On non-primitive Weierstrass points

TL;DR

This paper introduces the effective weight, a new combinatorial measure, to bound the codimension of varieties of marked curves with specific Weierstrass semigroups, extending prior results to non-primitive cases.

Contribution

It defines the effective weight and proves it provides accurate codimension bounds for non-primitive Weierstrass semigroups, generalizing previous work.

Findings

Effective weight bounds the codimension of $M^S_{g,1}$.

Non-primitive semigroups are included in the analysis.

Results align with known cases, supporting the effectiveness of the measure.

Abstract

We give an upper bound on the codimension in $M_{g,1}$ of the variety $M^S_{g,1}$ 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 it precisely when the semigroup is not primitive. We prove that whenever the effective weight is less than g, the variety $M^S_{g,1}$ is nonempty and has a component of the predicted codimension. These results extend previous results of Eisenbud, Harris, and Komeda to the case of non-primitive semigroups. We also survey other cases where the codimension of $M^S_{g,1}$ is known, as evidence that the effective weight estimate is correct in much wider circumstances.