Prym-Brill-Noether Theory for General Covers

Summary · qwen2.5:32b

The study establishes new bounds on the dimension of the Prym-Brill-Noether variety for étale double covers of k-gonal curves, disproving a conjecture by Creech et al., and proposes a new conjecture. A key method involves analyzing the Prym-Brill-Noether variety of a double cover of a specific tropical curve called the loop of loops, utilizing Coxeter group combinatorics to prove topological properties.

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Exploring Prym-Brill-Noether Theory for General Covers in mathematics, focusing on double covers of elliptic and gonal curves.

Excerpt

arXiv:2607.01173v2 Announce Type: replace Abstract: We bound the dimension of the Prym-Brill-Noether variety for an open subset of the moduli space of \'{e}tale double covers of k-elliptic curves. We also obtain new bounds on the dimension of the Prym-Brill-Noether variety for general \'{e}tale double covers of k-gonal curves, disproving a conjecture of Creech, Len, Ritter, and Wu, and provide a new conjecture for its dimension. To do this, we completely describe the Prym-Brill-Noether variety of a double cover of a certain tropical curve known as the loop of loops. We use the combinatorics of Coxeter groups to establish several topological properties of these tropical Prym-Brill-Noether varieties, and prove a lifting result when the edge lengths are generic.