Simple birational extensions of a polynomial ring k[x,y

Abstract

Introduction. Let k be an algebraically closed field of characteristic zero and let k[xy y] be a polynomial ring over k in two variables x and y. Let/ and g be two elements of k[xy y] without common nonconstant factors, and let A= k[xyyyfjg]. In the present article we consider the structures of the affine kdomain A under an assumption that V:=Sτpec(A) has only isolated singularities. In the first section we describe how V is obtained from A:=Spec(k[x, y\) and we see that if V has only isolated singularities V is a normal surface whose singular points (if any) are rational double points. The divisor class group Cl(V) can be explicitly determined (cf. Theorem 1.9); we obtain, therefore, necessary and sufficient conditions for A to be a unique factorization domain. If g is irreducible and if the curves / = 0 and g = 0 on A meet each other then A is a unique factorization domain if and only if the curves / = 0 and £ = 0 meet in only one point where both curves intersect transversally. We consider, in the same section, a problem: When is every invertible element of A constant? In the second section we prove the following: