Let $R$ be a complete DVR of characteristic $p$, say $R=\mathbb{F}_p[[t]]$, and $X$ be a reduced scheme of finite type over $R$. Let also $X_s$ denote the special fiber of $X$. If I understand correctly, resolution of singularities in characteristic p, says that there is a regular scheme $X'$ of finite type over $R$ and a blowup $f:X'\to X$ in a nowhere dense center $Z$.

However, it gives us no information on what the new special fiber $X'_s$ looks like.

Q: (a) Is there a (conjectured) version of resolution of singularities in characteristic $p$ that also ensures that $X'_s$ is of a "nice" form (normal crossings divisor)?

(b) If so, does this mean that $X'$ locally looks like
$$\operatorname{Spec}(R[x_1,\dotsc,x_d]/(f_1^{e_1}f_2^{e_2}\dotsb f_n^{e_n}-t))$$ where the $\mathbb{F}_p$-schemes cut out by the $\bar f_i$'s are as in the definition of normal crossings?

It seems that the corresponding statement in characteristic $0$ is true and well-known (Theorem 3.1.5 in Temkin - Absolute desingularization in characteristic zero). Also, any references that address this (or closely related) issues would be useful.

Some background: I am not an algebraic geometer, but I found it convenient to assume, at least temporarily, some form of RoS in characteristic $p$ for a problem I am trying to solve.


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