My questions are at the very end, first I'll describe the context.
Let $f:\mathbb{P}^3\to \mathbb{P}^3$ be an involution whose fixed locus consists of two disjoint lines $L, L' \subset \mathbb{P}^3$. Let P be any plane in $\mathbb{P}^3$ in general position with respect to the lines L and L', that is, $P\cap L=Q$ and $P\cap L'=Q'$. Then $P'=f(P)$ is again a plane, we don't want to look at the trivial case $P'=P$ so assume they are different. Then the line M joining Q and Q' is exactly $P\cap P'$.
Let R be a point of M not equal to Q or Q'. Then R':=f(R) is a point of M not equal to R, Q, and Q' because they are not contained in the lines L and L'.
Now let C be a smooth conic in P' with $C\cap M=\{R,R'\}$. Then C':=f(C) is a smooth conic in P' with $C'\cap M=\{R,R'\}$. Let U:=$\mathbb{P}^3\setminus\{R\}$, $U':=\mathbb{P}^3\setminus\{R'\}$, $\widetilde{U}:=B\ell_{\widetilde{C'}}(B\ell_{C}U)$, and $\widetilde{U'}:=B\ell_{\widetilde{C}}(B\ell_{C'}U)$ where we glue the last two along to get a smooth proper threefold X.
Let E be the exceptional locus of the morphism $B\ell_{C}U\to U$; here we note that $E\cong (C\setminus R)\times \mathbb{P}^{1}\cong \mathbb{A}^1\times \mathbb{P}^1$. Let E' be the exceptional locus of $B\ell_{C'}U\to U$ and let $A:=E\cap \widetilde{C'}$ we know that this is in $E_{R'}$ where $E_{R'}$ is defined accordingly.
Now let $\widetilde{f}:X\to X$ be the lifting of the map $f:\mathbb{P}^3\to \mathbb{P}^3$, and let Y:=X/f. Consider the quotient map $q:X\setminus\{R,R'\}\to Y\setminus\{S\}$ where S=q(R)=q(R'). By SGA 3 Expose 5, we can say that q is ``nice", i.e. it is an étale map and $Y\setminus\{S\}$ is a scheme.
Now my questions are as follows:
If $\mathfrak{m}_S$ is the unique of maximal of the stalk $\mathcal{O}_{Y,S}$, then what can we say about $dim_{\mathbb{C}}(\mathfrak{m}_S/\mathfrak{m}^2_S)$? (After going over some correspondences related to tangent space at R, $T_RX$; intuitively, I suppose that it might be 4 but I can't proceed formally.)
How can the map $\mathcal{O}_{Y,S}\to \mathcal{O}_{X,R}$ be described explicitly?
Thanks in advance for any help.