Rational singularities of normal T-varieties

Alvaro Liendo (1)

1 IF - Institut Fourier

A T-variety is an algebraic variety X with an effective regular action of an algebraic torus T. Altmann and Hausen gave a combinatorial description of an affine T-variety X by means of polyhedral divisors. In this paper we compute the higher direct images of the structure sheaf of a desingularization of X in terms of this combinatorial data. As a consequence, we give a criterion as to when a T-variety has rational singularities. We also provide a partial criterion for a T-variety to be Cohen-Macaulay. As an application we characterize in this terms quasihomogeneous elliptic singularities of surfaces.

Mots clés: elliptic surface singularities; torus actions; T-varieties; rational singularities; Cohen-Macaulay singularities; elliptic surface singularities.

Domaines: Géométrie algébrique [math.AG]

Alvaro Liendo. Rational singularities of normal T-varieties. 2009. ⟨hal-00419203⟩