zbMATH — the first resource for mathematics

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\((X)=\mathbb Z\).

14E05 Rational and birational maps
14E20 Coverings in algebraic geometry
14E07 Birational automorphisms, Cremona group and generalizations
Full Text: DOI