About relative normalization in Deligne’s definition of a “tangential morphism”

I’m reading Deligne’s paper “Le Groupe Fondamental de la Droite Projective Moins Trois Points”, specifically in the section “Theorie profinie” (sections 15.13 – 15.27)

I’m specifically interested in 15.19, where he says (I had to fix some apparent typos):

More generally, let $\overline{X}$ be a smooth morphism of relative dimension 1 over a normal scheme $B$, and $S$ a divisor of $X$ etale over $B$. Let $X := \overline{X} – S$,and $Y$ a tamely ramified cover of $X$. We may extend $Y$ to a finite map $\pi : \overline{Y}\rightarrow\overline{X}$, where $\overline{Y}$ is smooth over $B$, the preimage $S_Y$ of $S$ in $\overline{Y}$ is etale over $B$, and the “principal part” of $\pi : \overline{Y}\rightarrow\overline{X}$ along $S_Y$ is a finite etale morphism between relative tangent spaces punctured at the origin.

My question is: How does he construct this scheme $\overline{Y}$? It seems to me that $\overline{Y}$ should be the normalization of $\overline{X}$ in $Y$, but I don’t see (1) why this is smooth over $B$, and (2) why $S_Y$ is finite etale over $B$.

I would very much appreciate any insights into this.

I’d also appreciate any relevant references.

EDIT: I think I can make progress on this if I can prove that $\overline{Y}$ (defined as the normalization of $\overline{X}$ in $Y$) is flat and of finite presentation over $B$.

EDIT: Currently having difficulty regarding purity. Ie, is it possible for $\overline{Y}$ to be unramified over $\overline{X}$ in codimension 1, but is somehow ramified at codimension $\ge 2$ points lying over $S$? In fact, since genus is constant in flat families, if $\overline{Y}$ were to be flat over $B$, then this shouldn’t be able to happen (at least when $\overline{X}$ is an elliptic curve over $B$)

Answer

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Source : Link , Question Author : Will Chen , Answer Author : Community

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