Graded subrings of \({\mathbb{C}}[X,Y]\) (Q2639106)

Let \({\mathbb{C}}\) denote the complex numbers and let R be a \({\mathbb{C}}\)- subalgebra of the polynomial ring \({\mathbb{C}}[X,Y]\) which is normal and is generated as a \({\mathbb{C}}\)-algebra by finitely many homogeneous polynomials \(F_ 1,...,F_ n\). The main result of this paper is that if the g.c.d. of \((F_ 1,...,F_ n)\) has at most two distinct linear factors (possibly occurring with multiplicities), then R is isomorphic to the fixed ring of a finite subgroup \(W\subseteq GL(2,{\mathbb{C}})\) acting linearly on a polynomial ring \({\mathbb{C}}[U,V]\). If R is birational with \({\mathbb{C}}[X,Y]\), then W can be chosen to be cyclic. Several related results are also given as well as a discussion of the relationship of the above result to a conjecture of C.T.C. Wall on algebraic groups.