The Cox ring of an algebraic variety with torus action

Abstract

We investigate the Cox ring of a normal complete variety X with algebraic torus action. Our first results relate the Cox ring of X to that of a maximal geometric quotient of X. As a consequence, we obtain a complete description of the Cox ring in terms of generators and relations for varieties with torus action of complexity one. Moreover, we provide a combinatorial approach to the Cox ring using the language of polyhedral divisors. Applied to smooth K ∗ -surfaces, our results give a description of the Cox ring in terms of Orlik–Wagreich graphs. As examples, we explicitly compute the Cox rings of all Gorenstein del Pezzo K ∗ -surfaces with Picard number at most two and the Cox rings of projectivizations of rank two vector bundles as well as cotangent bundles over toric varieties in terms of Klyachko's description.