Embedded resolution

Let $Z$ be a reduced, excellent, Noetherian scheme and let $X$ be a regular excellent scheme. Suppose that $Z\hookrightarrow X$ is a closed immersion.

Embedded resolution. There exists a projective, surjective morphism

$$\pi:X'\to X$$

which is an isomorphism over $X\setminus Z$, such that $\pi^{-1}(Z)$ is an snc divisor in $X'$.

This is the embedded resolution of singularities conjecture. It is known when $\dim(Z)\leq 2$ and, together with resolution of singularities, implies the existence of suitable smooth snc compactifications in the paper's setting.