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 $C$.