2008/01/08 by David Swinarski, Swinarski, David
Mathematics · #14H10 #14L24 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.0801.1288
openalex publication_date 2008/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Mg,n and Hassett's moduli spaces of weighted pointed curves Mg,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 PN gives rise to weighted filtrations of H0 (C, O(1)) and H0 (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.