Results 1  10
of
25
Normal affine surfaces with C*actions
, 2002
"... A classification of affine surfaces admitting a C∗action was given in the work of Bia̷lynickiBirula, Fieseler and L. Kaup, Orlik and Wagreich, Rynes and others. We provide a simple alternative description of normal quasihomogeneous affine surfaces in terms of their graded rings as well as by def ..."
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Cited by 23 (6 self)
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A classification of affine surfaces admitting a C∗action was given in the work of Bia̷lynickiBirula, Fieseler and L. Kaup, Orlik and Wagreich, Rynes and others. We provide a simple alternative description of normal quasihomogeneous affine surfaces in terms of their graded rings as well as by defining equations. This is based on a generalization of the DolgachevPinkhamDemazure construction.
On the MakarLimanov, Derksen invariants, and finite automorphism groups of algebraic varieties
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Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity
, 2010
"... We say that a group G acts infinitely transitively on a set X if for every m ∈ N the induced diagonal action of G is transitive on the cartesian mth power X m \ ∆ with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely t ..."
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Cited by 15 (6 self)
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We say that a group G acts infinitely transitively on a set X if for every m ∈ N the induced diagonal action of G is transitive on the cartesian mth power X m \ ∆ with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of affine cones over flag varieties, the second of nondegenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups of a reinforced type.
Affine Tvarieties of complexity one and locally nilpotent derivations
, 2008
"... Let X = SpecA be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus of dimension n. Let also ∂ be a homogeneous locally nilpotent derivation on the normal affine Z ngraded domain A, so that ∂ generates a k+action on X. We pr ..."
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Cited by 11 (1 self)
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Let X = SpecA be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus of dimension n. Let also ∂ be a homogeneous locally nilpotent derivation on the normal affine Z ngraded domain A, so that ∂ generates a k+action on X. We provide a complete classification of pairs (X,∂) in two cases: for toric varieties (n = dim X) and in the case where n = dimX − 1. This generalizes previously known results for surfaces due to Flenner and Zaidenberg. As an application we show that ker ∂ is finitely generated. Thus the generalized Hilbert’s fourteenth problem has a positive answer in this particular case, which strengthen a result of Kuroda. As another application, we compute the homogeneous MakarLimanov invariant of such varieties. In particular we exhibit a family of nonrational varieties with trivial MakarLimanov invariant.
Rational curves and rational singularities
 Math. Zeitschrift
"... Abstract. We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a C ∗action. For varieties with an isolated singularity, we show that the presence of sufficiently many rational curves outside the singular point strongly affects the character of the sing ..."
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Cited by 11 (3 self)
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Abstract. We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a C ∗action. For varieties with an isolated singularity, we show that the presence of sufficiently many rational curves outside the singular point strongly affects the character of the singularity. This provides an explanation of classical results due to H. A. Schwartz and G. H. Halphen on polynomial solutions of the generalized Fermat equation. Contents
Gaactions of fiber type on affine Tvarieties
 J. Algebra
"... Abstract. Let X be a normal affine Tvariety, where T stands for the algebraic torus. We classify Gaactions on X arising from homogeneous locally nilpotent derivations of fiber type. We deduce that any variety with trivial MakarLimanov (ML) invariant is birationally decomposable as Y × P 2, for so ..."
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Cited by 6 (0 self)
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Abstract. Let X be a normal affine Tvariety, where T stands for the algebraic torus. We classify Gaactions on X arising from homogeneous locally nilpotent derivations of fiber type. We deduce that any variety with trivial MakarLimanov (ML) invariant is birationally decomposable as Y × P 2, for some Y. Conversely, given a variety Y, there exists an affine variety X with trivial ML invariant birational to Y × P 2. Finally, we introduce a new version of the ML invariant, called the FML invariant. According to our conjecture, the triviality of the FML invariant implies rationality. This conjecture holds in dimension at most 3. hal00430172, version 1 5 Nov 2009
GROUP ACTIONS ON AFFINE CONES
, 2009
"... We address the following question: Determine the affine cones over smooth projective varieties which admit an action of a connected algebraic group different from the standard C ∗action by scalar matrices and its inverse action. We show in particular that the affine cones over anticanonically emb ..."
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Cited by 6 (2 self)
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We address the following question: Determine the affine cones over smooth projective varieties which admit an action of a connected algebraic group different from the standard C ∗action by scalar matrices and its inverse action. We show in particular that the affine cones over anticanonically embedded smooth del Pezzo surfaces of degree ≥ 4 possess such an action. A question in [FZ1] whether this is also true for cubic surfaces, occurs to be out of reach for our methods. Nevertheless, we provide a general geometric criterion that could be helpful also in this case.
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.