Normal subalgebras of a polynomial ring

Abstract

Let A be a finitely generated subalgebra of a polynomial ring k[x,y,z] over the complex field k. Assuming that A is normal, we clarify the structure of A under additional assumptions if dim⁡A≥2. If dim⁡A=2 and A is regular, then Spec A has an A1-fibration over P1 or A1 with restrictions on the number of multiple fibers (see Theorem 3). If dim⁡A=3, we assume that A is cofinite, i.e., k[x,y,z] is a finite A-module, and A contains a coordinate x. Then either A is a polynomial ring (see Theorem 6) or the invariant subring k[x,u,v]G with respect to a small finite subgroup G of GL(2,k) (see section 4). The results in section 4 are generalized in the next section.