Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 

 

On the existence of Enriques-Fano threefolds of index greater than one


Authors: Luis Giraldo, Angelo Felice Lopez and Roberto Muñoz
Journal: J. Algebraic Geom. 13 (2004), 143-166
Published electronically: September 17, 2003
MathSciNet review: 2008718
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Abstract | References | Additional Information

Abstract: Let $X \subset \mathbb{P}^{N}$ be an irreducible threefold having a hyperplane section $Y$ that is a smooth Enriques surface and such that $X$ is not a cone over $Y$. In 1938 Fano claimed a classification of such threefolds; however, due to gaps in his proof, the problem still remains open. In this article we solve the case when $Y$ is the $r$-th Veronese embedding, for $r \geq 2$, of another Enriques surface, by proving that there are no such $X$. The latter is achieved, among other things, by a careful study of trisecant lines to Enriques surfaces. As another consequence we get precise information on the ideal of an Enriques surface. In a previous paper we had proved that any smooth linearly normal Enriques surface has homogeneous ideal generated by quadrics and cubics. Here we are able to specify when the quadrics are enough, at least scheme-theoretically.


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Additional Information

Luis Giraldo
Affiliation: Departamento de Álgebra, Universidad Complutense de Madrid Avenida Complutense, s/n 28040 Madrid, Spain
Address at time of publication: Departamento de Matematicas, Facultad de Ciencias, Universidad de Cadiz, Apartado 40, 11510 Puerto Real, Cadiz, Spain
Email: luis.giraldo@uca.es

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

Roberto Muñoz
Affiliation: ESCET, Universidad Rey Juan Carlos, 28933 Móstoles (Madrid), Spain
Email: r.munoz@escet.urjc.es

DOI: https://doi.org/10.1090/S1056-3911-03-00342-4
Received by editor(s): August 6, 2001
Published electronically: September 17, 2003
Additional Notes: The research of the first and third authors was partially supported by DGES research project, reference BFM2000-0621, and that of the second author was partially supported by the MURST national project “Geometria Algebrica"

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