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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Bounding families of ruled surfaces


Authors: Francisco Javier Gallego, Luis Giraldo and Ignacio Sols
Journal: Proc. Amer. Math. Soc. 124 (1996), 2943-2951
MSC (1991): Primary 14C05, 14J26
MathSciNet review: 1363420
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Abstract: In this paper we provide a sharp bound for the dimension of a family of ruled surfaces of degree $d$ in $\mathbf P_{\mathbf K}^3$. We also find the families with maximal dimension: the family of ruled surfaces containing two unisecant skew lines, when $d\ge 9$ and the family of rational ruled surfaces, when $d\le 9$.

The first tool we use is a Castelnuovo-type bound for the irregularity of ruled surfaces in $\mathbf P_{\mathbf K}^n$. The second tool is an exact sequence involving the normal sheaf of a curve in the grassmannian. This sequence is analogous to the one constructed by Eisenbud and Harris in 1992, where they deal with the problem of bounding families of curves in projective space. However, our construction is more general since we obtain the mentioned sequence by purely algebraic means, studying the geometry of ruled surfaces and of the grassmannian.


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Additional Information

Francisco Javier Gallego
Affiliation: Departamento de Algebra, Facultad Matematicas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: gallego@sunal1.mat.ucm.es

Luis Giraldo
Affiliation: Departamento de Algebra, Facultad Matematicas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: giraldo@eucmax.sim.ucm.es

Ignacio Sols
Affiliation: Departamento de Algebra, Facultad Matematicas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: sols@eucmax.sim.ucm.es

DOI: https://doi.org/10.1090/S0002-9939-96-03701-X
Received by editor(s): February 13, 1995
Additional Notes: The authors were partially supported by CICYT, no. PB90-0637.
Communicated by: Eric M. Friedlander
Article copyright: © Copyright 1996 American Mathematical Society