P-Divisors of Cox Rings

The Cox ring of a so-called Mori Dream Space (MDS) is finitely
generated and it is graded over the divisor class group. Hence the
spectrum of the Cox ring comes with an action of an algebraic torus
whose GIT quotient is the variety in question. We present the
associated description of this Cox ring as a polyhedral divisor of
a T-variety.
Via the shape of its polyhedral coefficients, it connects the equivariant structure of the Cox ring with the world of stable loci and stable multiplicities of linear systems.