If C* is acting on an affine variety X, then X is close to a cone
over a polarized projective variety (Y,L). For higher-dimensional
torus actions, the situation can be described by a polyhedral
divisor on some Y. This means that the coefficients of the divisor
are polyhedra in the dual of the character lattice of the torus T.

This polyhedral divisor encodes all information about the T-orbit
decomposition of X; deforming the divisor leads to deformations
of X, and X can be understood as the contraction of a degenerate
toric bundle over Y.

If X is the Cox ring of a Mori dream space Y, then we consider the
action of the Picard torus. The corresponding polyhedral divisor D
lives on this Y or a modification of it, and its combinatorial
structure reflects the birational geometry of X. If Y is a surface,
then D is related to the Zariski decomposition of the divisors on Y.