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GIT stability of weighted pointed curves


Author: David Swinarski
Journal: Trans. Amer. Math. Soc. 364 (2012), 1737-1770
MSC (2010): Primary 14L24, 14H10; Secondary 14D22
DOI: https://doi.org/10.1090/S0002-9947-2011-05360-2
Published electronically: November 15, 2011
MathSciNet review: 2869190
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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 $.


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Additional Information

David Swinarski
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30606
Address at time of publication: Department of Mathematics, Fordham University, New York, New York 10023
Email: davids@math.uga.edu, dswinarski@fordham.edu

DOI: https://doi.org/10.1090/S0002-9947-2011-05360-2
Keywords: Hilbert stability
Received by editor(s): August 25, 2009
Received by editor(s) in revised form: March 26, 2010
Published electronically: November 15, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.