Some results on codimension-one A1-fibrations

We prove a few results on the sufficiency of generic and codimension-one fibre conditions for determination of the structure of algebras of transcendence degree one. We first show that over a Noetherian normal domain R, a faithfully flat subalgebra of a finitely generated algebra whose generic and codimension-one fibres are A1 is necessarily the symmetric algebra of an invertible ideal of R. We next prove a structure theorem for a faithfully flat algebra over a locally factorial Krull domain R whose generic and codimension-one fibres are A1. For R local, we deduce a minimal sufficient condition for the algebra to be finitely generated and hence A1.