GIT stability of weighted pointed curves

Abstract:

Here we 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 $\overline {M}_{g,n}$ and Hassett’s moduli spaces of weighted pointed curves $\overline {{M}}_{g,\mathcal {A}}$. (The full construction of the moduli spaces is not given here, only the stability proof.) This proof follows Gieseker’s approach to reduce the GIT problem to a combinatorial problem, although the solution is very different. The action of any 1-PS $\lambda$ on a curve $C \subset \mathbf {P}^N$ gives rise to weighted filtrations of $H^{0}(C, \mathcal {O}_{C}(1))$ and $H^{}(C, \mathcal {O}_{C}(m))$. We 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 $\lambda$.