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Modular compactifications of the space of pointed elliptic curves II

Part of: Birational geometry Curves

Published online by Cambridge University Press:  30 August 2011

David Ishii Smyth
David Ishii Smyth*
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138, USA (email: [email protected])
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Abstract

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We prove that the moduli spaces of n-pointed m-stable curves introduced in our previous paper have projective coarse moduli. We use the resulting spaces to run an analogue of Hassett’s log minimal model program for .

Keywords

moduli of curveselliptic curvesminimal model program

MSC classification

Secondary: 14H10: Families, moduli (algebraic) 14H20: Singularities, local rings 14E30: Minimal model program (Mori theory, extremal rays)
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 6 , November 2011 , pp. 1843 - 1884
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[AC98]Arbarello, E. and Cornalba, M., Calculating cohomology groups of moduli spaces of curves via algebraic geometry, Publ. Math. Inst. Hautes Études Sci. 88 (1998), 97–127, 1998.CrossRefGoogle Scholar
[CH88]Cornalba, M. and Harris, J., Divisor classes associated to families of stable varieties, with applications to the moduli space of curves, Ann. Sci. Éc. Norm. Supér. (4) 21 (1988), 455–475.CrossRefGoogle Scholar
[Deb01]Debarre, O., Higher-dimensional algebraic geometry, Universitext (Springer, New York, 2001).CrossRefGoogle Scholar
[DM69]Deligne, P. and Mumford, D., The irreducibility of the space of curves of given genus, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 75–109.CrossRefGoogle Scholar
[FS10]Fedorchuk, M. and Smyth, D., Ample divisors on moduli spaces of pointed rational curves, J. Algebraic Geom., to appear, Preprint (2010), arXiv: 0810.1677.Google Scholar
[GLS07]Greuel, G.-M., Lossen, C. and Shustin, E., Introduction to singularities and deformations, Springer Monographs in Mathematics (Springer, Berlin, 2007).Google Scholar
[HM98]Harris, J. and Morrison, I., Moduli of curves, Graduate Texts in Mathematics, vol. 187 (Springer, New York, 1998).Google Scholar
[Har10]Hartshorne, R., Deformation theory, Graduate Texts in Mathematics, vol. 257 (Springer, New York, 2010).CrossRefGoogle Scholar
[Has05]Hassett, B., Classical and minimal models of the moduli space of curves of genus two, in Geometric methods in algebra and number theory, Progress in Mathematics, vol. 235 (Birkhäuser, Boston, MA, 2005), 169–192.CrossRefGoogle Scholar
[HH06]Hassett, B. and Hyeon, D., Log minimal model program for the moduli space of curves: the first flip, Preprint (2006), arXiv: 0806.3444.Google Scholar
[HH09]Hassett, B. and Hyeon, D., Log canonical models for the moduli space of curves: the first divisorial contraction, Trans. Amer. Math. Soc. 361 (2009), 4471–4489.CrossRefGoogle Scholar
[Kol96]Kollár, J., Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32 (Springer, Berlin, 1996).CrossRefGoogle Scholar
[Ser06]Sernesi, E., Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 334 (Springer, Berlin, 2006).Google Scholar
[Smy11]Smyth, D., Modular compactifications of the space of pointed elliptic curves I, Compositio Math. 147 (2011), 877–913.CrossRefGoogle Scholar
[Vis89]Vistoli, A., Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), 613–670.CrossRefGoogle Scholar