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Introduction

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2122)

Abstract

One of the first successful applications of Geometric Invariant Theory (GIT for short), and perhaps one of the major motivations for its development by Mumford and his co-authors (see [MFK94]), was the construction of the moduli space M g of smooth curves of genus g ≥ 2 and its compactification \(\overline{M}_{g}\) via stable curves (i.e. connected nodal projective curves with finite automorphism group), carried out by Mumford [Mum77] and Gieseker [Gie82].

Keywords

  • Modulus Space
  • Line Bundle
  • Hilbert Scheme
  • Stable Curf
  • Semistable Point

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Fig. 1.1
Fig. 1.2

Notes

  1. 1.

    Notice that Li-Wang worked more generally with polarized pointed weighted nodal curves.

  2. 2.

    In particular, when working with Hilb d , we will always consider the m-linearization for m ≫ 0; see Sect. 4.1 for details.

References

  1. J. Alper, M. Fedorchuk, D. Smyth, Singularities with \(\mathbb{G}_{m}\)-action and the log minimal model program for \(\overline{M}_{g}\). Journal für die reine und angewandte Mathematik. Preprint available at arXiv:1010.3751 (to appear)

    Google Scholar 

  2. J. Alper, M. Fedorchuk, D. Smyth, Finite Hilbert stability of (bi)canonical curves. Invent. Math. 191, 671–718 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. J. Alper, D. Hyeon, GIT construction of log canonical models of \(\overline{M}_{g}\), in Compact Moduli Spaces and Vector Bundles. Contemporary Mathematics, vol. 564 (American Mathematical Society, Providence, 2012), pp. 87–106

    Google Scholar 

  4. A.B. Altman, S.L. Kleiman, Bertini theorems for hypersurface sections containing a subscheme. Commun. Algebra 7, 775–790 (1979)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. G. Bini, M. Melo, F. Viviani, On GIT quotients of Hilbert and Chow schemes of curves. Electron. Res. Announc. Math. Sci. 19, 33–40 (2012)

    MATH  MathSciNet  Google Scholar 

  6. L. Caporaso, A compactification of the universal Picard variety over the moduli space of stable curves. J. Am. Math. Soc. 7, 589–660 (1994)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. L. Caporaso, Néron models and compactified Picard schemes over the moduli stack of stable curves. Am. J. Math. 130(1), 1–47 (2008)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. I. Dolgachev, Lectures on Invariant Theory. London Mathematical Society Lecture Note Series, vol. 296 (Cambridge University Press, Cambridge, 2003)

    Google Scholar 

  9. M. Fedorchuk, D.I. Smyth, Alternate compactifications of moduli space of curves, in Handbook of Moduli: Volume I, ed. by G. Farkas, I. Morrison, Advanced Lectures in Mathematics, vol. 24 (International Press of Boston, Inc, 2013), pp. 331–414

    Google Scholar 

  10. F. Felici, GIT for Hilbert and Chow schemes of curves. Ph.D. thesis, Roma Tre University, 2014

    Google Scholar 

  11. D. Gieseker, Lectures on Moduli of Curves. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 69 (Tata Institute of Fundamental Research, Bombay, 1982)

    Google Scholar 

  12. J. Harris, I. Morrison, Moduli of Curves. Graduate Text in Mathematics, vol. 187 (Springer, New York/Heidelberg, 1998)

    Google Scholar 

  13. B. Hassett, D. Hyeon, Log canonical models for the moduli space of curves: first divisorial contraction. Trans. Am. Math. Soc. 361, 4471–4489 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. B. Hassett, D. Hyeon, Log canonical models for the moduli space of curves: the first flip. Ann. Math. (2) 177, 911–968 (2013)

    Google Scholar 

  15. D. Hyeon, I. Morrison, Stability of tails and 4-canonical models. Math. Res. Lett. 17(4), 721–729 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  16. J. Li, X. Wang, Hilbert-Mumford criterion for nodal curves. Preprint available at arXiv: 1108.1727v1

    Google Scholar 

  17. M. Melo, Compactified Picard stacks over \(\overline{\mathcal{M}}_{g}\). Math. Zeit. 263(4), 939–957 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. I. Morrison, GIT constructions of moduli spaces of stable curves and maps, in Geometry of Riemann Surfaces and Their Moduli Spaces, ed. by L. Ji et al. Surveys in Differential Geometry, vol. 14 (International Press, Somerville, 2010), pp. 315–369

    Google Scholar 

  19. D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34, 3rd edn. (Springer, Berlin, 1994)

    Google Scholar 

  20. D. Mumford, Stability of projective varieties. Enseignement Math. (2) 23, 39–110 (1977)

    Google Scholar 

  21. D. Schubert, A new compactification of the moduli space of curves. Compositio Math. 78, 297–313 (1991)

    MATH  MathSciNet  Google Scholar 

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Bini, G., Felici, F., Melo, M., Viviani, F. (2014). Introduction. In: Geometric Invariant Theory for Polarized Curves. Lecture Notes in Mathematics, vol 2122. Springer, Cham. https://doi.org/10.1007/978-3-319-11337-1_1

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