Effective divisors on \ℴ𝓋ℯ𝓇𝓁𝒾𝓃ℯ{ℳ}_{ℊ}, curves on 𝒦3 surfaces, and the slope conjecture

Farkas, Gavril; Popa, Mihnea

doi:10.1090/S1056-3911-04-00392-3

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
		Online ISSN 1534-7486; Print ISSN 1056-3911

Effective divisors on $\overline{\mathcal{M}}_g$ , curves on surfaces, and the slope conjecture

Authors: Gavril Farkas and Mihnea Popa
Journal: J. Algebraic Geom. 14 (2005), 241-267
Posted: November 18, 2004
MathSciNet review: 2123229
Full-text PDF

Abstract | References | Additional Information

Abstract: We compute the class of the compactification of the divisor of curves sitting on a 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 .

[AC] Enrico Arbarello and Maurizio Cornalba, Calculating cohomology groups of moduli spaces of curves via algebraic geometry, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 97–127 (1999). MR 1733327 (2001h:14030)
[CR] Mei-Chu Chang and Ziv Ran, On the slope and Kodaira dimension of \𝑜𝑣𝑒𝑟𝑙𝑖𝑛𝑒𝑀_{𝑔} for small 𝑔, J. Differential Geom. 34 (1991), no. 1, 267–274. MR 1114463 (92h:14015)
[CH] Maurizio Cornalba and Joe Harris, Divisor classes associated to families of stable varieties, with applications to the moduli space of curves, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 3, 455–475. MR 974412 (89j:14019)
[Ck] Fernando Cukierman, Families of Weierstrass points, Duke Math. J. 58 (1989), no. 2, 317–346. MR 1016424 (90g:14013), http://dx.doi.org/10.1215/S0012-7094-89-05815-8
[CU] Fernando Cukierman and Douglas Ulmer, Curves of genus ten on 𝐾3 surfaces, Compositio Math. 89 (1993), no. 1, 81–90. MR 1248892 (94m:14047)
[EH1] David Eisenbud and Joe Harris, Limit linear series: basic theory, Invent. Math. 85 (1986), no. 2, 337–371. MR 846932 (87k:14024), http://dx.doi.org/10.1007/BF01389094
[EH2] David Eisenbud and Joe Harris, Irreducibility of some families of linear series with Brill-Noether number -1, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 33–53. MR 985853 (90a:14035)
[EH3] David Eisenbud and Joe Harris, The Kodaira dimension of the moduli space of curves of genus ≥23, Invent. Math. 90 (1987), no. 2, 359–387. MR 910206 (88g:14027), http://dx.doi.org/10.1007/BF01388710
[EH4] D. Eisenbud and J. Harris, A simpler proof of the Gieseker-Petri theorem on special divisors, Invent. Math. 74 (1983), no. 2, 269–280. MR 723217 (85e:14039), http://dx.doi.org/10.1007/BF01394316
[F] Gavril Farkas, The geometry of the moduli space of curves of genus 23, Math. Ann. 318 (2000), no. 1, 43–65. MR 1785575 (2001f:14048), http://dx.doi.org/10.1007/s002080000108
[GH] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725 (80b:14001)
[HH] R. Hartshorne and A. Hirschowitz, Smoothing algebraic space curves, Algebraic geometry, Sitges (Barcelona), 1983, Lecture Notes in Math., vol. 1124, Springer, Berlin, 1985, pp. 98–131. MR 805332 (87h:14023), http://dx.doi.org/10.1007/BFb0074998
[Ha] J. Harris, On the Kodaira dimension of the moduli space of curves. II. The even-genus case, Invent. Math. 75 (1984), no. 3, 437–466. MR 735335 (86j:14024), http://dx.doi.org/10.1007/BF01388638
[Hu] Klaus Hulek, Projective geometry of elliptic curves, Astérisque 137 (1986), 143 (English, with French summary). MR 845383 (88c:14046)
[HM] J. Harris and I. Morrison, Slopes of effective divisors on the moduli space of stable curves, Invent. Math. 99 (1990), no. 2, 321–355. MR 1031904 (91d:14009), http://dx.doi.org/10.1007/BF01234422
[La] Robert Lazarsfeld, Brill-Noether-Petri without degenerations, J. Differential Geom. 23 (1986), no. 3, 299–307. MR 852158 (88b:14019)
[Log] Adam Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math. 125 (2003), no. 1, 105–138. MR 1953519 (2003j:14035)
[Loo] Eduard Looijenga, Compactifications defined by arrangements. I. The ball quotient case, Duke Math. J. 118 (2003), no. 1, 151–187. MR 1978885 (2004i:14042a), http://dx.doi.org/10.1215/S0012-7094-03-11816-5
[M1] Shigeru Mukai, Fano 3-folds, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 255–263. MR 1201387 (94a:14042), http://dx.doi.org/10.1017/CBO9780511662652.018
[M2] Shigeru Mukai, Curves and 𝐾3 surfaces of genus eleven, Moduli of vector bundles (Sanda, 1994; Kyoto, 1994) Lecture Notes in Pure and Appl. Math., vol. 179, Dekker, New York, 1996, pp. 189–197. MR 1397987 (97g:14031)
[PR] Kapil Paranjape and S. Ramanan, On the canonical ring of a curve, Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo, 1988, pp. 503–516. MR 977775 (90b:14024)
[Ta] Sheng-Li Tan, On the slopes of the moduli spaces of curves, Internat. J. Math. 9 (1998), no. 1, 119–127. MR 1612259 (99k:14042), http://dx.doi.org/10.1142/S0129167X98000087
[V1] Claire Voisin, Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri, Acta Math. 168 (1992), no. 3-4, 249–272 (French). MR 1161267 (93b:14045), http://dx.doi.org/10.1007/BF02392980
[V2] Claire Voisin, Green’s generic syzygy conjecture for curves of even genus lying on a 𝐾3 surface, J. Eur. Math. Soc. (JEMS) 4 (2002), no. 4, 363–404. MR 1941089 (2003i:14040), http://dx.doi.org/10.1007/s100970200042
[V3] C. Voisin, Green's canonical syzygy conjecture for generic curves of odd genus, math.AG/0301359.
[W] Jonathan M. Wahl, The Jacobian algebra of a graded Gorenstein singularity, Duke Math. J. 55 (1987), no. 4, 843–871. MR 916123 (89a:14042), http://dx.doi.org/10.1215/S0012-7094-87-05540-2

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
Email: mpopa@math.harvard.edu

DOI: http://dx.doi.org/10.1090/S1056-3911-04-00392-3
PII: S 1056-3911(04)00392-3
Received by editor(s): May 16, 2003
Posted: 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

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|