Abstract:: (N,gamma)-hyperelliptic semigroups were introduced by Fernando Torres to encapsulate the most salient properties of Weierstrass semigroups associated to totally-ramified points of N-fold covers of curves of genus gamma. He proved that when the genus g is suficiently large relative to the target genus gamma, every (N,gamma)-hyperelliptic semigroup of genus g is realizable by a cover of pointed curves. Understanding which cover realizes a given (N,gamma)-hyperelliptic semigroups is, however, a delicate question. We are particularly interested in the realizability problem for (N,gamma)-hyperelliptic semigroups of maximal weight, or equivalently, those for which the canonical linear series of the source curve is maximally inflected in the source point. We give a conjectural formula for this maximal weight when N is prime, and we describe work in progress on the realizability problem for (N,gamma)-hyperelliptic semigroups arising from cyclic covers.
