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Factorization of certain birational morphisms. (Factorisation de certains morphismes birationnels.) (French) Zbl 0829.14005
Let \(G\) be a connected complex reductive group acting on an algebraic variety \(X\). If \(X\) is normal and a Borel subgroup \(B\) of \(G\) has a dense open orbit in \(X\), then \(X\) is called spherical. The minimum of codimensions of orbits of the unipotent radical of \(B\) on \(X\) is called the rank of \(X\). The author proves the following theorem:
Let \(X\) be a smooth spherical variety of rank 2 and \(A \subset X\) its irreducible \(G\)-invariant subvariety. Then the blow-up of \(X\) along \(A\) is smooth and its exceptional divisor is irreducible and reduced.
Any equivariant birational proper morphism of spherical smooth varieties of rank 2 is a composition of blowing-ups along \(G\)-invariant irreducible subvarieties.
Let \(X\) and \(X'\) be smooth complete spherical varieties of rank 2 and \(\varphi : X' \to X\) an equivariant birational mapping. Then there exists a smooth complete spherical variety \(X''\) and the birational morphisms \(p : X'' \to X\) and \(p' : X'' \to X'\) such that \(p = \varphi \circ p'\) and \(p,p'\) are the compositions of blowing-ups along the \(G\)-invariant smooth centers.
In the second part of the paper the author describes the singularities of \(G\)-invariant irreducible subvarieties of codimension \(> 1\) of smooth complete spherical varieties of rank 2.
Reviewer: V.L.Popov (Moskva)

14E05 Rational and birational maps
14L30 Group actions on varieties or schemes (quotients)
14L24 Geometric invariant theory
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14M17 Homogeneous spaces and generalizations
Full Text: Numdam EuDML
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