Duke Mathematical Journal

Compact moduli of plane curves

Paul Hacking

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Abstract

We construct a compactification $\mathcal{M}$d of the moduli space of plane curves of degree d. We regard a plane curve C ⊂ℙ2 as a surface-divisor pair (ℙ2,C), and we define $\mathcal{M}$d as a moduli space of pairs (X,D), where X is a degeneration of the plane. We show that, if d is not divisible by 3, the stack $\mathcal{M}$d is smooth and the degenerate surfaces X can be described explicitly.

Article information

Source
Duke Math. J. Volume 124, Number 2 (2004), 213-257.

Dates
First available: 5 August 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1091735975

Digital Object Identifier
doi:10.1215/S0012-7094-04-12421-2

Mathematical Reviews number (MathSciNet)
MR2078368

Zentralblatt MATH identifier
02108047

Subjects
Primary: 14H10: Families, moduli (algebraic) 14J10: Families, moduli, classification: algebraic theory
Secondary: 14E30: Minimal model program (Mori theory, extremal rays)

Citation

Hacking, Paul. Compact moduli of plane curves. Duke Mathematical Journal 124 (2004), no. 2, 213--257. doi:10.1215/S0012-7094-04-12421-2. http://projecteuclid.org/euclid.dmj/1091735975.


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References

  • V. Alexeev, ``Moduli spaces $M_g,n(W)$ for surfaces'' in Higher-Dimensional Complex Varieties (Trento, Italy, 1994), de Gruyter, Berlin, 1996, 1--22.
  • M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165--189.
  • M. Artin, A. Grothendieck, and J. L. Verdier, eds., Théorie des topos et cohomologie étale des schémas, 1: Théorie des topos, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 4), Lecture Notes in Math. 269, Springer, Berlin, 1972.
  • --------, Théorie des topos et cohomologie étale des schémas, 2, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 4), Lecture Notes in Math. 270, Springer, Berlin, 1972.
  • --------, Théorie des topos et cohomologie étale des schémas, 3, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 4), Lecture Notes in Math. 305, Springer, Berlin, 1973.
  • W. Fulton, Introduction to Toric Varieties, Ann. of Math. Stud. 131, Princeton Univ. Press, Princeton, 1993.
  • P. Hacking, A compactification of the space of plane curves, Ph.D. dissertation, Cambridge Univ., Cambridge, 2001, preprint.
  • --. --. --. --., Semistable divisorial contractions, J. Algebra 278 (2004), 173--186.
  • B. Hassett, Stable log surfaces and limits of quartic plane curves, Manuscripta Math. 100 (1999), 469--487.
  • L. Illusie, Complexe cotangent et déformations, I, Lecture Notes in Math. 239, Springer, Berlin, 1971.
  • --------, Complexe cotangent et déformations, II, Lecture Notes in Math. 283, Springer, Berlin, 1972.
  • L. Illusie, ``Cotangent complex and deformations of torsors and group schemes'' in Toposes, Algebraic Geometry and Logic (Halifax, Canada, 1971), Lecture Notes in Math 274, Springer, Berlin, 1972, 159--189.
  • Y. Kawamata, $D$-equivalence and $K$-equivalence, J. Differential Geom. 61 (2002), 147--171.
  • J. Kollár, Toward moduli of singular varieties, Compositio Math. 56 (1985), 369--398.
  • J. Kollár, D. Abramovich, et al., Flips and Abundance for Algebraic Threefolds (Salt Lake City, Utah, 1991), Astérisque 211, Soc. Math. France, Montrouge, 1992.
  • J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, 1998.
  • J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), 299--338.
  • E. Looijenga, Riemann-Roch and smoothings of singularities, Topology 25 (1986), 293--302.
  • M. Manetti, Normal degenerations of the complex projective plane, J. Reine Angew. Math. 419 (1991), 89--118.
  • H. Matsumura, Commutative Ring Theory, 2nd ed., Cambridge Stud. Adv. Math. 8, Cambridge Univ. Press, Cambridge, 1989.
  • L. J. Mordell, Diophantine Equations, Pure Appl. Math. 30, Academic Press, London, 1969.
  • D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, 3rd ed., Ergeb. Math. Grenzgeb. (2) 34, Springer, Berlin, 1994.
  • H. C. Pinkham, Deformations of Algebraic Varieties with $\mathbbG_m$ Action, Astérisque 20, Soc. Math. France, Montrouge, 1974.
  • M. Reid, ``Young person's guide to canonical singularities'' in Algebraic Geometry (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46, Part 1, Amer. Math. Soc., Providence, 1987, 345--414.
  • J. H. M. Steenbrink, ``Mixed Hodge structure on the vanishing cohomology'' in Real and Complex Singularities (Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, Netherlands, 1977, 525--563.
  • --. --. --. --., ``Mixed Hodge structures associated with isolated singularities'' in Singularities (Arcata, Calif., 1981), Proc. Sympos. Pure Math. 40, Amer. Math. Soc., Providence, 1983, 513--536.