zbMATH — the first resource for mathematics

Nonrationality of the general Enriques variety. (English. Russian original) Zbl 0589.14041
Math. USSR, Sb. 51, 267-273 (1985); translation from Mat. Sb., Nov. Ser. 123(165), No. 2, 269-275 (1984).
This paper contains the proof of the following result: the general Enriques threefold \(V\subset {\mathbb{P}}^ 4({\mathbb{C}})\) is nonrational. This proof was announced by the author in a previous note and is founded on the following preliminary and nice result: the general Enriques threefold V admits a standard model \(\pi_ W:\quad W\to S_ W\) where \(S_ W\) is the blow-up of the product \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) in four general points and the degeneracy curve \(C_ W\) is a smooth curve of genus 5 (proposition (2.1)). The notion of standard model was defined by V. G. Sarkisov [Russ. Math. Surv. 34, No.4, 183-184 (1979); translation from Usp. Mat. Nauk. 34, No.4, 207-208 (1979; Zbl 0426.14017)]. The result of the present paper is founded on a result of Shokurov on the intermediate Jacobian \(J_ 3(W)\) and a result of C. H. Clemens and P. A. Griffiths [Ann. Math., II. Ser. 95, 281-356 (1972; Zbl 0214.483)].
Reviewer: M.Becheanu

14M20 Rational and unirational varieties
14J30 \(3\)-folds
14K30 Picard schemes, higher Jacobians
Full Text: DOI