Birational contractions of $\overline {M}_{3,1}$ and $\overline {M}_{4,1}$
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Abstract:
We study the birational geometry of $\overline {M}_{3,1}$ and $\overline {M}_{4,1}$. In particular, we pose a pointed analogue of the Slope Conjecture and prove it in these low-genus cases. Using variation of GIT, we construct birational contractions of these spaces in which certain divisors of interest β the pointed Brill-Noether divisors β are contracted. As a consequence, we see that these pointed Brill-Noether divisors generate extremal rays of the effective cones for these spaces.References
- Michela Artebani, A compactification of $\scr M_3$ via $K3$ surfaces, Nagoya Math. J. 196 (2009), 1β26. MR 2591089, DOI 10.1017/S0027763000009776
- Fernando Cukierman and Lung-Ying Fong, On higher Weierstrass points, Duke Math. J. 62 (1991), no.Β 1, 179β203. MR 1104328, DOI 10.1215/S0012-7094-91-06208-3
- Sebastian Casalaina-Martin and Radu Laza, The moduli space of cubic threefolds via degenerations of the intermediate Jacobian, J. Reine Angew. Math. 633 (2009), 29β65. MR 2561195, DOI 10.1515/CRELLE.2009.059
- Fernando Cukierman, Determinant of complexes and higher Hessians, Math. Ann. 307 (1997), no.Β 2, 225β251. MR 1428872, DOI 10.1007/s002080050032
- Igor V. Dolgachev and Yi Hu, Variation of geometric invariant theory quotients, Inst. Hautes Γtudes Sci. Publ. Math. 87 (1998), 5β56. With an appendix by Nicolas Ressayre. MR 1659282, DOI 10.1007/BF02698859
- David Eisenbud and Joe Harris, The Kodaira dimension of the moduli space of curves of genus $\geq 23$, Invent. Math. 90 (1987), no.Β 2, 359β387. MR 910206, DOI 10.1007/BF01388710
- Gavril Farkas, Koszul divisors on moduli spaces of curves, Amer. J. Math. 131 (2009), no.Β 3, 819β867. MR 2530855, DOI 10.1353/ajm.0.0053
- Gavril Farkas and Mihnea Popa, Effective divisors on $\overline {\scr M}_g$, curves on $K3$ surfaces, and the slope conjecture, J. Algebraic Geom. 14 (2005), no.Β 2, 241β267. MR 2123229, DOI 10.1090/S1056-3911-04-00392-3
- 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.Β 8, 4471β4489. MR 2500894, DOI 10.1090/S0002-9947-09-04819-3
- Brendan Hassett and Donghoon Hyeon. Log canonical models for the moduli space of curves: the first flip. preprint, 2009.
- Donghoon Hyeon and Yongnam Lee, Log minimal model program for the moduli space of stable curves of genus three, Math. Res. Lett. 17 (2010), no.Β 4, 625β636. MR 2661168, DOI 10.4310/MRL.2010.v17.n4.a4
- Joe Harris and David Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no.Β 1, 23β88. With an appendix by William Fulton. MR 664324, DOI 10.1007/BF01393371
- Adam Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math. 125 (2003), no.Β 1, 105β138. MR 1953519, DOI 10.1353/ajm.2003.0005
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
- William Frederick Rulla, The birational geometry of moduli space M(3) and moduli space M(2,1), ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)βThe University of Texas at Austin. MR 2701950
- Michael Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), no.Β 3, 691β723. MR 1333296, DOI 10.1090/S0894-0347-96-00204-4
Additional Information
- David Jensen
- Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794
- Received by editor(s): October 19, 2010
- Received by editor(s) in revised form: February 3, 2011, and March 6, 2011
- Published electronically: November 27, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 2863-2879
- MSC (2010): Primary 14H10, 14E30
- DOI: https://doi.org/10.1090/S0002-9947-2012-05581-4
- MathSciNet review: 3034451