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

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-01081134
Contributor : Ronan Terpereau <>
Submitted on : Friday, December 23, 2016 - 11:40:21 AM
Last modification on : Wednesday, July 18, 2018 - 8:08:02 AM
Long-term archiving on : Tuesday, March 21, 2017 - 7:11:33 AM

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### Identifiers

• HAL Id : hal-01081134, version 2
• ARXIV : 1411.2480

### Citation

Kevin Langlois, Ronan Terpereau. On the geometry of normal horospherical $G$-varieties of complexity one. Journal of Lie Theory, 2016, 26 (1), pp.49-78. ⟨hal-01081134v2⟩

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