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Nonrationality of a four-dimensional smooth complete intersection of a quadric and a quadric not containing a plane. (English. Russian original) Zbl 1071.14017

Sb. Math. 194, No. 11, 1679-1699 (2003); translation from Mat. Sb. 194, No. 11, 95-116 (2003).
A terminal \(\mathbb Q\)-factorial Fano manifold \(X\) is called birationally superrigid if it admits no non-biregular birational automorphisms and it can be birationally transformed neither into a fibration whose generic fibre has Kodaira dimension \(-\infty\) nor into a \(\mathbb Q\)-factorial terminal Fano variety with Picard group \(\mathbb Z\) not biregular to \(X\). It follows in particular that all birational automorphisms of \(X\) are biregular and \(X\) is not rational. Let \(V\subset \mathbb P^{6}\) be a smooth complete intersection of a quadric and a quartic that contains no 2-dimensional linear space. In this paper the author proves that such a \(V\) is superrigid and it is not birationally isomorphic to an elliptic fibration. The main tools used in the proof are the properties of movable log-pairs (i.e. pairs \((X,M_{X})\) where \(X\) is a variety and \(M_{X}= \sum_{i=1}^{n}a_{i}M_{i}\) is a formal linear combination of linear systems \(M_{i}\) on \(X\) without fixed components, \(a_{i}\in \mathbb Q_{\geq 0}\)). The result is obtained from a detailed study of the set of centres of canonical singularities of an effective movable log pair \((V,M_{V})\) such that \(K_{V}+M_{V}\sim_{\mathbb Q} 0\).

MSC:

14E05 Rational and birational maps
14E07 Birational automorphisms, Cremona group and generalizations
14E08 Rationality questions in algebraic geometry
14J35 \(4\)-folds
14J45 Fano varieties

Keywords:

log-pairs
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