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Flexible affine cones and flexible coverings. (English) Zbl 1408.14199

Take \(X\) an affine variety over an algebraically closed \(k\). The special automorphism group of \(X\) is the subgroup of the automorphims of \(X\) generated by all the \((k,+)\)-actions. The variety \(X\) is said to be flexible when the special automorphism group acts transitively on the regular locus of \(X\).
The first result in the paper under review is a criterion for flexibility of affine cones. To be concrete (see Theorem 1.4) the affine cone over a normal projective variety covered by flexible affine open subsets whose complement in \(Y\) is the support of a \(\mathbb{Q}\)-divisor linearly equivalent to a hyperplane section of \(Y\) is flexible. This is applied to show (see Theorem 2.20) that the affine cone over the secant (resp. tangential) variety of a Segre-Veronese (resp. Segre) variety is flexible. Moreover, see Theorem 4.5, affine cones over Fano 3-folds are considered. Finally (cf. Theorem 5.4), the total coordinate spaces (see Definition 5.1) of smooth Del Pezzo surfaces (among others, see Theorem 5.9) are shown to be flexible.

MSC:

14R20 Group actions on affine varieties
14J50 Automorphisms of surfaces and higher-dimensional varieties
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