Weierstrass semigroups from cyclic covers of hyperelliptic curves

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 $\mathrm{S}\subset \mathbb{N}$ with finite complement are {\it realizable} as Weierstrass semigroups $\mathrm{S}=\mathrm{S}(C,p)$. In this note, we establish realizability results for cyclic covers $\pi :(C,p)\to (B,q)$ 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.

arXiv comment30 pages. Final version, to appear in Bulletin of the Brazilian Mathematical Society