GIT stability of weighted pointed curves
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- Trans. Amer. Math. Soc. 364 (2012), 1737-1770 Request permission
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 $\lambda$.References
- Elizabeth Baldwin and David Swinarski, A geometric invariant theory construction of moduli spaces of stable maps, Int. Math. Res. Pap. IMRP 1 (2008), Art. ID rp. 004, 104. MR 2431236
- Igor V. Dolgachev and Yi Hu, Variation of geometric invariant theory quotients, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 5–56. With an appendix by Nicolas Ressayre. MR 1659282
- D. Gieseker, Lectures on moduli of curves, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 69, Published for the Tata Institute of Fundamental Research, Bombay by Springer-Verlag, Berlin-New York, 1982. MR 691308
- D. Gieseker, Geometric invariant theory and applications to moduli problems, Invariant theory (Montecatini, 1982) Lecture Notes in Math., vol. 996, Springer, Berlin, 1983, pp. 45–73. MR 718126, DOI 10.1007/BFb0063235
- Joe Harris and Ian Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998. MR 1631825
- Brendan Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), no. 2, 316–352. MR 1957831, DOI 10.1016/S0001-8708(02)00058-0
- Brendan Hassett and Donghoon Hyeon, Log canonical models for the moduli space of curves: the first divisorial contraction, Trans. Amer. Math. Soc. 361 (2009), no. 8, 4471–4489. MR 2500894, DOI 10.1090/S0002-9947-09-04819-3
- Ian Morrison, Projective stability of ruled surfaces, Invent. Math. 56 (1980), no. 3, 269–304. MR 561975, DOI 10.1007/BF01390049
- David Mumford and John Fogarty, Geometric invariant theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 34, Springer-Verlag, Berlin, 1982. MR 719371, DOI 10.1007/978-3-642-96676-7
- David Mumford, Stability of projective varieties, Enseign. Math. (2) 23 (1977), no. 1-2, 39–110. MR 450272
- Francis Ronald Allaire, ON REDUCIBLE CONFIGURATIONS FOR THE FOUR-COLOUR PROBLEM, ProQuest LLC, Ann Arbor, MI, 1977. Thesis (Ph.D.)–University of Manitoba (Canada). MR 2626882
- Michael Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), no. 3, 691–723. MR 1333296, DOI 10.1090/S0894-0347-96-00204-4
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
- MR Author ID: 844110
- Email: davids@math.uga.edu, dswinarski@fordham.edu
- Received by editor(s): August 25, 2009
- Received by editor(s) in revised form: March 26, 2010
- Published electronically: November 15, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2869190