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On the Automorphisms of Moduli Spaces of Curves

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Automorphisms in Birational and Affine Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 79))

Abstract

In the last years the biregular automorphisms of Deligne–Mumford’s and Hassett’s compactifications of the moduli space of n-pointed genus g smooth curves have been extensively studied by A. Bruno and the authors. In this paper we give a survey of these recent results and extend our techniques to some moduli spaces appearing as intermediate steps of Kapranov’s and Keel’s realizations of \(\overline{M}_{0,n}\), and to the degenerations of Hassett’s spaces obtained by allowing zero weights.

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Notes

  1. 1.

    Partially supported by Progetto PRIN 2010 “Geometria sulle varietà algebriche” MIUR and GRIFGA

References

  1. A. Bruno, M. Mella, On some fibrations of \(\overline{M}_{0,n}\). (2011) http://arxiv.org/abs/1105.3293v1 [arXiv:1105.3293v1]

  2. A. Bruno, M. Mella, The automorphism group of \(\overline{M}_{0,n}\). J. Eur. Math. Soc. 15(3), 949–968 (2013)

    Google Scholar 

  3. P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus. Inst. Hautes \(\mathrm{\acute{E}}\) tudes Sci. Publ. Math. 36, 75109 (1969)

    Google Scholar 

  4. I.V. Dolgachev, V.A. Iskovskikh, Finite subgroups of the plane Cremona group, in Algebra, Arithmetic, and Geometry: In Honor of Y.I. Manin, Vol. I. Progress in Mathematics, vol. 269 (Birkhäuser, Boston, 2009), pp. 443–548

    Google Scholar 

  5. G. Farkas, The global geometry of the moduli space of curves. Proc. Symp. Pure Math. 80, 125–147 (2009)

    Article  MathSciNet  Google Scholar 

  6. A. Gibney, S. Keel, I. Morrison, Towards the ample cone of \(\overline{M}_{g,n}\). J. Am. Math. Soc. 15, 273–294 (2002)

    Google Scholar 

  7. B. Hassett, Moduli spaces of weighted pointed stable curves. Adv. Math. 173(2), 316–352 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Kapranov, Veronese curves and Grothendieck-Knudsen moduli spaces \(\overline{M}_{0,n}\). J. Algebr. Geom. 2, 239–262 (1993)

    Google Scholar 

  9. S. Keel, Intersection theory of moduli space of stable N-pointed curves of genus zero. Trans. Am. Math. Soc. 330(2), 545–574 (1992)

    MATH  MathSciNet  Google Scholar 

  10. S. Keel, J. McKernan, Contractible extremal rays of \(\overline{M}_{0,n}\). (1997) http://arxiv.org/abs/alg-geom/9607009v1 [arXiv:alg-geom/9607009v1]

  11. A. Massarenti, The automorphism group of \(\overline{M}_{g,n}\). J. Lond. Math. Soc. (2013). doi: 10.1112/jlms/jdt057

    Google Scholar 

  12. A. Massarenti, M. Mella, On the automorphisms of Hassett’s moduli spaces. http://arxiv.org/abs/1307.6828v1 [arXiv:1307.6828v1]

  13. S. Mochizuki, Correspondences on hyperbolic curves. J. Pure Appl. Algebr. 131, 227–244 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. H. Moon, Log canonical models for \(\overline{M}_{g,n}\). http://arxiv.org/abs/1111.5354 [arXiv:1111.5354]

  15. H.L. Royden, Automorphisms and isometries of Teichmüller spaces, in Advances in the Theory of Riemann Surfaces, ed. by L.V. Ahlfors, L. Bers, H.M. Farkas, R.C. Gunning, I. Kra, H.E. Rauch. Annals of Mathematics Studies, vol. 66 (Princeton University Press, Princeton, 1971), pp. 369–383

    Google Scholar 

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Correspondence to Alex Massarenti .

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Massarenti, A., Mella, M. (2014). On the Automorphisms of Moduli Spaces of Curves. In: Cheltsov, I., Ciliberto, C., Flenner, H., McKernan, J., Prokhorov, Y., Zaidenberg, M. (eds) Automorphisms in Birational and Affine Geometry. Springer Proceedings in Mathematics & Statistics, vol 79. Springer, Cham. https://doi.org/10.1007/978-3-319-05681-4_9

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