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 g,1 of the variety G,1S 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 G,1S 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 G,1S 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