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.

Mori dream spaces

“…This conjecture has been proved if the vector bundle has rank two, see [11] and [8]. Note that it is not expected that X is always a log Fano.…”

Contributions to Algebraic Geometry

“…The crucial idea to approach this problem for a T -variety TV(S) is to relate its presentation to an appropriate quotient of TV(S) by the torus action. This ansatz was pursued in [HS10] whose key result states that the Cox ring of TV(S) is equal to a finitely generated algebra over the Cox ring of Y…”

“…For a broader discussion and detailed proofs of facts related to toric geometry which are merely mentioned here, the reader is asked to consult any of the standard textbooks like [KKMSD73], [Dan78], [Ful93], and [Oda88]. The subjects of non-toric T -varieties are covered in [AH03], [AH06], [AHS08], [AH08], [IS10], [Süß], [PS], [IS], [IVa], [HI], [Vol10], [HS10], [AW], and [AP]. There are also applications of the theory of T -varieties and polyhedral divisors in affine geometry [Liea, Lieb, Liec] and coding theory [IS10] which are not covered by this survey.…”

$F$-split and $F$-regular varieties with a diagonalizable group action

“…In [HS10], separated quotients of T -varieties are produced by considering the inverse limit of GIT-quotients. In this setting, the image of the quotient map into the GIT-limit gives a separation of X • /T and the distinguished component of the limit which contains the image coincides with the Chow-quotient introduced in [AH06].…”