On nonprimitive Weierstrass points

Abstract

We give an upper bound for the codimension in ${\mathcal{ℳ}}_{g,1}$ of the variety ${\mathcal{ℳ}}_{G,1}^S$ of marked curves $\left(C,p\right)$ with a given Weierstrass semigroup. The bound is a combinatorial quantity which we call the effective weight of the semigroup; it is a refinement of the weight of the semigroup, and differs from the weight precisely when the semigroup is not primitive. We prove that whenever the effective weight is less than $g$, the variety ${\mathcal{ℳ}}_{G,1}^S$ is nonempty and has a component of the predicted codimension. These results extend previous results of Eisenbud, Harris, and Komeda to the case of nonprimitive semigroups. We also survey other cases where the codimension of ${\mathcal{ℳ}}_{G,1}^S$ is known, as evidence that the effective weight estimate is correct in wider circumstances.