Rational maps between moduli spaces of curves and Gieseker-Petri divisors

Farkas, Gavril

doi:10.1090/S1056-3911-09-00510-4

Author: Gavril Farkas
Journal: J. Algebraic Geom. 19 (2010), 243-284
DOI: https://doi.org/10.1090/S1056-3911-09-00510-4
Published electronically: June 2, 2009
MathSciNet review: 2580676
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Abstract | References | Additional Information

Abstract: We study contractions of the moduli space of stable curves beyond the minimal model of $\overline {\mathcal {M}}_{g’}$ by giving a complete enumerative description of the rational map between two moduli spaces of curves $\overline {\mathcal {M}}_g \dashrightarrow \overline {\mathcal {M}}_{g’}$ which associates to a curve $C$ of genus $g$ its Brill–Noether locus of special divisors in the case this locus is a curve. As an application we construct many examples of moving effective divisors on $\overline {\mathcal {M}}_g$ of small slope, which in turn can be used to show that various moduli space of curves with level structure are of general type. For low $g’$ our calculation can be used to study the intersection theory of the moduli space of Prym varieties of dimension $5$.

References

E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932
Enrico Arbarello and Maurizio Cornalba, A few remarks about the variety of irreducible plane curves of given degree and genus, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 3, 467–488 (1984). MR 740079
Enrico Arbarello and Maurizio Cornalba, Footnotes to a paper of Beniamino Segre: “On the modules of polygonal curves and on a complement to the Riemann existence theorem” (Italian) [Math. Ann. 100 (1928), 537–551; Jbuch 54, 685], Math. Ann. 256 (1981), no. 3, 341–362. MR 626954, DOI https://doi.org/10.1007/BF01679702
Edoardo Ballico, Cinzia Casagrande, and Claudio Fontanari, Moduli of Prym curves, Doc. Math. 9 (2004), 265–281. MR 2117416
Arnaud Beauville, Prym varieties and the Schottky problem, Invent. Math. 41 (1977), no. 2, 149–196. MR 572974, DOI https://doi.org/10.1007/BF01418373
Ciro Ciliberto, Joe Harris, and Montserrat Teixidor i Bigas, On the endomorphisms of ${\rm Jac}(W^1_d(C))$ when $\rho =1$ and $C$ has general moduli, Classification of irregular varieties (Trento, 1990) Lecture Notes in Math., vol. 1515, Springer, Berlin, 1992, pp. 41–67. MR 1180337, DOI https://doi.org/10.1007/BFb0098337
Steven Diaz, Exceptional Weierstrass points and the divisor on moduli space that they define, Mem. Amer. Math. Soc. 56 (1985), no. 327, iv+69. MR 791679, DOI https://doi.org/10.1090/memo/0327
Ron Donagi, The fibers of the Prym map, Curves, Jacobians, and abelian varieties (Amherst, MA, 1990) Contemp. Math., vol. 136, Amer. Math. Soc., Providence, RI, 1992, pp. 55–125. MR 1188194, DOI https://doi.org/10.1090/conm/136/1188194
Ron Donagi and Roy Campbell Smith, The structure of the Prym map, Acta Math. 146 (1981), no. 1-2, 25–102. MR 594627, DOI https://doi.org/10.1007/BF02392458
David Eisenbud and Joe Harris, Limit linear series: basic theory, Invent. Math. 85 (1986), no. 2, 337–371. MR 846932, DOI https://doi.org/10.1007/BF01389094
David Eisenbud and Joe Harris, The Kodaira dimension of the moduli space of curves of genus $\geq 23$, Invent. Math. 90 (1987), no. 2, 359–387. MR 910206, DOI https://doi.org/10.1007/BF01388710
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, DOI https://doi.org/10.1007/BF01394316
Eduardo Esteves, Compactifying the relative Jacobian over families of reduced curves, Trans. Amer. Math. Soc. 353 (2001), no. 8, 3045–3095. MR 1828599, DOI https://doi.org/10.1090/S0002-9947-01-02746-5

G. Farkas, Koszul divisors on moduli spaces of curves, math.AG/0607475, to appear in the American Journal of Mathematics 131 (2009).

Gavril Farkas, Gaussian maps, Gieseker-Petri loci and large theta-characteristics, J. Reine Angew. Math. 581 (2005), 151–173. MR 2132674, DOI https://doi.org/10.1515/crll.2005.2005.581.151

G. Farkas and K. Ludwig, The Kodaira dimension of the moduli space of Prym varieties, Journal of European Math. Society (2009), to appear, arXiv:0804.4616

Gavril Farkas and Mihnea Popa, Effective divisors on $\overline {\scr M}_g$, curves on $K3$ surfaces, and the slope conjecture, J. Algebraic Geom. 14 (2005), no. 2, 241–267. MR 2123229, DOI https://doi.org/10.1090/S1056-3911-04-00392-3

D. Gieseker, Stable curves and special divisors: Petri’s conjecture, Invent. Math. 66 (1982), no. 2, 251–275. MR 656623, DOI https://doi.org/10.1007/BF01389394

D. Hyeon and Y. Lee, Log minimal model for the moduli space of stable curves of genus $3$, math.AG/07003093.

J. Harris and L. Tu, Chern numbers of kernel and cokernel bundles, Invent. Math. 75 (1984), no. 3, 467–475. MR 735336, DOI https://doi.org/10.1007/BF01388639

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, DOI https://doi.org/10.1007/BF01234422

Yi Hu and Sean Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786494, DOI https://doi.org/10.1307/mmj/1030132722

Shigeru Mukai, Curves and symmetric spaces. I, Amer. J. Math. 117 (1995), no. 6, 1627–1644. MR 1363081, DOI https://doi.org/10.2307/2375032

Gian Pietro Pirola and Montserrat Teixidor i Bigas, Generic Torelli for $W^r_d$, Math. Z. 209 (1992), no. 1, 53–54. MR 1143212, DOI https://doi.org/10.1007/BF02570819

Beniamino Segre, Sui moduli delle curve poligonali, e sopra un complemento al teorema di esistenza di Reimann, Math. Ann. 100 (1928), no. 1, 537–551 (Italian). MR 1512501, DOI https://doi.org/10.1007/BF01448862

Montserrat Teixidor i Bigas, The divisor of curves with a vanishing theta-null, Compositio Math. 66 (1988), no. 1, 15–22. MR 937985

Sheng-Li Tan, On the slopes of the moduli spaces of curves, Internat. J. Math. 9 (1998), no. 1, 119–127. MR 1612259, DOI https://doi.org/10.1142/S0129167X98000087

Jonathan Wahl, Gaussian maps on algebraic curves, J. Differential Geom. 32 (1990), no. 1, 77–98. MR 1064866

Additional Information

Gavril Farkas
Affiliation: Humboldt-Universität zu Berlin, Institut für Mathematik, 10099 Berlin, Germany
Email: farkas@math.hu-berlin.de

Received by editor(s): September 19, 2007
Received by editor(s) in revised form: December 19, 2007
Published electronically: June 2, 2009
Additional Notes: Research was partially supported by an Alfred P. Sloan Fellowship, the NSF Grants DMS-0450670 and DMS-0500747 and a 2006 Texas Summer Research Assignment. Most of this paper was written while visiting the Institut Mittag-Leffler in Djursholm in the Spring of 2007. Support from the institute is gratefully acknowledged.