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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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 | References | Similar Articles | Additional Information

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 $\mathbb{P}^{4}$.


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Additional Information

Luca Chiantini
Affiliation: Dipartimento di Matematica, Università di Siena, Via del Capitano 15, 53100 Siena, Italy
Email: chiantini@unisi.it

Angelo Felice Lopez
Affiliation: Dipartimento di Matematica, Università di Roma Tre, Largo San Leonardo Murialdo 1, 00146 Roma, Italy
Email: lopez@matrm3.mat.uniroma3.it

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05407-6
PII: S 0002-9939(99)05407-6
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