On nonprimitive Weierstrass points

Pflueger, Nathan

doi:10.2140/ant.2018.12.1923

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On nonprimitive Weierstrass points

Nathan Pflueger

Vol. 12 (2018), No. 8, 1923–1947

DOI: 10.2140/ant.2018.12.1923

Abstract

We give an upper bound for the codimension in $ℳ_{g, 1}$ of the variety $ℳ_{G, 1}^{S}$ of marked curves $(C, p)$ 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 $ℳ_{G, 1}^{S}$ 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 $ℳ_{G, 1}^{S}$ is known, as evidence that the effective weight estimate is correct in wider circumstances.

Keywords

Weierstrass points, numerical semigroups, algebraic curves, limit linear series

Mathematical Subject Classification 2010

Primary: 14H55

Milestones

Received: 2 September 2016

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

Vol. 12, No. 8, 2018