The Cox ring of a toric variety X can be viewed as a generalisation of the homogeneous coordinate ring of projective n-space. Over the complex numbers, the theory is outlined in The Homogeneous Coordinate Ring of a Toric Variety.

We associate to X a graded polynomial ring S and an ideal B. The space $V=Spec(S)$ is isomorphic to $\mathbb{C}^n$ and has a subvariety Z cut out by the ideal B. There is an action of a torus G on V such that V - Z is G-invariant with quotient isomorphic to X. Furthermore, when X is smooth and complete, the automorphism group Aut(X) of the toric variety has a cover $\widetilde{Aut}(X)$ fitting into the exact sequence

$1 \to G \to \widetilde{Aut}(X) \to Aut(X) \to 1$.

The construction can be generalised to other varieties and base fields, often under the name of "universal torsors." Unfortunately, this added generality lacks the wonderful combinatorial description in Cox's paper (for example, the Demazure root system).

Toric varieties over general base fields can be studied by considering the actions of Galois groups on the fan. It should be fairly straghtforward to extend these ideas to the Cox ring, as well. Rather then reinvent the wheel, I'd like to see if this has already been done.

Does there exist a reference? A combinatorial description of Cox rings of toric varieties over arbitrary base fields, especially descriptions of analogs of $\widetilde{Aut}(X)$.


1 Answer 1


Parametrizations of toric varieties over any field: also available via google


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