On nonnegatively graded Weierstrass points

TL;DR

This paper introduces a new lower bound for the dimension of moduli spaces of smooth pointed curves with prescribed Weierstrass semigroups, improving previous bounds and establishing non-emptiness and pure dimension for certain cases.

Contribution

It provides a novel lower bound derived from advanced deformation theory and demonstrates its sharpness for specific symmetric semigroups of multiplicity six.

Findings

New lower bound for moduli space dimension

Improved over previous bounds by Pflueger

Existence and pure dimension of moduli spaces for certain semigroups

Abstract

We provide a new lower bound for the dimension of the moduli space of smooth pointed curves with prescribed Weierstrass semigroup at the marked point, derived from the Deligne-Greuel formula and Pinkham's equivariant deformation theory. Using Buchweitz's description of the first cohomology module of the cotangent complex for monomial curves, we show that our lower bound improves a recently one given by Pflueger. By allowing semigroups running over suitable families of symmetric semigroups of multiplicity six, we show that this new lower bound is attained, and that the corresponding moduli spaces are non-empty and of pure dimension.