#### Vol. 12, No. 8, 2018

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On nonprimitive Weierstrass points

### Nathan Pflueger

Vol. 12 (2018), No. 8, 1923–1947
##### Abstract

We give an upper bound for the codimension in ${\mathsc{ℳ}}_{g,1}$ of the variety ${\mathsc{ℳ}}_{G,1}^{S}$ 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 ${\mathsc{ℳ}}_{G,1}^{S}$ 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 ${\mathsc{ℳ}}_{G,1}^{S}$ is known, as evidence that the effective weight estimate is correct in wider circumstances.

##### Keywords
Weierstrass points, numerical semigroups, algebraic curves, limit linear series
Primary: 14H55
##### Milestones
Revised: 5 January 2018
Accepted: 10 March 2018
Published: 4 December 2018
##### Authors
 Nathan Pflueger Department of Mathematics and Statistics Amherst College Amherst, MA United States