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Let $X$ be a scheme which is smooth, integral, of finite type and separated over $\mathbb{Z}$, equipped with a smooth morphism $h: X \to \mathbb{A}^1_\mathbb{Z}$. Let $Y \subsetneq X$ be a closed subscheme and $\beta: X' \to X$ the blow-up of $Y$.

Suppose that the restriction of $h$ to $Y$ is not dominant and $Y \otimes \mathbb{Q}$ is non-empty. If I understand things correctly, then this means that $h\mid_Y \otimes \mathbb{Q}$ is constant with some value $\nu$. Is it true that the multiplicity of $\beta^{-1}(Y)$ in the divisor of the function $(h \circ \beta) - \nu$ then has to be equal to $1$?

Moreover, if $Z$ is the intersection of $\beta^{-1}(Y)$ with the union of the other components of the divisor of $(h \circ \beta) - \nu$, is it true that $h \circ \beta$ is smooth at every point of $\beta^{-1}(Y) \setminus Z$?

I'm not sure how I should look at this situation...

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    OK but then the question becomes a little bit too technical...2012-05-30

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