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arXiv · 2505.19890

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

Gavril Farkas·Soheyla Feyzbakhsh·Andrés Rojas
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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.

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Gavril Farkas, Soheyla Feyzbakhsh, Andrés Rojas. 2025-05-26. Hurwitz-Brill-Noether theory via K3 surfaces and stability conditions. https://arxiv.org/abs/2505.19890

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