Cox rings, semigroups and automorphisms of affine algebraic varieties
We study the Cox realization of an affine variety, that is, a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi quasitorus. The realization is described explicitly for the quotient space of a linear action of a finite group. A universal property of this realization is proved, and some results in the divisor theory of an abstract semigroup emerging in this context are given. We show that each automorphism of an affine variety can be lifted to an automorphism of the Cox ring normalizing the grading. It follows that the automorphism group of an affine toric variety of dimension ≥ 2 without nonconstant invertible regular functions has infinite dimension. We obtain a wild automorphism of the three-dimensional quadratic cone that rises to the Anick automorphism of the polynomial algebra in four variables.