Cox construction

The Cox construction presents a toric variety as a quotient of an open subset of affine space by a quasitorus determined by its divisor class group.

The Cox ring of a toric variety is the polynomial ring with one variable $x_{\rho }$ for each ray, graded by the divisor class group through $\mathrm{deg}{x}_{\rho }=[D_{\rho }]$.

A geometric quotient $X\to X/G$ has fibers equal to group orbits and carries precisely the invariant regular functions locally on the quotient.