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Contractions and flips for varieties with group action of small complexity

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Titel: Contractions and flips for varieties with group action of small complexity
Autoren: Brion, Michel, Knop, Friedrich
Verlagsinformationen: University of Tokyo, Graduate School of Mathematical Sciences, Tokyo
Schlagwörter: flipping birational morphisms, Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), NE, complexity of action on algebraic variety, convex cone of effective one-cycles, Minimal model program (Mori theory, extremal rays), Rational and birational maps, equivariant completion
Beschreibung: Let \(X\) be a normal algebraic variety over an algebraically closed field and \(G\) a connected reductive transformation group on \(X\). The minimal codimension of a \(B\)-orbit in \(X\), where \(B \subset G\) is a Borel subgroup, is called the complexity of the \(G\)-action on \(X\). Now let \(X\) be projective and unirational with a \(G\)-action of complexity at most one. Then the algebra of regular functions on \(X\) is finitely generated [see \textit{F. Knop}, Math. Z. 213, No. 1, 33-36 (1993; Zbl 0788.14042)]. Based on this result and generalizing results of \textit{M. Brion} [Duke Math. J. 72, No. 2, 369-405 (1993; Zbl 0821.14029)] the authors study the convex cone \(\text{NE} (X)\) of effective one-cycles in \(X\) and show that this cone is finitely generated. Moreover each face of \(\text{NE} (X)\) can be contracted by a morphism \(\varphi : X \to X'\), where \(X'\) is a projective, normal algebraic variety. Using the technique of flipping birational morphisms the authors prove that every sequence of directed flips is finite if \(X\) is \(\mathbb{Q}\)-factorial. As a consequence, if \(H\) is a closed subgroup of \(G\) such that the complexity of the \(G\)-action on \(Y : = G/H\) is at most one, then \(Y\) admits a \(G\)-equivariant completion with an ample anticanonical divisor.
Publikationsart: Article
Dateibeschreibung: application/xml
Zugangs-URL: https://zbmath.org/770311
Dokumentencode: edsair.c2b0b933574d..13adc28df3cab26b7d0c0ea7ccd8715e
Datenbank: OpenAIRE
Beschreibung
Abstract:Let \(X\) be a normal algebraic variety over an algebraically closed field and \(G\) a connected reductive transformation group on \(X\). The minimal codimension of a \(B\)-orbit in \(X\), where \(B \subset G\) is a Borel subgroup, is called the complexity of the \(G\)-action on \(X\). Now let \(X\) be projective and unirational with a \(G\)-action of complexity at most one. Then the algebra of regular functions on \(X\) is finitely generated [see \textit{F. Knop}, Math. Z. 213, No. 1, 33-36 (1993; Zbl 0788.14042)]. Based on this result and generalizing results of \textit{M. Brion} [Duke Math. J. 72, No. 2, 369-405 (1993; Zbl 0821.14029)] the authors study the convex cone \(\text{NE} (X)\) of effective one-cycles in \(X\) and show that this cone is finitely generated. Moreover each face of \(\text{NE} (X)\) can be contracted by a morphism \(\varphi : X \to X'\), where \(X'\) is a projective, normal algebraic variety. Using the technique of flipping birational morphisms the authors prove that every sequence of directed flips is finite if \(X\) is \(\mathbb{Q}\)-factorial. As a consequence, if \(H\) is a closed subgroup of \(G\) such that the complexity of the \(G\)-action on \(Y : = G/H\) is at most one, then \(Y\) admits a \(G\)-equivariant completion with an ample anticanonical divisor.