Regular components of moduli spaces of stable maps

Farkas, Gavril

doi:10.1090/S0002-9939-02-06814-4

Regular components of moduli spaces of stable maps

Author: Gavril Farkas
Journal: Proc. Amer. Math. Soc. 131 (2003), 2027-2036
MSC (2000): Primary 14H10
DOI: https://doi.org/10.1090/S0002-9939-02-06814-4
Published electronically: November 4, 2002
MathSciNet review: 1963746
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Abstract: We construct regular components of the moduli space of stable maps from curves of genus to a product of two projective spaces. These components are generically smooth and have the expected dimension predicted by deformation theory. This result can be seen as a general position theorem for loci in consisting of curves carrying exceptional linear series.

[ACV] D. Abramovich, A. Corti, A. Vistoli, Twisted bundles and admissible covers, math.AG/0106211 preprint.
[AC] E. Arbarello, M. Cornalba, Footnotes to a paper of Beniamino Segre, Math. Annalen 256(1981), 341-362. MR 83d:14016
[ACGH] E. Arbarello, M. Cornalba, P. Griffiths, J. Harris, Geometry of algebraic curves, Grundlehren der Mathematischen Wissenschaften, 267, Springer Verlag, 1985. MR 86h:14019
[BE] E. Ballico, Ph. Ellia, On the existence of curves with maximal rank in $\mathbb P^n$ , J. Reine Angew. Math. 397(1989), 1-22. MR 90b:14033
[CGGH] J. Carlson, M. Green, P. Griffiths, J. Harris, Infinitesimal variations of Hodge structure (I), Compositio Math. 50(1983), no. 2-3, 109-205. MR 86e:32026a
[CR] M.C. Chang, Z. Ran, Deformations and smoothing of complete linear systems on reducible curves, in: Algebraic geometry-Bowdoin 1985 (S. Bloch ed.), Proc. Symp. Pure Math. 46, Part 1, 1987, 63-75. MR 89f:14020
[Co] M. Coppens, Some remarks on the schemes , Ann. Mat. Pura Appl. (4) 157(1990), no.1, 183-197. MR 92h:14018
[CKM] M. Coppens, C. Keem, G. Martens, The primitive length of a general -gonal curve, Indag. Math., N.S., 5(1994), 145-159. MR 95c:14021
[CM] M. Coppens, G. Martens, Linear series on a general -gonal curve, Abh. Math. Sem. Univ. Hamburg 69(1999), 347-371. MR 2000i:14043
[FP] W. Fulton, R. Pandharipande, Notes on stable maps and quantum cohomology, in: Algebraic geometry-Santa Cruz 1995 (J. Kollar, R. Lazarsfeld, D. Morrison, eds.), Proc. Symp. Pure Math., 62, Part 2, 1997, 45-96. MR 98m:14025
[Fu] W. Fulton, Hurwitz schemes and irreducibility of moduli of algebraic curves, Annals of Math. 90(1969), 542-575. MR 41:5375
[HM] J. Harris, D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67(1982), no.1, 23-88. MR 83i:14018
[Lo] A. Lopez, On the existence of components of the Hilbert scheme with the expected number of moduli, Comm. Algebra 27(1999), no. 7, 3485-3493. MR 2001d:14006
[Ran] Z. Ran, Deformations of maps, in: Algebraic Curves and Projective Geometry: Proceedings, Trento 1988, Lecture Notes in Math. 1389, Springer Verlag, 1989, 246-253. MR 91f:32021
[Se] E. Sernesi, On the existence of certain families of curves, Invent. Math. 75(1984), no.1, 25-57. MR 85e:14035
[St] F. Steffen, A generalized principal ideal theorem with applications to Brill-Noether theory, Invent. Math. 132(1998), no.1, 73-89. MR 99c:14035