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?