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INFINITELY TRANSITIVE ACTIONS ON REAL AFFINE SUSPENSIONS
 JOURNAL OF PURE AND APPLIED ALGEBRA 216 (2012) 21062112
, 2012
"... A group G acts infinitely transitively on a set Y if for every positive integer m, its action is mtransitive on Y. Given a real affine algebraic variety Y of dimension greater than or equal to 2, we show that, under a mild restriction, if the special automorphism group of Y (the group generated by ..."
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A group G acts infinitely transitively on a set Y if for every positive integer m, its action is mtransitive on Y. Given a real affine algebraic variety Y of dimension greater than or equal to 2, we show that, under a mild restriction, if the special automorphism group of Y (the group generated by oneparameter unipotent subgroups) is infinitely transitive on each connected component of the smooth locus Yreg, then for any real affine suspension X over Y, the special automorphism group of X is infinitely transitive on each connected component of Xreg. This generalizes a recent result by Arzhantsev, Kuyumzhiyan, and Zaidenberg over the field of real numbers.
DEMAZURE ROOTS AND SPHERICAL VARIETIES: THE EXAMPLE OF HORIZONTAL SL2ACTIONS
"... Abstract. Let G be a connected reductive group, and let X be an affine Gspherical variety. We show that the classification of Gaactions on X normalized by G can be reduced to the description of quasiaffine homogeneous spaces under the action of a semidirect product Ga ⋊ G with the following prop ..."
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Abstract. Let G be a connected reductive group, and let X be an affine Gspherical variety. We show that the classification of Gaactions on X normalized by G can be reduced to the description of quasiaffine homogeneous spaces under the action of a semidirect product Ga ⋊ G with the following property. The induced Gaction is spherical and the complement of the open orbit is either empty or a Gorbit of codimension one. These homogeneous spaces are parametrized by a subset Rt(X) of the character lattice X(G) of G, which we call the set of Demazure roots of X. We give a complete description of the set Rt(X) when G is a semidirect product of SL2 and an algebraic torus; we show particularly that Rt(X) can be obtained explicitly as the intersection of a finite union of polyhedra in Q ⊗Z X(G) and a sublattice of X(G). We conjecture that Rt(X) can be described in a similar combinatorial way for an arbitrary affine spherical variety X.
FLEXIBLE BUNDLES OVER RIGID AFFINE SURFACES
"... Abstract. We construct a smooth rational affine surface S with finite automorphism group but with the property that the group of automorphisms of the cylinder S×A 2 acts infinitely transitively on the complement of a closed subset of codimension at least two. Such a surface S is in particular rigid ..."
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Abstract. We construct a smooth rational affine surface S with finite automorphism group but with the property that the group of automorphisms of the cylinder S×A 2 acts infinitely transitively on the complement of a closed subset of codimension at least two. Such a surface S is in particular rigid but not stably rigid with respect to the MakarLimanov invariant. hal00813497, version 1 15 Apr 2013
Infinite transitivity on affine varieties
 In: Birational Geometry, Rational Curves, and
, 2013
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ON ORBITS OF THE AUTOMORPHISM GROUP ON A COMPLETE TORIC VARIETY
"... Abstract. Let X be a complete toric variety and Aut(X) be the automorphism group. We give an explit description of Aut(X)orbits on X. In particular, we show that Aut(X) acts on X transitively if and only if X is a product of projective spaces. 1. ..."
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Abstract. Let X be a complete toric variety and Aut(X) be the automorphism group. We give an explit description of Aut(X)orbits on X. In particular, we show that Aut(X) acts on X transitively if and only if X is a product of projective spaces. 1.
Acyclic curves and group actions . . .
, 2011
"... We show that every irreducible, simply connected curve on a toric affine surface X over C is an orbit closure of a Gmaction on X. It follows that up to the action of the automorphism group Aut(X) there are only finitely many nonequivalent embeddings of the affine line A¹ in X. A similar descriptio ..."
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We show that every irreducible, simply connected curve on a toric affine surface X over C is an orbit closure of a Gmaction on X. It follows that up to the action of the automorphism group Aut(X) there are only finitely many nonequivalent embeddings of the affine line A¹ in X. A similar description is given for simply connected curves in the quotients of the affine plane by small finite linear groups. We provide also an analog of the Jungvan der Kulk theorem for affine toric surfaces, and apply this to study actions of algebraic groups on such surfaces.