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Journal of Lie Theory 26 (2016), No. 1, 049--078
Copyright Heldermann Verlag 2016

On the Geometry of Normal Horospherical G-Varieties of Complexity One

Kevin Langlois
Instituto de Ciencias Matematicas, Campus Cantoblanco, UAM, Universidad Autůnoma de Madrid, Madrid 28049, Spain

Ronan Terpereau
Fachbereich Physik, Mathematik und Informatik, Johannes-Gutenberg-Universitšt, 55099 Mainz, Germany

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.

Keywords: Luna-Vust theory, colored polyhedral divisors, normal G-varieties.

MSC: 14L30, 14M27, 14M17

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