\begin{thebibliography}{BCHM06}

\bibitem[BB63]{Bia63}
Andrzej Bia{\l}ynicki-Birula.
\newblock On homogeneous affine spaces of linear algebraic groups.
\newblock {\em Amer. J. Math.}, 85:577--582, 1963.

\bibitem[BB73]{Bia}
Andrzej Bia{\l}ynicki-Birula.
\newblock Some theorems on actions of algebraic groups.
\newblock {\em Ann. of Math. (2)}, 98:480--497, 1973.

\bibitem[BCHM06]{BCHM}
Caucher Birkar, Paolo Cascini, Christopher~D. Hacon, and James McKernan.
\newblock Existence of minimal models for varieties of log general type, 2006.
\newblock arXiv.org:math/0610203.

\bibitem[CH88]{CH}
Maurizio Cornalba and Joe Harris.
\newblock Divisor classes associated to families of stable varieties, with
  applications to the moduli space of curves.
\newblock {\em Ann. Sci. \'Ecole Norm. Sup. (4)}, 21(3):455--475, 1988.

\bibitem[DH98]{DolHu}
Igor~V. Dolgachev and Yi~Hu.
\newblock Variation of geometric invariant theory quotients.
\newblock {\em Inst. Hautes \'Etudes Sci. Publ. Math.}, (87):5--56, 1998.
\newblock With an appendix by Nicolas Ressayre.

\bibitem[DM69]{DM}
Pierre Deligne and David Mumford.
\newblock The irreducibility of the space of curves of given genus.
\newblock {\em Inst. Hautes \'Etudes Sci. Publ. Math.}, (36):75--109, 1969.

\bibitem[EH86]{EH}
David Eisenbud and Joe Harris.
\newblock Limit linear series: basic theory.
\newblock {\em Invent. Math.}, 85(2):337--371, 1986.

\bibitem[Eis05]{EisGoS}
David Eisenbud.
\newblock {\em The geometry of syzygies}, volume 229 of {\em Graduate Texts in
  Mathematics}.
\newblock Springer-Verlag, New York, 2005.
\newblock A second course in commutative algebra and algebraic geometry.

\bibitem[Fab96]{F4}
Carel Faber.
\newblock Intersection-theoretical computations on {$\overline {\scr M}\sb g$}.
\newblock In {\em Parameter spaces (Warsaw, 1994)}, volume~36 of {\em Banach
  Center Publ.}, pages 71--81. Polish Acad. Sci., Warsaw, 1996.

\bibitem[Far06]{Far}
Gavril Farkas.
\newblock The global geometry of the moduli space of curves, 2006.
\newblock arXiv:math/0612251v1.

\bibitem[Fog69]{Fog}
John Fogarty.
\newblock Truncated {H}ilbert functors.
\newblock {\em J. Reine Angew. Math.}, 234:65--88, 1969.

\bibitem[Gie82]{Gies}
David Gieseker.
\newblock {\em Lectures on moduli of curves}, volume~69 of {\em Tata Institute
  of Fundamental Research Lectures on Mathematics and Physics}.
\newblock Published for the Tata Institute of Fundamental Research, Bombay,
  1982.

\bibitem[GKM02]{GKM}
Angela Gibney, Sean Keel, and Ian Morrison.
\newblock Towards the ample cone of {$\overline M\sb {g,n}$}.
\newblock {\em J. Amer. Math. Soc.}, 15(2):273--294 (electronic), 2002.

\bibitem[Has05]{Has}
Brendan Hassett.
\newblock Classical and minimal models of the moduli space of curves of genus
  two.
\newblock In {\em Geometric methods in algebra and number theory}, volume 235
  of {\em Progress in mathematics}, pages 160--192. Birkh\"auser, Boston, 2005.

\bibitem[HHL07]{HHL}
Brendan Hassett, Donghoon Hyeon, and Yongnam Lee.
\newblock Stability computation via {G}r\"obner basis, 2007.
\newblock Preprint.

\bibitem[HHar]{HH1}
Brendan Hassett and Donghoon Hyeon.
\newblock Log canonical models for the moduli space of curves: First divisorial
  contraction.
\newblock {\em Trans. Amer. Math. Soc.}, to appear.
\newblock arXiv:math.AG/0607477.

\bibitem[HL07a]{HL2}
Donghoon Hyeon and Yongnam Lee.
\newblock Log minimal model program for the moduli space of stable curves of
  genus three, 2007.
\newblock arXiv.org:math/0703093.

\bibitem[HL07b]{HL1}
Donghoon Hyeon and Yongnam Lee.
\newblock Stability of tri-canonical curves of genus two.
\newblock {\em Math. Ann.}, 337(2):479--488, 2007.

\bibitem[HM82]{HM}
Joe Harris and David Mumford.
\newblock On the {K}odaira dimension of the moduli space of curves.
\newblock {\em Invent. Math.}, 67(1):23--88, 1982.
\newblock With an appendix by William Fulton.

\bibitem[Kem78]{Kempf}
George~R. Kempf.
\newblock Instability in invariant theory.
\newblock {\em Ann. of Math. (2)}, 108(2):299--316, 1978.

\bibitem[KM76]{Kn}
Finn~Faye Knudsen and David Mumford.
\newblock The projectivity of the moduli space of stable curves. {I}.
  {P}reliminaries on ``det'' and ``{D}iv''.
\newblock {\em Math. Scand.}, 39(1):19--55, 1976.

\bibitem[KM98]{KM}
J{\'a}nos Koll{\'a}r and Shigefumi Mori.
\newblock {\em Birational geometry of algebraic varieties}, volume 134 of {\em
  Cambridge Tracts in Mathematics}.
\newblock Cambridge University Press, Cambridge, 1998.
\newblock With the collaboration of C. H. Clemens and A. Corti, Translated from
  the 1998 Japanese original.

\bibitem[Kol96]{Kol}
J{\'a}nos Koll{\'a}r.
\newblock {\em Rational curves on algebraic varieties}, volume~32 of {\em
  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]}.
\newblock Springer-Verlag, Berlin, 1996.

\bibitem[Mat60]{Mat}
Yoz{\^o} Matsushima.
\newblock Espaces homog\`enes de {S}tein des groupes de {L}ie complexes.
\newblock {\em Nagoya Math. J}, 16:205--218, 1960.

\bibitem[MFK94]{GIT}
David Mumford, John Fogarty, and Frances Kirwan.
\newblock {\em Geometric invariant theory}, volume~34 of {\em Ergebnisse der
  Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related
  Areas (2)]}.
\newblock Springer-Verlag, Berlin, third edition, 1994.

\bibitem[Mor98]{Mor}
Atsushi Moriwaki.
\newblock Relative {B}ogomolov's inequality and the cone of positive divisors
  on the moduli space of stable curves.
\newblock {\em J. Amer. Math. Soc.}, 11(3):569--600, 1998.

\bibitem[Mum66]{MumCAS}
David Mumford.
\newblock {\em Lectures on curves on an algebraic surface}.
\newblock With a section by G. M. Bergman. Annals of Mathematics Studies, No.
  59. Princeton University Press, Princeton, N.J., 1966.

\bibitem[Mum77]{M}
David Mumford.
\newblock Stability of projective varieties.
\newblock {\em Enseignement Math. (2)}, 23(1-2):39--110, 1977.

\bibitem[Rei89]{Rei}
Zinovy Reichstein.
\newblock Stability and equivariant maps.
\newblock {\em Invent. Math.}, 96(2):349--383, 1989.

\bibitem[SB06]{SB}
Nicholas~I. Shepherd-Barron.
\newblock Perfect forms and the moduli space of abelian varieties.
\newblock {\em Invent. Math.}, 163(1):25--45, 2006.

\bibitem[Sch91]{Sch}
David Schubert.
\newblock A new compactification of the moduli space of curves.
\newblock {\em Compositio Math.}, 78(3):297--313, 1991.

\bibitem[Ses72]{Ses1}
C.~S. Seshadri.
\newblock Quotient spaces modulo reductive algebraic groups.
\newblock {\em Ann. of Math. (2)}, 95:511--556; errata, ibid. (2) 96 (1972),
  599, 1972.

\bibitem[Ses77]{Ses}
C.~S. Seshadri.
\newblock Geometric reductivity over arbitrary base.
\newblock {\em Advances in Math.}, 26(3):225--274, 1977.

\bibitem[Tha96]{thaddeus}
Michael Thaddeus.
\newblock Geometric invariant theory and flips.
\newblock {\em J. Amer. Math. Soc.}, 9(3):691--723, 1996.

\bibitem[Vie89]{V}
Eckart Viehweg.
\newblock Weak positivity and the stability of certain {H}ilbert points.
\newblock {\em Invent. Math.}, 96(3):639--667, 1989.

\bibitem[Vie95]{Vbook}
Eckart Viehweg.
\newblock {\em Quasi-projective moduli for polarized manifolds}, volume~30 of
  {\em Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in
  Mathematics and Related Areas (3)]}.
\newblock Springer-Verlag, Berlin, 1995.

\end{thebibliography}
