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Factorization of certain birational morphisms

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Názov: Factorization of certain birational morphisms
Autori: Brion, Michel
Informácie o vydavateľovi: Cambridge University Press, Cambridge; London Mathematical Society, London
Predmety: Homogeneous spaces and generalizations, invariant subvariety, spherical variety, Group actions on varieties or schemes (quotients), Geometric invariant theory, singularities of invariant irreducible subvarieties, blow- up, Global theory and resolution of singularities (algebro-geometric aspects), factorization of birational morphisms, exceptional divisor, Rational and birational maps
Popis: 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.
Druh dokumentu: Article
Popis súboru: application/xml
Prístupová URL adresa: https://zbmath.org/591032
Prístupové číslo: edsair.c2b0b933574d..4dce10485f83c67324d5eb0fdc5921a2
Databáza: OpenAIRE
Popis
Abstrakt: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.