On one-parameter families of affine spaces

Details On one-parameter families of affine spaces On one-parameter families of affine spaces

2012-03-08

arxiv.org/abs/1203.1691v1

Mathematics

Commutative Algebra

5 pages

Scientific paper

The following result is proved : Let $k$ be an algebraically closed field, and let both $X$ and $S$ are $k$-schemes of finite type. Let $\varphi : X \rightarrow S$ be a flat, affine morphism; assume that $S$ is normal, integral and $\dim(S) = 1$, and that the fiber $\varphi^{-1}(p)$ of $\varphi$ above each point $p$ of $S$ is isomorphic to the affine $n$-space $\mathbb{A}^n_{k(p)}$ over the residue field $k(p)$ of $p$. Then $X$ is an $\mathbb{A}^n_S$-bundle over $S$ with respect to the Zariski topology on $S$.

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