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arXiv · 2507.04902

Brill--Noether loci in genus $\leq 12$

Richard Haburcak
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Abstract

A refined Brill--Noether theory seeks to determine which linear series are admitted by a ``general'' curve in a particular Brill--Noether locus. However, as Brill--Noether loci are not irreducible in general, a coarse answer is given by the relative positions of Brill--Noether loci. Via an analysis of unstable Lazarsfeld--Mukai bundles on K3 surfaces, we distinguish Brill--Noether loci and provide expectations for the relative positions of Brill--Noether loci in general. Together with classical results, the refined Brill--Noether theory for curves of fixed gonality and on Hirzebruch surfaces, and explicit constructions, we identify the relative positions of all Brill--Noether loci in genus $g\leq 12$.

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Richard Haburcak. 2025-07-07. Brill--Noether loci in genus $\leq 12$. https://arxiv.org/abs/2507.04902

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