Birational contractions of \overline{𝑀}_{3,1} and \overline{𝑀}_{4,1}

Jensen, David

doi:10.1090/S0002-9947-2012-05581-4

Birational contractions of $\overline {M}_{3,1}$ and $\overline {M}_{4,1}$
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by David Jensen PDF

Trans. Amer. Math. Soc. 365 (2013), 2863-2879 Request permission

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 – the pointed Brill-Noether divisors – 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), 1–26. MR 2591089, DOI 10.1017/S0027763000009776
Fernando Cukierman and Lung-Ying Fong, On higher Weierstrass points, Duke Math. J. 62 (1991), no. 1, 179–203. 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), 29–65. MR 2561195, DOI 10.1515/CRELLE.2009.059
Fernando Cukierman, Determinant of complexes and higher Hessians, Math. Ann. 307 (1997), no. 2, 225–251. MR 1428872, DOI 10.1007/s002080050032
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, 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. 2, 359–387. MR 910206, DOI 10.1007/BF01388710
Gavril Farkas, Koszul divisors on moduli spaces of curves, Amer. J. Math. 131 (2009), no. 3, 819–867. 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. 2, 241–267. 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. 8, 4471–4489. 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. 4, 625–636. 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. 1, 23–88. 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. 1, 105–138. 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.)–The University of Texas at Austin. MR 2701950
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