Effective divisors on $\overline {\mathcal {M}}_g$, curves on $K3$ surfaces, and the slope conjecture
Authors:
Gavril Farkas and Mihnea Popa
Journal:
J. Algebraic Geom. 14 (2005), 241-267
DOI:
https://doi.org/10.1090/S1056-3911-04-00392-3
Published electronically:
November 18, 2004
MathSciNet review:
2123229
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Abstract |
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Additional Information
Abstract: We compute the class of the compactification of the divisor of curves sitting on a $K3$ surface and show that it violates the Harris-Morrison Slope Conjecture. We carry this out using the fact that this divisor has four distinct incarnations as a geometric subvariety of the moduli space of curves. We also give a counterexample to a hypothesis raised by Harris and Morrison that the Brill-Noether divisors are essentially the only effective divisors on the moduli space of curves having minimal slope $6+12/(g+1)$.
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[AC]arbarello-cornalba E. Arbarello and M. Cornalba, Calculating cohomology groups of moduli spaces of curves via algebraic geometry, Inst. Hautes Etudes Sci. Publ. Math. 88 (1998), 97–127.
[CR]chang-ran M.-C. Chang and Z. Ran, On the slope and Kodaira dimension of $\overline {\mathcal {M}}_g$ for small $g$, J. Diff. Geom. 34 (1991), 267–274.
[CH]CH M. Cornalba and J. Harris, Divisor classes associated to stable varieties with applications to the moduli space of curves, Ann. Sci. Ec. Norm. Sup. 21 (1988), 455-475.
[Ck]cukierman F. Cukierman, Families of Weierstrass points, Duke Math. J. 58 (1989), 317–346.
[CU]cukierman-ulmer F. Cukierman and D. Ulmer, Curves of genus $10$ on $K3$ surfaces, Compositio Math. 89 (1993), 81–90.
[EH1]EH1 D. Eisenbud and J. Harris, Limit linear series: basic theory, Invent. Math. 85 (1986), 337–371.
[EH2]EH2 D. Eisenbud and J. Harris, Irreducibility of some families of linear series with Brill-Noether number, I, Ann. Scient. Ec. Norm. Sup.(4) 22 (1989), 33–53.
[EH3]EH3 D. Eisenbud and J. Harris, The Kodaira dimension of the moduli space of curves of genus $\geq 23,$ Invent. Math. 90 (1987), 359–387.
[EH4]EH4 D. Eisenbud and J. Harris, A simpler proof of the Gieseker-Petri Theorem on special divisors, Invent. Math. 74 (1983), 269–280.
[F]F G. Farkas, The geometry of the moduli space of curves of genus $23$, Math. Ann. 318 (2000), 43-65.
[GH]griffiths-harris P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley-Interscience, 1978.
[HH]hh R. Hartshorne and A. Hirschowitz, Smoothing algebraic space curves, in: Algebraic Geometry: Sitges, Barcelona, Lecture Notes in Mathematics 1124 (1983), 98–131.
[Ha]Ha J. Harris, On the Kodaira dimension of the moduli space of curves II: The even genus case, Invent. Math. 75 (1984), 437–466.
[Hu]Hu K. Hulek, Projective geometry of elliptic curves, Asterisque 137 (1986).
[HM]harris-morrison J. Harris and I. Morrison, Slopes of effective divisors on the moduli space of stable curves, Invent. Math. 99 (1990), 321–355.
[La]Laz R. Lazarsfeld, Brill-Noether-Petri without degenerations, J. Diff. Geom. 23 (1986), 299-307.
[Log]Lo A. Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. of Math. 125 (2003), 105–138.
[Loo]Loo E. Looijenga, Compactifications defined by arrangements II: locally symmetric varieties of type IV, Duke Math. J. 119 (2003), 527–588.
[M1]Mukai1 S. Mukai, Fano $3$-folds, in: Complex Projective Geometry, London Math. Soc. Lecture Notes Ser. 179, Cambridge University Press (1992), 255–263.
[M2]Mukai2 S. Mukai, Curves and $K3$ surfaces of genus eleven, in: Moduli of vector bundles, Lecture Notes in Pure and Appl. Math. 179, Dekker (1996), 189–197.
[PR]pr K. Paranjape and S. Ramanan, On the canonical ring of a curve, in: Algebraic Geometry and Commutative Algebra, Kinokuniya, Tokyo, 1988, 503–516.
[Ta]tan S.-L. Tan, On the slopes of the moduli spaces of curves, Int. J. Math. 9 (1998), 119–127.
[V1]Voisin1 C. Voisin, Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri, Acta Math. 168 (1992), 249–272.
[V2]Voisin2 C. Voisin, Green’s generic syzygy conjecture for curves of even genus lying on a $K3$ surface, J. Eur. Math. Soc. 4 (2002), 363-404.
[V3]Voisin3 C. Voisin, Green’s canonical syzygy conjecture for generic curves of odd genus, math.AG/0301359.
[W]Wahl J. Wahl, The Jacobian algebra of a graded Gorenstein singularity, Duke Math. J. 55 (1987), 843–871.
Additional Information
Gavril Farkas
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544
Address at time of publication:
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712
Email:
gfarkas@math.princeton.edu
Mihnea Popa
Affiliation:
Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138
MR Author ID:
653676
Email:
mpopa@math.harvard.edu
Received by editor(s):
May 16, 2003
Published electronically:
November 18, 2004
Additional Notes:
The first author’s research was partially supported by NSF Grant DMS-0140520. The second author’s research was partially supported by NSF Grant DMS-0200150
