algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety higher-dimensional algebraic varieties; birational geometry; birational classification theory; minimal model program; Mori theory; cohomological vanishing theorems; cohomological nonvanishing theorems; Cartier divisors; morphisms from curves; varieties with many rational curves; rational quotient of a variety; cone theorem; contraction theorem; extremal rays 