Weierstrass semigroups from cyclic covers of hyperelliptic curves

TL;DR

This paper investigates which numerical semigroups can be realized as Weierstrass semigroups at points on algebraic curves, focusing on cyclic covers of hyperelliptic curves and their relation to divisor classes in Jacobians.

Contribution

It provides new realizability results for Weierstrass semigroups on cyclic covers of hyperelliptic curves, linking realizability to divisor class behavior in hyperelliptic Jacobians.

Findings

Realizability depends on divisor class behavior under multiplication in Jacobians.

Established realizability results for cyclic covers of hyperelliptic curves.

Connected semigroup realizability with algebraic properties of Jacobian divisor classes.

Abstract

The {\it Weierstrass semigroup} of pole orders of meromorphic functions in a point $p$ of a smooth algebraic curve $C$ is a classical object of study; a celebrated problem of Hurwitz is to characterize which semigroups ${\rm S} \subset \mathbb{N}$ with finite complement are {\it realizable} as Weierstrass semigroups ${\rm S}= {\rm S}(C,p)$. In this note, we establish realizability results for cyclic covers $\pi: (C,p) \rightarrow (B,q)$ of hyperelliptic targets $B$ marked in hyperelliptic Weierstrass points; and we show that realizability is dictated by the behavior under $j$-fold multiplication of certain divisor classes in hyperelliptic Jacobians naturally associated to our cyclic covers, as $j$ ranges over all natural numbers.