Let \(\varphi: X\to Y\) be an affine faithfully flat morphism of finite type between locally noetherian schemes. The aim of this paper is to investigate sufficient fibre conditions (which would in some sense be minimal) for \(X\) to be an \(\mathbb{A}^1\)-bundle over \(Y\) relative to the Zariski topology, or at least an \(\mathbb{A}^1\)-fibration over \(Y\).
Theorem A. Let \(\varphi: X\to Y\) as above such that
(i) the fibre of \(\varphi\) at the generic point of \(Y\) is \(\mathbb{A}^1\);
(ii) the fibre of \(\varphi\) at the generic point of each irreducible reduced closed subscheme of \(Y\) of codimension one is geometrically integral.
Then \(X\) is an \(\mathbb{A}^1\)-bundle over \(Y\). In particular if \(Y\) is an affine scheme then \(X\) is a line bundle over \(Y\).
This result has been proved earlier by T. Kambayashi and M. Miyanishi [Illinois J. Math. 22, 662-671 (1978; Zbl 0406.14012); theorem 1].
Theorem B. Let \(\varphi: X\to Y\) be as above such that
(i) the fibre of \(\varphi\) at the generic point of every irreducible component of \(Y\) is \(\mathbb{A}^1\);
(ii) as in theorem A.
Then all the fibres of \(\varphi\) are \(\mathbb{A}^1\)-forms. Thus if \(Y\) is a \(\mathbb{Q}\)-scheme then \(\varphi\) is actually an \(\mathbb{A}^1\)-fibration.