universal torsor

Proposition 2.2. Let X be a normal variety with Γ(X, O∗) = K∗ and finitely generated divisor class group. Let R be a sheaf of Cl(X)-graded algebras as constructed before, assume that the Cox ring R(X) is finitely generated, denote by p : ̂X → X the associated universal torsor and by X′ ⊆ X the set of smooth points. (i) The Cox ring R(X) is normal and every homogeneous invertible f ∈ R(X) is constant. (ii) The canonical morphism ̂X → X ̄ ̄ is an open embedding; in particular, ̂X is quasiaffine. (iii) The complement X ̄ ̄ \ p−1(X′) is small in X ̄ ̄, and the group H acts freely on the set p−1(X′) ⊆ ̂X. (iv) Every H-invariant Weil divisor on the total coordinate space X ̄ ̄ is principal. (page) (Hausen, 2008, p. 715)