Focal loci of families and the genus of curves on surfaces
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- by Luca Chiantini and Angelo Felice Lopez
- Proc. Amer. Math. Soc. 127 (1999), 3451-3459
- DOI: https://doi.org/10.1090/S0002-9939-99-05407-6
- Published electronically: July 23, 1999
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Corrigendum: Proc. Amer. Math. Soc. 137 (2009), 3951-3951.
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 $g > 1 + \operatorname {deg} C (d - 5) / 2$. 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
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Bibliographic Information
- Luca Chiantini
- Affiliation: Dipartimento di Matematica, Università di Siena, Via del Capitano 15, 53100 Siena, Italy
- MR Author ID: 194958
- ORCID: 0000-0001-5776-1335
- Email: chiantini@unisi.it
- Angelo Felice Lopez
- Affiliation: Dipartimento di Matematica, Università di Roma Tre, Largo San Leonardo Murialdo 1, 00146 Roma, Italy
- MR Author ID: 289566
- ORCID: 0000-0003-4923-6885
- Email: lopez@matrm3.mat.uniroma3.it
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 3451-3459
- MSC (1991): Primary 14J29; Secondary 32H20, 14C20
- DOI: https://doi.org/10.1090/S0002-9939-99-05407-6
- MathSciNet review: 1676295