# Blow up along a section of a smooth morphism

Let $C$ be a ground locally notherian and quasiprojective scheme. Let $\pi:S\to B$ be a $C$-morphism of finite type $C$-schemes. We call $\pi$ a $C$-smooth morphism if the morphisms $S\to C$ and $B\to C$ are smooth.

My question is (edited by @Ben's comment): Given a $C$-section $\sigma:B \to S$ of $\pi$ and the blow up $b:\tilde S\to S$ of $S$ along $\sigma(B)$, is the morphism $\tilde \pi:= \pi\circ b:\tilde S\to B$ smooth or at least $C$-smooth?

In Intersection theory of W.Fulton there are referents to EGA to see that $\sigma$ is local complete intersection morphism.

I'm asking for some kind of generalization of Theorem 22.3.10 (if $X\to Y$ is a closed embedding of smooth varieties over k, then $Bl_XY$ is smooth) in Vakil's notes to a ground scheme $C$.

• Take $B=C=\mathrm{Spec}(k)$, $k$ algebraically closed, and e.g. $S=\mathbb{P}^2$. You are asking whether the blow up morphism of a point in $\mathbb{P}^2$ is smooth. I hope you realize this is completely false.
– abx
Commented Sep 21, 2015 at 13:27
• Isn't $\tilde{\pi}$ just the map to $\mathrm{Spec}(k)$ in this case? Commented Sep 21, 2015 at 13:37
• Thanks for your coment @abx, I edited my question. I'm sorry. Commented Sep 21, 2015 at 13:56
• mathoverflow.net/questions/32396/good-reduction-and-blow-ups Commented Sep 21, 2015 at 15:45