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On the number of moduli of extendable canonical curves

Published online by Cambridge University Press:  22 January 2016

Ciro Ciliberto  and
Angelo Felice Lopez
Ciro Ciliberto
Affiliation:
Dipartimento di Matematica Universitè di Roma “Tor Vergata”, Viale della Ricerca Scientifica 00133 Roma, Italy, cilibert@axp.mat.uniroma2.it
Angelo Felice Lopez
Affiliation:
Dipartimento di Matematica Universitè di Roma Tre, Largo San Leonardo Murialdo 1 00146 Roma, Italy, lopez@matrm3.mat.uniroma3.it
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Abstract

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Let C ⊂ ℙg−1 be a canonical curve of genus g. In this article we study the problem of extendability of C, that is when there is a surface S ⊂ ℙg different from a cone and having C as hyperplane section. Using the work of Epema we give a bound on the number of moduli of extendable canonical curves. This for example implies that a family of large dimension of curves that are cover of another curve has general member nonextendable. Using a theorem of Wahl we prove the surjectivity of the Wahl map for the general k-gonal curve of genus g when k = 5, g ≥ 15 or k = 6, g ≥ 13 or k ≥ 7, g ≥ 12.

Keywords

14H1014J1014J26
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 167 , 2002 , pp. 101 - 115
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2002

References

[AC] Arbarello, E. and Cornalba, M., Footnotes to a paper of Beniamino Segre. The number of on a general d-gonal curve, and the unirationality of the Hurwitz spaces of 4-gonal and 5-gonal curves, Math. Ann., 256 (1981), 341–362.CrossRefGoogle Scholar
[ACGH] Arbarello, E., Cornalba, M., Griffiths, P.A. and Harris, J., Geometry of algebraic curves, Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 267, Springer-Verlag, New York-Berlin, 1985.Google Scholar
[B1] Brawner, J.N., The Gaussian-Wahl map for trigonal curves, Proc. Amer. Math. Soc., 123 (1995), 1357–1361.Google Scholar
[B2] Brawner, J.N., Tetragonal curves, scrolls, and K3 surfaces., Trans. Amer. Math. Soc, 349 (1997), 3075–3091.Google Scholar
[CHM] Ciliberto, C., Harris, J. and Miranda, R., On the surjectivity of the Wahl map, Duke Math. J., 57 (1988), 829–858.Google Scholar
[CLM] Ciliberto, C., Lopez, A.F. and Miranda, R., On the Wahl map of plane nodal curves, In: Proceedings of the Symposium on Complex Analysis and Algebraic Geometry in memory of Michael Schneider, Bayreuth 1998, De Gruyter, Berlin-New York: 2000, 155–163.Google Scholar
[CM] Ciliberto, C. and Miranda, R., Gaussian maps for certain families of canonical curves, In:Complex projective geometry (Trieste/Bergen, 1989), London Math. Soc. Lecture Note Ser. 179, 106-127. Cambridge Univ. Press, Cambridge, 1992.Google Scholar
[D] Duflot, J., Gaussian maps for double coverings, Manuscripta Math., 82 (1994), 71–87.CrossRefGoogle Scholar
[E1] Epema, D.H.J., Surfaces with canonical hyperplane sections, CWI Tract 1. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1984.Google Scholar
[E2] Epema, D.H.J., Surfaces with canonical hyperplane sections, Nederl. Akad. Wetensch. Indag. Math., 45 (1983), 173–184.Google Scholar
[G] Green, M., Koszul cohomology and the geometry of projective varieties, J. Diff. Geom., 19 (1984), 125–171.Google Scholar
[H] Horowitz, T., Varieties of low degree, Brown University Ph. D. Thesis, (1982) and Varieties of low Δ-genus, Duke Math. J. 50 (1983), 667–683.Google Scholar
[L] Lopez, A.F., Surjectivity of gaussian maps on curves in ℙr with general moduli, J. Algebraic Geom., 5 (1996), 609–631.Google Scholar
[Lv] L’vovskii, S.M., Extensions of projective varieties and deformations, Michigan Math. J., 39 (1992), 65–70.Google Scholar
[Ma] Martens, R., Uber den Clifford-index algebraischer kurven, J. Reine Angew. Math., 320 (1980), 68–85.Google Scholar
[Mi] Miranda, R., Triple covers in algebraic geometry, Amer. J. Math., 107 (1985), 1123–1158.Google Scholar
[P] Paoletti, R., Generalized Wahl maps and adjoint line bundles on a general curve, Pacific J. Math., 168 (1995), 313–334.Google Scholar
[S] Schreyer, F.O., A standard basis approach to syzygies of canonical curves, J. Reine Angew. Math., 421 (1991), 83–123.Google Scholar
[SD] Saint-Donat, B., Projective models of K3 surfaces, Amer. J. Math., 96 (1974), 602–639.Google Scholar
[V1] Voisin, C., Courbes tetragonales et cohomologie de Koszul, J. Reine Angew. Math., 387 (1988), 111–121.Google Scholar
[V2] Voisin, C., Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri, Acta Math., 168 (1992), 249–272.Google Scholar
[W1] Wahl, J., The Jacobian algebra of a graded Gorenstein singularity, Duke Math. J., 55 (1987), 843–871.CrossRefGoogle Scholar
[W2] Wahl, J., Gaussian maps on algebraic curves, J. Diff. Geom., 32 (1990), 77–98.Google Scholar
[W3] Wahl, J., On cohomology of the square of an ideal sheaf J. Algebraic Geom., 6 (1997), 481–511.Google Scholar
[W4] Wahl, J., Hyperplane sections of Calabi-Yau varieties., preprint.Google Scholar
[Z] Zak, F.L., Some properties of dual varieties and their application in projective geometry, In: Algebraic Geometry, Proceedings Chicago 1989. Lecture Notes in Math. 1479. Springer, Berlin-New York: 1991, 273–280.Google Scholar