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Relatively minimal quasihomogeneous projective 3-folds. (English) Zbl 0964.14041

Let \(X\) be a smooth projective threefold and \(G\) a connected algebraic group acting algebraically on \(X\). By minimal model theory, there exists a sequence of extremal ray contractions and flips such that the resulting variety \(X'\) has at most \(\mathbb{Q}\)-factorial terminal singularities and either the canonical sheaf \(K_{X'}\) is numerically effective, or \(X'\) allows an extremal ray contraction of fiber type.
It has been shown in a previous paper [S. Kebekus, Doc. Math., J. DMV, 3, 15-26 (1998; Zbl 0940.14029)] that all steps of the minimal model program are equivariant with respect to the action of \(G\). If one assumes additionally that \(G\) acts almost transitively, which is to say that the \(G\)-action has an open orbit, then it is shown that the minimal model program always ends with a contraction of fiber type. The aim of the present paper is a classification of these varieties, more precisely, a classification of 3-folds \(X\) which are quasihomogeneous under the action of a linear, non-solvable algebraic group and allow an extremal ray contraction of fiber type.

MSC:

14M17 Homogeneous spaces and generalizations
14J30 \(3\)-folds
14L30 Group actions on varieties or schemes (quotients)
32M12 Almost homogeneous manifolds and spaces
14E30 Minimal model program (Mori theory, extremal rays)

Citations:

Zbl 0940.14029

References:

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