  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.
