On the geometry of normal horospherical \(G\)-varieties of complexity one (Q2810920)

Let \(G\) be a connected simply connected reductive algebraic group and let \(X\) be a normal algebraic \(G\)-variety such that the isotropy group of any point contains a maximal unipotent subgroup of \(G\) and \(\dim X-\max_{x\in X} \dim G\cdot x=1\) (the algebraically closed base field is supposed to be of characteristic zero). The authors describe the class group of \(X\) by generators and relations, obtain a factoriality criterion for \(X\), give a representative of a canonical class, and prove a criterion to determine whether the singularities of \(X\) are rational or log-terminal.