Weierstrass Semigroups from Cyclic Covers of Hyperelliptic Curves

The 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 S ⊂ N with finite complement are realizable as Weierstrass semigroups S = S ( C , p ) . In this note, we establish realizability results for cyclic covers π : ( C , p ) → ( 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.