algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety Mori dream space; Cox ring; class group; toric varieties; gale duality; the secondary fan; GKZ decomposition; good and geometric quotient; fan matrix; weight matrix; nef cone; moving cone; pseudo-effective cone; Picard number; bunch of cones; irrelevant ideal and locus; completion; completion of fans; minimal model program; small modification; rational contraction 