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Reye congruences. (English) Zbl 0541.14034

In this paper the author studies the congruences, called Reye congruences, of lines which are included in two distinct quadrics of a web without basepoints \(\lambda_ 1Q_ 1+\lambda_ 2Q_ 2+\lambda_ 3Q_ 3+\lambda_ 4Q_ 4=0,\) where \(Q_ i\) is a quadric in \({\mathbb{P}}^ 3_{{\mathbb{C}}}\), \({\mathbb{C}}=complex\) field, \(\lambda_ i\in {\mathbb{C}}\). These congruences were introduced by T. Reye (1882) and studied by G. Fano (1901-1910) who showed, in particular, that the generic Reye congruence is an Enriques surface. The purpose of this paper is to understand the special features held by an Enriques surface which is a Reye congruence. The author analyses the Picard group of the Reye congruence associated to a generic web W and finds the following results: Theorem 1. The Reye congruences form a nine-dimensional family of Enriques surfaces which coincides generically with the family of Enriques surfaces of special type, i.e. which contains an elliptic pencil \(| P|\) and a smooth rational curve \(\theta\) such that \(P\cdot \theta =2.\)- Theorem 2. Let R be an Enriques surface which contains an elliptic pencil \(| P|\) and a smooth rational curve \(\theta\) such that \(P\cdot \theta =6\). Then R is the minimal desingularization of a Reye congruence. - Moreover the author proves the following result due to A. Cayley (1894): Theorem 3. A K3 surface is the étale double cover of a generic Reye congruence if and only if it can be realized as a double cover of \({\mathbb{P}}^ 2\) branched over a sextic curve which splits into two cubics which intersect transversally and have a totally tangent smooth conic which does not contain any of the points of intersection of the cubics. - In section 1 the author gives a description of the space of quadrics in \({\mathbb{P}}^ 2\) and \({\mathbb{P}}^ 3\), referring for the proofs to A. Tyurin [Usp. Mat. Nauk 30, 51-99 (1975; Zbl 0328.14014)].
The author does not say anything about the ground field.
Reviewer: E.Stagnaro

MSC:

14J25 Special surfaces
14N05 Projective techniques in algebraic geometry
14C21 Pencils, nets, webs in algebraic geometry
14C22 Picard groups
14M15 Grassmannians, Schubert varieties, flag manifolds

Citations:

Zbl 0328.14014
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References:

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