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

Kevin Langlois (1) , Ronan Terpereau (2)

1 ICMAT - Instituto de Ciencias Matemàticas [Madrid]
2 Fachbereich Physik, Mathematik und Informatik [Mainz]

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.

Mots clés: Luna-Vust theory; Colored polyhedral divisors; Normal G-varieties

Domaines: Géométrie algébrique [math.AG]; Théorie des représentations [math.RT]

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⟩