, providing novel examples and proofs for the existence and non-existence of certain curves.

Contribution

It develops a Bridgeland stability framework for curves on elliptic K3 surfaces and constructs explicit examples of general Hurwitz-Brill-Noether curves over number fields.

Findings

Curves on elliptic K3 surfaces are the first known smooth k-gonal curves that are general in Hurwitz-Brill-Noether theory.

New proofs of key non-existence and existence results in Hurwitz-Brill-Noether theory.

Explicit construction of curves over number fields that are general in Hurwitz-Brill-Noether theory.

Abstract

We develop a novel approach to the Brill-Noether theory of curves endowed with a degree k cover of the projective line via Bridgeland stability conditions on elliptic K3 surfaces. We first develop the Brill-Noether theory on elliptic K3 surfaces via the notion of Bridgeland stability type for objects in their derived category. As a main application, we show that curves on elliptic K3 surfaces serve as the first known examples of smooth k-gonal curves which are general from the viewpoint of Hurwitz-Brill-Noether theory. In particular, we provide new proofs of the main non-existence and existence results in Hurwitz-Brill-Noether theory. Finally, using degree-k Halphen surfaces, we construct explicit examples of curves defined over number fields which are general from the perspective of Hurwitz-Brill-Noether theory.