143 research outputs found
Horizontal Ga-actions on affine T-varieties of complexity one
We classify the -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
Let be a connected reductive group, and let be an affine
-spherical variety. We show that the classification of
-actions on normalized by can be reduced to the
description of quasi-affine homogeneous spaces under the action of a
semi-direct product with the following property. The
induced -action is spherical and the complement of the open orbit is either
empty or a -orbit of codimension one. These homogeneous spaces are
parametrized by a subset of the character lattice
of , which we call the set of Demazure roots of . We give a complete
description of the set when is a semi-direct product of and an algebraic torus; we show particularly that can be
obtained explicitly as the intersection of a finite union of polyhedra in
and a sublattice of
. We conjecture that can be described in a similar
combinatorial way for an arbitrary affine spherical variety .Comment: Added Section 4; modified main result, Theorem 5.18 now; other
change
On the geometry of normal horospherical G-varieties of complexity one
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
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
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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