Lüroth semigroups of hyperelliptic curves over number fields (Jasmine Camero, Pomona College)

Let $C$ be a smooth, projective, geometrically integral hyperelliptic curve of genus $g \ge 2$ over a number field $k$. We investigate the structure of the $\mathbb{P}^1$-parameterized density degree set $\delta_{\mathbb{P}^1}(C/k)$ (equivalently, the Lüroth semigroup) for such curves, consisting of degrees arising from morphisms to $\mathbb{P}^1$. Building on foundational frameworks for parameterized points, we provide a comprehensive classification of these sets. Specifically, we establish a master classification theorem that links the admissibility of odd degrees directly to Brill-Noether loci on the Jacobian variety. Our results characterize both the extreme behaviors, dictated by whether the rational points of the Jacobian are Zariski dense or finite, and the intermediate spectrum of possible minimum odd integers.