Flat families of affine lines are affine-line bundles
By a ''fibration'' over a scheme s we mean a faithfully flat morphism \(X\to S\) of finite type, and one such morphisms will be called an \({\mathbb{A}}^ n\)-bundle over S if the S-scheme X is locally isomorphic to the affine n- space \({\mathbb{A}}^ n_ S\) on S relative to the Zariski topology. The purpose of this note is to establish the following theorem: If \(X\to S\) is a fibration such that for each point \(s\in S\) the fiber of s in X is isomorphic to the affine line \({\mathbb{A}}^ 1_{k(s)}\) and if the base S is a Noetherian normal scheme, then X is an \({\mathbb{A}}^ 1\)-bundle over S. Earlier, Kambayashi and Miyanishi proved this theorem under the additional hypotheses that the morphism \(X\to S\) be affine and the base S be locally factorial [\textit{T. Kambayashi} and \textit{M. Miyanishi}, Ill. J. Math. 22, 662-671 (1978; Zbl 0406.14012); theorem 1]. On the other hand, they assumed only the generic fiber to be an \({\mathbb{A}}^ 1\) and all other fibers to be geometrically integral. The proof in the present paper reduces to the case where S is the spectrum of a local ring, then proceeds by induction on the dimension of S.
scientific article; zbMATH DE number 3966278