Weierstrass semigroups from cyclic covers of hyperelliptic curves

Ethan Cotterill; Nathan Pflueger; Naizhen Zhang

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arXiv · 2201.00033

Weierstrass semigroups from cyclic covers of hyperelliptic curves

Ethan Cotterill·Nathan Pflueger·Naizhen Zhang

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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.

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BibTeX RIS

Ethan Cotterill, Nathan Pflueger, Naizhen Zhang. 2021-12-31. Weierstrass semigroups from cyclic covers of hyperelliptic curves. https://arxiv.org/abs/2201.00033

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