GIT stability of weighted pointed curves

TL;DR

This paper provides a direct proof that smooth pointed curves are asymptotically Hilbert stable, facilitating the construction of moduli spaces of stable pointed curves and weighted pointed curves.

Contribution

It introduces a novel combinatorial approach to GIT stability of weighted pointed curves, extending Gieseker's method with a different solution.

Findings

Smooth curves with distinct marked points are asymptotically Hilbert stable.

The proof applies to a wide range of parameter spaces and linearizations.

The method enables the construction of moduli spaces of stable pointed curves.

Abstract

Here I give a direct proof that smooth curves with distinct marked points are asymptotically Hilbert stable with respect to a wide range of parameter spaces and linearizations. This result can be used to construct the coarse moduli space of Deligne-Mumford stable pointed curves \bar M_g,n and Hassett's moduli spaces of weighted pointed curves \bar M_g,A (though the full construction of the moduli spaces is not contained in this paper, only the stability proof). My proof follows Gieseker's approach to reduce to the GIT problem to a combinatorial problem, though the solution is very different. The action of any 1-PS lambda on a curve C in P^N gives rise to weighted filtrations of H^0 (C, O(1)) and H^0 (C, O(m)), and I give a recipe in terms of the combinatorics of the base loci of the stages of these filtrations for showing that C is stable with respect to lambda.