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Regularization of birational automorphisms. (English. Russian original) Zbl 1059.14019
Math. Notes 76, No. 2, 264-275 (2004); translation from Mat. Zametki 76, No. 2, 286-299 (2004).
Let $$X$$ be an algebraic variety and let $$S$$ be a set of birational automorphisms of $$X$$. One says that $$S$$ is regularizable on a variety $$V$$ if there is a birational map $$f: X\dashrightarrow V$$ such that $$f\circ S\circ f^{-1}$$ is a set of biregular automorphisms of $$V$$. For instance, the Mori minimal model program implies that if $$X$$ is a variety of general type of dimension $$2$$ or $$3$$ then the group $$\text{Bir}(X)$$ of birational automorphisms is regularizable on a certain variety $$V$$. The aim of the paper under review is to prove various results concerning regularization of birational automorphisms. For instance, any finite subgroup $$G\subset \text{Bir}(X)$$ is regularizable. Another result asserts that if $$X$$ is a birationally rigid Fano variety of dimension $$2$$ or $$3$$ and if $$G\subset \text{Bir}(X)$$ is a finite subgroup of $$\text{Bir}(X)$$ then there exists a birational map $$f\colon X\dashrightarrow V$$, with $$V$$ a Fano variety with terminal singularities, such that $$f\circ G\circ f^{-1}$$ is a subgroup of biregular automorphisms of $$V$$. As an application of his results the author answers a question raised by Manin concerning the birational automorphisms of a cubic surface $$X$$ over a (non algebraically closed field) $$k$$ with Pic$$(X)=\mathbb Z$$.

##### MSC:
 1.4e+06 Rational and birational maps 1.4e+21 Coverings in algebraic geometry 1.4e+08 Birational automorphisms, Cremona group and generalizations
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