143 research outputs found

    Horizontal Ga-actions on affine T-varieties of complexity one

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    We classify the Ga\mathbb{G}_{a}-actions on normal affine varieties defined over any field that are horizontal with respect to a torus action of complexity one. This generalizes previous results that were available for perfect ground fields (cf. [Flenner-Zaidenberg2005, Liendo2010, L-Liendo2016]).Comment: 14 pages. To appear in Le Matematich

    Demazure roots and spherical varieties: the example of horizontal SL(2)-actions

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    Let GG be a connected reductive group, and let XX be an affine GG-spherical variety. We show that the classification of Ga\mathbb{G}_{a}-actions on XX normalized by GG can be reduced to the description of quasi-affine homogeneous spaces under the action of a semi-direct product Ga⋊G\mathbb{G}_{a}\rtimes G with the following property. The induced GG-action is spherical and the complement of the open orbit is either empty or a GG-orbit of codimension one. These homogeneous spaces are parametrized by a subset Rt(X){\rm Rt}(X) of the character lattice X(G)\mathbb{X}(G) of GG, which we call the set of Demazure roots of XX. We give a complete description of the set Rt(X){\rm Rt}(X) when GG is a semi-direct product of SL2{\rm SL}_{2} and an algebraic torus; we show particularly that Rt(X){\rm Rt}(X) can be obtained explicitly as the intersection of a finite union of polyhedra in Q⊗ZX(G)\mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{X}(G) and a sublattice of X(G)\mathbb{X}(G). We conjecture that Rt(X){\rm Rt}(X) can be described in a similar combinatorial way for an arbitrary affine spherical variety XX.Comment: Added Section 4; modified main result, Theorem 5.18 now; other change

    On the geometry of normal horospherical G-varieties of complexity one

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    Let G be a connected simply-connected reductive algebraic group. In this article, we consider the normal algebraic varieties equipped with a horospherical G-action such that the quotient of a G-stable open subset is a curve. Let X be such a G-variety. Using the combinatorial description of Timashev, we describe the class group of X by generators and relations and we give a representative of the canonical class. Moreover, we obtain a smoothness criterion for X and a criterion to determine whether the singularities of X are rational or log-terminal respectively.Comment: 29 pages, final version, to appear in J. Lie Theor

    Additive group actions on affine T-varieties of complexity one in arbitrary characteristic

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    Let X be a normal affine T-variety of complexity at most one over a perfect field k, where T stands for the split algebraic torus. Our main result is a classification of additive group actions on X that are normalized by the T-action. This generalizes the classification given by the second author in the particular case where k is algebraically closed and of characteristic zero. With the assumption that the characteristic of k is positive, we introduce the notion of rationally homogeneous locally finite iterative higher derivations which corresponds geometrically to additive group actions on affine T-varieties normalized up to a Frobenius map. As a preliminary result, we provide a complete description of these additive group actions in the toric situation.Comment: 31 page

    The Cox ring of a complexity-one horospherical variety

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    Cox rings are intrinsic objects naturally generalizing homogeneous coordinate rings of projective spaces. A complexity-one horospherical variety is a normal variety equipped with a reductive group action whose general orbit is horospherical and of codimension one. In this note, we provide a presentation by generators and relations for the Cox rings of complete rational complexity-one horospherical varieties.Comment: 9 pages, to appear in Arch. Mat
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