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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

##### MSC:
 14M20 Rational and unirational varieties 14J30 $$3$$-folds 14K30 Picard schemes, higher Jacobians
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