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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)

MSC:
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
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References:
[1] N. Bourbaki , Groupes et algèbres de Lie, Chap. VIII , C.C.L.S., Paris, 1975. · Zbl 0329.17002
[2] M. Brion , D. Luna , and T. Vust , Espaces homogènes sphériques , Invent. Math. 84 (1986) 617-632. · Zbl 0604.14047 · doi:10.1007/BF01388749 · eudml:143354
[3] M. Brion and F. Pauer , Valuations des espaces homogènes sphériques , Comment. Math. Helv. 62 (1987) 265-285. · Zbl 0627.14038 · doi:10.1007/BF02564447 · eudml:140085
[4] M. Brion , Sur l’image de l’application moment, dans: Séminaire d’algèbre (M. P. Malliavin, ed.) 177-192, Lecture Note in Math. 1296, Springer-Verlag, 1987. · Zbl 0667.58012
[5] M. Brion , Groupe de Picard et nombres caractéristiques des variétés sphériques , Duke Math. J. 58(2) (1989) 397-424. · Zbl 0701.14052 · doi:10.1215/S0012-7094-89-05818-3
[6] L. Ein , Varieties with small dual varieties I , Invent. Math. 86 (1986) 63 - 74. · Zbl 0603.14025 · doi:10.1007/BF01391495 · eudml:143387
[7] F. Kirwan , Cohomology of quotients in symplectic and algebraic geometry , Mathematical Note 31, Princeton University Press, 1984. · Zbl 0553.14020
[8] F. Knop , The Luna-Vust theory of spherical embeddings , Proceedings of the Hyderabad conference on algebraic groups (S. Ramanan, ed.), 225-249, Manoj Prakashan, 1991. · Zbl 0812.20023
[9] D. Luna , Slices étales , Mémoire de la S.M.F. 33 (1973), 81-105. · Zbl 0286.14014 · numdam:MSMF_1973__33__81_0 · eudml:94648
[10] T. Oda , Convex bodies and algebraic geometry (An introduction to the theory of toric varieties) , Ergebnisse der Math. 15, Springer-Verlag, 1988. · Zbl 0628.52002 · eudml:203658
[11] M. Sato and T. Kimura , A classification of irreducible prehomogeneous vector spaces and their relative invariants , Nagoya Math. J. 65 (1977), 1-155. · Zbl 0321.14030 · doi:10.1017/S0027763000017633
[12] D. Snow , The nef value and defect of homogeneous line bundles , à paraître. · Zbl 0808.14042 · doi:10.2307/2154553
[13] M. Steinsieck , Transformation groups on homogeneous-rational manifolds , Math. Ann. 260 (1982) 423-435. · Zbl 0503.32017 · doi:10.1007/BF01457022 · eudml:163678
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