If $S$ is a scheme, $X$ is a smooth quasi-projective $S$-scheme, is the $S$-projective closure of $X$ a **smooth** $S$-scheme with $X$ an **open** subscheme?



Closure of quasi projective scheme
I don't know what you mean by *the* $S$-projective closure of $X$, but for most locally closed immersions¹ $X \hookrightarrow \mathbb P^n\_S$, the closure of $X$ in $\mathbb P^n\_S$ will not be smooth (even if $S = \operatorname{Spec} k$).


For example, if $k$ is a field of characteristic $p > 0$ and $X$ is a smooth quasi-projective $k$-variety, it is not even known if there exists *any* smooth proper variety $X'$ that is birational to $X$ (this is a very weak form of resolution of singularities).


It is always true that the map from $X$ to its closure in $\mathbb P^n\_S$ is an open immersion. In the Stacks project, this follows from the definition of an ample line bundle on a quasi-compact $S$-scheme, plus part (4) of [Tag 01VJ](https://stacks.math.columbia.edu/tag/01VJ).


Alternatively, you can use that *quasi-compact* immersions factor as an open followed by a closed [Tag [01QV](https://stacks.math.columbia.edu/tag/01QV)]. The immersion $X \hookrightarrow \mathbb P^n\_S$ is quasi-compact because $X \to S$ is quasi-compact.




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¹The existence of an immersion $X \hookrightarrow \mathbb P^n\_S$ means that $X \to S$ is what the Stacks project [calls](https://stacks.math.columbia.edu/tag/01VW) *H-quasi-projective* (H for Hartshorne, as opposed to the more general definition given in EGA). For simplicity, I am sticking to this definition here.


