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As far as I understand, embedded resolution of singularities means the following: given a variety $X$ over an algebraically closed field, and a closed subvariety $Y$, there exists a birational map $f:X'\to X$ such that $X'$ and the proper transform $Y'$ of $Y$ are smooth. Sometimes a bit more is required (e.g. $Y'$ transverse to the exceptional locus), so I'm a bit confused about the terminology.

We know that we can do this if either (1) the base field is of characteristic zero, or (2) $X$ is a surface and $Y$ is a curve, and maybe in some other low-dimensional situations.

Question. Can we always find an embedded resolution when $X$ is smooth and $Y$ is a curve?

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This should be fine at least for the weaker notion you mention. One can always resolve the singularities of a curve by blowing up a sequence of closed points (on the curve). Hence if you embed the curve in a regular ambient space $X$, you can always do the following. Choose the point on $Y \subseteq X$ you were going to blowup, blowup that closed point on $X$, and notice that the resulting $X_1$ is still regular (since you blew up a regular center). Repeat to get $\ldots \to X_3 \to X_2 \to X_1 \to X$.

Indeed, as long as your resolution algorithm always blows up only smooth centers, the same result is obtained (say for surfaces embedded in any dimensional space). My impression is that this is even true for threefolds.

However, you could ask for more as you said! In particular that the exceptional divisor of $X' \to X$ is SNC and meets $Y'$ also transversely. I don't know if this is true in the generality you want.

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