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 and Hassett's moduli spaces of weighted pointed curves . (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 on a curve gives rise to weighted filtrations of and . We give a recipe in terms of the combinatorics of the base loci of the stages of these filtrations for showing that is stable with respect to .

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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.