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Focal loci of families
and the genus of curves on surfaces

Authors: Luca Chiantini and Angelo Felice Lopez
Journal: Proc. Amer. Math. Soc. 127 (1999), 3451-3459
MSC (1991): Primary 14J29; Secondary 32H20, 14C20
Published electronically: July 23, 1999
Corrigendum: Proc. Amer. Math. Soc. 137 (2009), 3951-3951.
MathSciNet review: 1676295
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Abstract: In this article we apply the classical method of focal loci of families to give a lower bound for the genus of curves lying on general surfaces. First we translate and reprove Xu's result that any curve $C$ on a general surface in $\mathbb{P}^{3}$ of degree $d \geq 5$ has geometric genus $g > 1 + \hbox {deg} C (d - 5) / 2$. Then we prove a similar lower bound for the curves lying on a general surface in a given component of the Noether-Lefschetz locus in $\mathbb{P}^{3}$ and on a general projectively Cohen-Macaulay surface in $\mathbb{P}^{4}$.

References [Enhancements On Off] (What's this?)

  • [B] Brody,R.: Compact manifolds in hyperbolicity. Trans. Amer. Math. Soc. 235, (1978) 213-219. MR 57:10010
  • [BG] Brody,R., Green,M.: A family of smooth hyperbolic hypersurfaces in $\mathbb P^{3}$. Duke Math. J. 44, (1977) 873-874. MR 56:12331
  • [CC] Chiantini,L., Ciliberto,C.: A few remarks on the lifting problem. Asterisque 218, (1993) 95-109. MR 95c:14072
  • [Ch] Chang,M.C.: A filtered Bertini type theorem. J. Reine Angew. Math. 397, (1989) 214-219. MR 90i:14054
  • [Cl] Clemens,H.: Curves on generic hypersurfaces. Ann. Sci. École Norm. Sup. 19, (1986) 629-636. MR 88c:14037
  • [CLR] Chiantini,L., Lopez,A.F., Ran, Z.: Subvarieties of generic hypersurfaces in any variety. Preprint alg-geom AG/9901083.
  • [CS] Ciliberto,C., Sernesi,E.: Singularities of the theta divisor and congruences of planes. J. Alg. Geom. 1, (1992) 231-250. MR 92j:14034
  • [D] Demailly,J.P.: Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. Notes of the AMS Summer Institute, Santa Cruz 1995. MR 99b:32037
  • [DEG] Demailly,J.P., El Goul,J.: Hyperbolicity of generic surfaces of high degree in projective 3-space. Preprint alg-geom AG/9804129.
  • [E1] Ein,L.: Subvarieties of generic complete intersections. Invent. Math. 94, (1988) 163-169. MR 89i:14002
  • [E2] Ein,L.: Subvarieties of generic complete intersections II. Math. Ann. 289, (1991) 465-471. MR 92h:14002
  • [EG] El Goul,J.: Algebraic families of smooth hyperbolic surfaces of low degree in $\mathbb P^{3}_{\mathbb{C}}$. Manuscripta Math. 90, (1996) 521-532. MR 97k:32039
  • [K] Kobayashi,S.: Hyperbolic manifolds and holomorphic mappings. Pure and Applied Mathematics 2. Marcel Dekker, New York 1970. MR 43:3503
  • [La] Lang,S.: Hyperbolic and Diophantine analysis. Bull. Amer. Math. Soc. 14, (1986) 159-205. MR 87h:32051
  • [Lo] Lopez,A.F.: Noether-Lefschetz theory and the Picard group of projective surfaces. Mem. Amer. Math. Soc. 89, vol. 438 (1991). MR 91f:14030
  • [N] Nadel,A.: Hyperbolic surfaces in $\mathbb P^{3}$. Duke Math. J. 58, (1989) 749-771. MR 91a:32036
  • [S] Segre,C.: Sui fuochi di $2^{o}$ ordine dei sistemi infiniti di piani, e sulle curve iperspaziali con una doppia infinità di piani plurisecanti. Atti R. Accad. Lincei 30, vol. 5, (1921) 67-71.
  • [V] Voisin,C.: On a conjecture of Clemens on rational curves on hypersurfaces. J. Differential Geom. 44, (1996) 200-213; correction, J. Differential Geom. 49, (1998) 601-611. MR 97j:14047; CMP 99:07
  • [V1] Voisin,C.: private communication.
  • [X1] Xu,G.: Subvarieties of general hypersurfaces in projective space. J. Differential Geom. 39, (1994) 139-172. MR 95d:14043
  • [X2] Xu,G.: Divisors on hypersurfaces. Math. Z. 219, (1995) 581-589. MR 96f:14054
  • [X3] Xu,G.: Divisors on general complete intersections in projective space. Trans. Amer. Math. Soc. 348, (1996) 2725-2736. MR 96i:14041

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Additional Information

Luca Chiantini
Affiliation: Dipartimento di Matematica, Università di Siena, Via del Capitano 15, 53100 Siena, Italy

Angelo Felice Lopez
Affiliation: Dipartimento di Matematica, Università di Roma Tre, Largo San Leonardo Murialdo 1, 00146 Roma, Italy

Received by editor(s): February 2, 1998
Published electronically: July 23, 1999
Additional Notes: This research was partially supported by the MURST national project “Geometria Algebrica"; the authors are members of GNSAGA of CNR
Communicated by: Ron Donagi
Article copyright: © Copyright 1999 American Mathematical Society

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