On the existence of Enriques-Fano threefolds of index greater than one. (English) Zbl 1059.14051
The authors revisit a problem studied by Fano, whose argument was incomplete. In particular, they prove (among many other things):
1. Let \(X \subset \mathbb P^N\) be an irreducible non-degenerate 3-fold such that \(X\) has a smooth hyperplane section \(Y\) which is the \(r\)-th Veronese embedding, for \(r \geq 2\), of a linearly normal Enriques surface \(S \subset \mathbb P^{g-1}\). Then \(X\) is a cone over \(Y\).
2. Let \(S \subset \mathbb P^{g-1}\) be a smooth linearly normal non-degenerate Enriques surface with \(g \geq 11\). Then:
(2a) If \(S\) does not contain a plane cubic curve, then \(S\) is scheme-theoretically cut out by quadrics.
(2b) If \(S\) contains a plane cubic curve, then there are exactly two of them, and the intersection of the quadrics containing \(S\) is the union of \(S\) and the two planes in which the cubic curves lie.
The authors also include a very large list of references which should be very useful.
1. Let \(X \subset \mathbb P^N\) be an irreducible non-degenerate 3-fold such that \(X\) has a smooth hyperplane section \(Y\) which is the \(r\)-th Veronese embedding, for \(r \geq 2\), of a linearly normal Enriques surface \(S \subset \mathbb P^{g-1}\). Then \(X\) is a cone over \(Y\).
2. Let \(S \subset \mathbb P^{g-1}\) be a smooth linearly normal non-degenerate Enriques surface with \(g \geq 11\). Then:
(2a) If \(S\) does not contain a plane cubic curve, then \(S\) is scheme-theoretically cut out by quadrics.
(2b) If \(S\) contains a plane cubic curve, then there are exactly two of them, and the intersection of the quadrics containing \(S\) is the union of \(S\) and the two planes in which the cubic curves lie.
The authors also include a very large list of references which should be very useful.
Reviewer: De-Qi Zhang (Singapore)
MSC:
| 14J30 | \(3\)-folds |
| 14J28 | \(K3\) surfaces and Enriques surfaces |
| 14J45 | Fano varieties |
| 14N05 | Projective techniques in algebraic geometry |
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