Log minimal model program for the moduli space of stable curves: the first flip

Abstract

We give a geometric invariant theory (GIT) construction of the log canonical model $\bar M_g(\alpha)$ of the pairs $(\bar M_g, \alpha \delta)$ for $\alpha \in (7/10 – \epsilon, 7/10]$ for small $\epsilon \in \mathbb Q_+$. We show that $\bar M_g(7/10)$ is isomorphic to the GIT quotient of the Chow variety of bicanonical curves; $\bar M_g(7/10-\epsilon)$ is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions; e.g., they do not contain elliptic chains. We show that there is a small contraction $\Psi: \bar M_g(7/10+\epsilon) \to \bar M_g(7/10)$ that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction $\Psi^+ : \bar M_g(7/10-\epsilon) \to \bar M_g(7/10)$ that is the Mori flip of $\Psi$.

  • [AH] Go to document J. Alper and D. Hyeon, "GIT constructions of log canonical models of $\overline {M}_g$," in Compact Moduli Spaces and Vector Bundles, Providence, RI: Amer. Math. Soc., 2012, vol. 564, pp. 87-106.
    @incollection {AH, MRKEY = {2895185},
      AUTHOR = {Alper, Jarod and Hyeon, Donghoon},
      TITLE = {G{IT} constructions of log canonical models of {$\overline {M}_g$}},
      BOOKTITLE = {Compact Moduli Spaces and Vector Bundles},
      SERIES = {Contemp. Math.},
      VOLUME = {564},
      PAGES = {87--106},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2012},
      MRCLASS = {14H10 (14D23 14E30 14L24)},
      MRNUMBER = {2895185},
      MRREVIEWER = {Hsian-Hua Tseng},
      DOI = {10.1090/conm/564/11152},
      ZBLNUMBER = {06052325},
      }
  • [Bia] Go to document A. Białynicki-Birula, "Some theorems on actions of algebraic groups," Ann. of Math., vol. 98, pp. 480-497, 1973.
    @article {Bia, MRKEY = {0366940},
      AUTHOR = {Bia{\l}ynicki-Birula, A.},
      TITLE = {Some theorems on actions of algebraic groups},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {98},
      YEAR = {1973},
      PAGES = {480--497},
      ISSN = {0003-486X},
      MRCLASS = {14M15 (14L99)},
      MRNUMBER = {0366940},
      MRREVIEWER = {Birger Iversen},
      ZBLNUMBER = {0275.14007},
      DOI = {10.2307/1970915},
     }
  • [BCHM] Go to document C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, "Existence of minimal models for varieties of log general type," J. Amer. Math. Soc., vol. 23, iss. 2, pp. 405-468, 2010.
    @article {BCHM, MRKEY = {2601039},
      AUTHOR = {Birkar, Caucher and Cascini, Paolo and Hacon, Christopher D. and McKernan, James},
      TITLE = {Existence of minimal models for varieties of log general type},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {23},
      YEAR = {2010},
      NUMBER = {2},
      PAGES = {405--468},
      ISSN = {0894-0347},
      MRCLASS = {14E30 (14E05)},
      MRNUMBER = {2601039},
      MRREVIEWER = {Mark Gross},
      DOI = {10.1090/S0894-0347-09-00649-3},
      ZBLNUMBER = {1210.14019},
      }
  • [DolHu] Go to document I. V. Dolgachev and Y. Hu, "Variation of geometric invariant theory quotients," Inst. Hautes Études Sci. Publ. Math., vol. 87, pp. 5-56, 1998.
    @article {DolHu, MRKEY = {1659282},
      AUTHOR = {Dolgachev, Igor V. and Hu, Yi},
      TITLE = {Variation of geometric invariant theory quotients},
      NOTE = {with an appendix by Nicolas Ressayre},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {87},
      YEAR = {1998},
      PAGES = {5--56},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {14L24},
      MRNUMBER = {1659282},
      MRREVIEWER = {P. E. Newstead},
      URL = {http://www.numdam.org/item?id=PMIHES_1998__87__5_0},
      ZBLNUMBER = {1001.14018},
      }
  • [DM] Go to document P. Deligne and D. Mumford, "The irreducibility of the space of curves of given genus," Inst. Hautes Études Sci. Publ. Math., vol. 36, pp. 75-109, 1969.
    @article {DM, MRKEY = {0262240},
      AUTHOR = {Deligne, P. and Mumford, D.},
      TITLE = {The irreducibility of the space of curves of given genus},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {36},
      YEAR = {1969},
      PAGES = {75--109},
      ISSN = {0073-8301},
      MRCLASS = {14.20},
      MRNUMBER = {0262240},
      MRREVIEWER = {Manfred Herrmann},
      ZBLNUMBER = {0181.48803},
      DOI = {10.1007/BF02684599},
     }
  • [EH] Go to document D. Eisenbud and J. Harris, "Limit linear series: basic theory," Invent. Math., vol. 85, iss. 2, pp. 337-371, 1986.
    @article {EH, MRKEY = {0846932},
      AUTHOR = {Eisenbud, David and Harris, Joe},
      TITLE = {Limit linear series: basic theory},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {85},
      YEAR = {1986},
      NUMBER = {2},
      PAGES = {337--371},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14H10 (14C20)},
      MRNUMBER = {0846932},
      MRREVIEWER = {Andrea Del Centina},
      DOI = {10.1007/BF01389094},
      ZBLNUMBER = {0598.14003},
      }
  • [F4] C. Faber, "Intersection-Theoretical Computations on $\overline {M}_g$," in Parameter Spaces, Warsaw: Polish Acad. Sci., 1996, vol. 36, pp. 71-81.
    @incollection {F4, MRKEY = {1481481},
      AUTHOR = {Faber, Carel},
      TITLE = {Intersection-Theoretical Computations on {$\overline {M}_g$}},
      BOOKTITLE = {Parameter Spaces},
      VENUE={{W}arsaw, 1994},
      SERIES = {Banach Center Publ.},
      VOLUME = {36},
      PAGES = {71--81},
      PUBLISHER = {Polish Acad. Sci.},
      ADDRESS = {Warsaw},
      YEAR = {1996},
      MRCLASS = {14H10 (14C15 14C17)},
      MRNUMBER = {1481481},
      MRREVIEWER = {Montserrat Teixidor i Bigas},
      ZBLNUMBER = {0870.14018},
      }
  • [Far] G. Farkas, "The global geometry of the moduli space of curves," in Algebraic Geometry—Seattle 2005. Part 1, Providence, RI: Amer. Math. Soc., 2009, vol. 80, pp. 125-147.
    @incollection {Far, MRKEY = {2483934},
      AUTHOR = {Farkas, Gavril},
      TITLE = {The global geometry of the moduli space of curves},
      BOOKTITLE = {Algebraic Geometry---{S}eattle 2005. {P}art 1},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME = {80},
      PAGES = {125--147},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2009},
      MRCLASS = {14H10 (14C20 14E08 14E30)},
      MRNUMBER = {2483934},
      MRREVIEWER = {Ethan G. Cotterill},
      ZBLNUMBER = {1169.14309},
      }
  • [Fog] J. Fogarty, "Truncated Hilbert functors," J. Reine Angew. Math., vol. 234, pp. 65-88, 1969.
    @article {Fog, MRKEY = {0244268},
      AUTHOR = {Fogarty, John},
      TITLE = {Truncated {H}ilbert functors},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {234},
      YEAR = {1969},
      PAGES = {65--88},
      ISSN = {0075-4102},
      MRCLASS = {14.55},
      MRNUMBER = {0244268},
      MRREVIEWER = {M. Miyanishi},
      ZBLNUMBER = {0197.17101},
      }
  • [Gies] D. Gieseker, Lectures on Moduli of Curves, Published for the Tata Institute of Fundamental Research, Bombay, 1982, vol. 69.
    @book {Gies, MRKEY = {0691308},
      AUTHOR = {Gieseker, D.},
      TITLE = {Lectures on Moduli of Curves},
      SERIES = {Tata Inst. Fund. Res. Lect. Math. and Phys.},
      VOLUME = {69},
      PUBLISHER = {Published for the Tata Institute of Fundamental Research, Bombay},
      YEAR = {1982},
      PAGES = {iii+99},
      ISBN = {3-540-11953-1},
      MRCLASS = {14H10 (14D25 32J05)},
      MRNUMBER = {0691308},
      MRREVIEWER = {Ciro Ciliberto},
      ZBLNUMBER = {0534.14012},
      }
  • [GKM] Go to document A. Gibney, S. Keel, and I. Morrison, "Towards the ample cone of $\overline M_{g,n}$," J. Amer. Math. Soc., vol. 15, iss. 2, pp. 273-294, 2002.
    @article {GKM, MRKEY = {1887636},
      AUTHOR = {Gibney, Angela and Keel, Sean and Morrison, Ian},
      TITLE = {Towards the ample cone of {$\overline M\sb {g,n}$}},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {15},
      YEAR = {2002},
      NUMBER = {2},
      PAGES = {273--294},
      ISSN = {0894-0347},
      MRCLASS = {14H10 (14C17 14E30)},
      MRNUMBER = {1887636},
      MRREVIEWER = {Dan Avritzer},
      DOI = {10.1090/S0894-0347-01-00384-8},
      ZBLNUMBER = {0993.14009},
      }
  • [Gotzmann] G. Gotzmann, "Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes," Math. Z., vol. 158, iss. 1, pp. 61-70, 1978.
    @article {Gotzmann, MRKEY = {0480478},
      AUTHOR = {Gotzmann, Gerd},
      TITLE = {Eine {B}edingung für die {F}lachheit und das {H}ilbertpolynom eines graduierten {R}inges},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {158},
      YEAR = {1978},
      NUMBER = {1},
      PAGES = {61--70},
      ISSN = {0025-5874},
      MRCLASS = {13C10 (14C99)},
      MRNUMBER = {0480478},
      MRREVIEWER = {Knud Lonsted},
      ZBLNUMBER={352.13009},
      }
  • [Has] Go to document B. Hassett, "Classical and minimal models of the moduli space of curves of genus two," in Geometric Methods in Algebra and Number Theory, Boston, MA: Birkhäuser, 2005, vol. 235, pp. 169-192.
    @incollection {Has, MRKEY = {2166084},
      AUTHOR = {Hassett, Brendan},
      TITLE = {Classical and minimal models of the moduli space of curves of genus two},
      BOOKTITLE = {Geometric Methods in Algebra and Number Theory},
      SERIES = {Progr. Math.},
      VOLUME = {235},
      PAGES = {169--192},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {2005},
      MRCLASS = {14H10 (14D22 14E30)},
      MRNUMBER = {2166084},
      MRREVIEWER = {Arvid Siqveland},
      DOI = {10.1007/0-8176-4417-2_8},
      ZBLNUMBER = {1094.14017},
      }
  • [HH1] Go to document B. Hassett and D. Hyeon, "Log canonical models for the moduli space of curves: the first divisorial contraction," Trans. Amer. Math. Soc., vol. 361, iss. 8, pp. 4471-4489, 2009.
    @article {HH1, MRKEY = {2500894},
      AUTHOR = {Hassett, Brendan and Hyeon, Donghoon},
      TITLE = {Log canonical models for the moduli space of curves: the first divisorial contraction},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {361},
      YEAR = {2009},
      NUMBER = {8},
      PAGES = {4471--4489},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {14H10 (14E30)},
      MRNUMBER = {2500894},
      MRREVIEWER = {Hsian-Hua Tseng},
      DOI = {10.1090/S0002-9947-09-04819-3},
      ZBLNUMBER = {1172.14018},
      }
  • [HHL] Go to document B. Hassett, D. Hyeon, and Y. Lee, "Stability computation via Gröbner basis," J. Korean Math. Soc., vol. 47, iss. 1, pp. 41-62, 2010.
    @article {HHL, MRKEY = {2591024},
      AUTHOR = {Hassett, Brendan and Hyeon, Donghoon and Lee, Yongnam},
      TITLE = {Stability computation via {G}röbner basis},
      JOURNAL = {J. Korean Math. Soc.},
      FJOURNAL = {Journal of the Korean Mathematical Society},
      VOLUME = {47},
      YEAR = {2010},
      NUMBER = {1},
      PAGES = {41--62},
      ISSN = {0304-9914},
      CODEN = {JKMSDG},
      MRCLASS = {14C05 (13P10 14H10 14L24 14Q05)},
      MRNUMBER = {2591024},
      MRREVIEWER = {Thomas E. Markwig},
      DOI = {10.4134/JKMS.2010.47.1.041},
      ZBLNUMBER = {1185.14022},
     }
  • [HL1] Go to document D. Hyeon and Y. Lee, "Stability of tri-canonical curves of genus two," Math. Ann., vol. 337, iss. 2, pp. 479-488, 2007.
    @article {HL1, MRKEY = {2262795},
      AUTHOR = {Hyeon, Donghoon and Lee, Yongnam},
      TITLE = {Stability of tri-canonical curves of genus two},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {337},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {479--488},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {14H10 (14E30 14L24)},
      MRNUMBER = {2262795},
      MRREVIEWER = {Arvid Siqveland},
      DOI = {10.1007/s00208-006-0046-2},
      ZBLNUMBER = {1111.14017},
      }
  • [HL2] D. Hyeon and Y. Lee, "Log minimal model program for the moduli space of stable curves of genus three," Math. Res. Lett., vol. 17, iss. 4, pp. 625-636, 2010.
    @article {HL2, MRKEY = {2661168},
      AUTHOR = {Hyeon, Donghoon and Lee, Yongnam},
      TITLE = {Log minimal model program for the moduli space of stable curves of genus three},
      JOURNAL = {Math. Res. Lett.},
      FJOURNAL = {Mathematical Research Letters},
      VOLUME = {17},
      YEAR = {2010},
      NUMBER = {4},
      PAGES = {625--636},
      ISSN = {1073-2780},
      MRCLASS = {14H10 (14E30)},
      MRNUMBER = {2661168},
      MRREVIEWER = {Gianfranco Casnati},
      ZBLNUMBER = {1230.14035},
      }
  • [HM] Go to document J. Harris and D. Mumford, "On the Kodaira dimension of the moduli space of curves," Invent. Math., vol. 67, iss. 1, pp. 23-88, 1982.
    @article {HM, MRKEY = {0664324},
      AUTHOR = {Harris, Joe and Mumford, David},
      TITLE = {On the {K}odaira dimension of the moduli space of curves},
      NOTE = {with an appendix by William Fulton},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {67},
      YEAR = {1982},
      NUMBER = {1},
      PAGES = {23--88},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14H10},
      MRNUMBER = {0664324},
      MRREVIEWER = {Yujiro Kawamata},
      DOI = {10.1007/BF01393371},
      ZBLNUMBER = {0506.14016},
      }
  • [Kempf] Go to document G. R. Kempf, "Instability in invariant theory," Ann. of Math., vol. 108, iss. 2, pp. 299-316, 1978.
    @article {Kempf, MRKEY = {0506989},
      AUTHOR = {Kempf, George R.},
      TITLE = {Instability in invariant theory},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {108},
      YEAR = {1978},
      NUMBER = {2},
      PAGES = {299--316},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {20G05 (14L30 14M99 15A72)},
      MRNUMBER = {0506989},
      MRREVIEWER = {V. L. Popov},
      DOI = {10.2307/1971168},
      ZBLNUMBER = {0406.14031},
      }
  • [Kn] F. F. Knudsen and D. Mumford, "The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”," Math. Scand., vol. 39, iss. 1, pp. 19-55, 1976.
    @article {Kn, MRKEY = {0437541},
      AUTHOR = {Knudsen, Finn Faye and Mumford, David},
      TITLE = {The projectivity of the moduli space of stable curves. {I}. {P}reliminaries on ``det'' and ``{D}iv''},
      JOURNAL = {Math. Scand.},
      FJOURNAL = {Mathematica Scandinavica},
      VOLUME = {39},
      YEAR = {1976},
      NUMBER = {1},
      PAGES = {19--55},
      ISSN = {0025-5521},
      MRCLASS = {14H10 (14F05 14C05)},
      MRNUMBER = {0437541},
      MRREVIEWER = {P. E. Newstead},
      ZBLNUMBER = {0343.14008},
     }
  • [KM] Go to document J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge: Cambridge Univ. Press, 1998, vol. 134.
    @book {KM, MRKEY = {1658959},
      AUTHOR = {Koll{á}r, J{á}nos and Mori, Shigefumi},
      TITLE = {Birational Geometry of Algebraic Varieties},
      SERIES = {Cambridge Tracts in Math.},
      VOLUME = {134},
      NOTE = {with the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1998},
      PAGES = {viii+254},
      ISBN = {0-521-63277-3},
      MRCLASS = {14E30},
      MRNUMBER = {1658959},
      MRREVIEWER = {Mark Gross},
      DOI = {10.1017/CBO9780511662560},
      ZBLNUMBER = {0926.14003},
      }
  • [Kol] J. Kollár, Rational Curves on Algebraic Varieties, New York: Springer-Verlag, 1996, vol. 32.
    @book {Kol, MRKEY = {1440180},
      AUTHOR = {Koll{á}r, J{á}nos},
      TITLE = {Rational Curves on Algebraic Varieties},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {32},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1996},
      PAGES = {viii+320},
      ISBN = {3-540-60168-6},
      MRCLASS = {14-02 (14C05 14E05 14F17 14J45)},
      MRNUMBER = {1440180},
      MRREVIEWER = {Yuri G. Prokhorov},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0877.14012},
      }
  • [GIT] D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, Third ed., New York: Springer-Verlag, 1994, vol. 34.
    @book {GIT, MRKEY = {1304906},
      AUTHOR = {Mumford, David and Fogarty, J. and Kirwan, F.},
      TITLE = {Geometric Invariant Theory},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {34},
      EDITION = {Third},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1994},
      PAGES = {xiv+292},
      ISBN = {3-540-56963-4},
      MRCLASS = {14D25 (58E05 58F05)},
      MRNUMBER = {1304906},
      MRREVIEWER = {Yi Hu},
      ZBLNUMBER = {0797.14004},
      }
  • [Morrison] Go to document I. Morrison, "Projective stability of ruled surfaces," Invent. Math., vol. 56, iss. 3, pp. 269-304, 1980.
    @article {Morrison, MRKEY = {0561975},
      AUTHOR = {Morrison, Ian},
      TITLE = {Projective stability of ruled surfaces},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {56},
      YEAR = {1980},
      NUMBER = {3},
      PAGES = {269--304},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14D25 (14C05)},
      MRNUMBER = {0561975},
      MRREVIEWER = {T. Oda},
      DOI = {10.1007/BF01390049},
      ZBLNUMBER = {0423.14005},
      }
  • [Mor] Go to document A. Moriwaki, "Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves," J. Amer. Math. Soc., vol. 11, iss. 3, pp. 569-600, 1998.
    @article {Mor, MRKEY = {1488349},
      AUTHOR = {Moriwaki, Atsushi},
      TITLE = {Relative {B}ogomolov's inequality and the cone of positive divisors on the moduli space of stable curves},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {11},
      YEAR = {1998},
      NUMBER = {3},
      PAGES = {569--600},
      ISSN = {0894-0347},
      MRCLASS = {14H10 (14C20 14G40)},
      MRNUMBER = {1488349},
      MRREVIEWER = {Montserrat Teixidor i Bigas},
      DOI = {10.1090/S0894-0347-98-00261-6},
      ZBLNUMBER = {0893.14004},
      }
  • [Mukai] S. Mukai, An Introduction to Invariants and Moduli, Cambridge: Cambridge University Press, 2003, vol. 81.
    @book{Mukai, MRKEY={2004218},
      AUTHOR={Mukai, S.},
      TITLE={An Introduction to Invariants and Moduli},
      SERIES={Cambridge Stud. Adv. Math.},
      VOLUME={81},
      PUBLISHER={Cambridge University Press},
      ADDRESS={Cambridge},
      YEAR={2003},
      NOTE={translated from the 1998 and 2000 Japanese editions by W. M. Oxbury},
      MRNUMBER = {2004218},
      ZBLNUMBER = {1033.14008},
     }
  • [M] D. Mumford, "Stability of projective varieties," Enseignement Math., vol. 23, iss. 1-2, pp. 39-110, 1977.
    @article {M, MRKEY = {0450272},
      AUTHOR = {Mumford, David},
      TITLE = {Stability of projective varieties},
      JOURNAL = {Enseignement Math.},
      FJOURNAL = {L'Enseignement Mathématique. Revue Internationale. IIe Série},
      VOLUME = {23},
      YEAR = {1977},
      NUMBER = {1-2},
      PAGES = {39--110},
      ISSN = {0013-8584},
      MRCLASS = {14D20},
      MRNUMBER = {0450272},
      MRREVIEWER = {P. E. Newstead},
      ZBLNUMBER = {0363.14003},
      }
  • [SB] Go to document N. I. Shepherd-Barron, "Perfect forms and the moduli space of abelian varieties," Invent. Math., vol. 163, iss. 1, pp. 25-45, 2006.
    @article {SB, MRKEY = {2208417},
      AUTHOR = {Shepherd-Barron, N. I.},
      TITLE = {Perfect forms and the moduli space of abelian varieties},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {163},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {25--45},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14K10},
      MRNUMBER = {2208417},
      MRREVIEWER = {David Lehavi},
      DOI = {10.1007/s00222-005-0453-0},
      ZBLNUMBER = {1088.14011},
      }
  • [Sch] Go to document D. Schubert, "A new compactification of the moduli space of curves," Compositio Math., vol. 78, iss. 3, pp. 297-313, 1991.
    @article {Sch, MRKEY = {1106299},
      AUTHOR = {Schubert, David},
      TITLE = {A new compactification of the moduli space of curves},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {78},
      YEAR = {1991},
      NUMBER = {3},
      PAGES = {297--313},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14H10},
      MRNUMBER = {1106299},
      MRREVIEWER = {R. F. Lax},
      URL = {http://www.numdam.org/item?id=CM_1991__78_3_297_0},
      ZBLNUMBER = {0735.14022},
      }
  • [Ses1] Go to document C. S. Seshadri, "Quotient spaces modulo reductive algebraic groups," Ann. of Math., vol. 95, pp. 511-556; errata, ibid. (2) 96 (1972), 599 (\zbl0248.14013, \doi10.2307/1970828), 1972.
    @article {Ses1, MRKEY = {0309940},
      AUTHOR = {Seshadri, C. S.},
      TITLE = {Quotient spaces modulo reductive algebraic groups},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {95},
      YEAR = {1972},
      PAGES = {511--556; errata, ibid. (2) 96 (1972), 599 (\zbl{0248.14013},
      \doi{10.2307/1970828})},
      ISSN = {0003-486X},
      MRCLASS = {14F05 (14C99 14D20)},
      MRNUMBER = {0309940},
      MRREVIEWER = {S. L. Kleiman},
      ZBLNUMBER = {0241.14024},
      DOI = {10.2307/1970870},
     }

Authors

Brendan Hassett

Department of Matheamtics-MS136, Rice University, 6100 S. Main St., Houston, TX 77251-1892

Donghoon Hyeon

Department of Mathematics, POSTECH San 31, Hyojadong, Namgu, Pohang, Gyungbu 790-784, Republic of Korea