Let $X$ be a variety over a perfect field $k$. The multiplicity is an invariant that stratifies $X$ into locally closed sets. $X$ is regular at a point $\xi$ if and only if $\mathrm{mult}(\xi) = 1$. Dade proved that, if
\begin{equation} 
(1)\qquad X
	\leftarrow X_1
	\leftarrow \dotsb
	\leftarrow X_l
\end{equation}
is a sequence of blow-ups along closed regular equimultiple centers, then
$$
	\max\mathrm{mult}(X_l) \leq \max\mathrm{mult}(X).
$$
If one can find a sequence like (1) so that $\max\mathrm{mult}(X_l) < \max\mathrm{mult}(X)$, then a resolution of singularities of $X$ can be constructed by iteration of this process.

The study of the multiplicity has been historically linked to that of finite morphisms. Zariski's formula says that, if $\beta : X' \to X$ is a finite and dominant morphism of varieties of generic rank $r$, then
\begin{equation}
(2) \qquad	\max\mathrm{mult}(X') \leq r \cdot \max\mathrm{mult}(X).
\end{equation}
Let $\underline{\mathrm{Max}}\mathrm{mult}(X)$ denote the stratum of maximum multiplicity of $X$. When the equality holds in (2), $\underline{\mathrm{Max}}\mathrm{mult}(X')$ is homeomorphic to its image in $X$, which is contained in $\underline{\mathrm{Max}}\mathrm{mult}(X)$. In our work, we study conditions under which
\[
	\beta \bigl( \underline{\mathrm{Max}}\mathrm{mult}(X') \bigr) \cong \underline{\mathrm{Max}}\mathrm{mult}(X),
\]
and such that this homeomorphism is preserved by <em>permissible</em> blow-ups. These conditions provide a relation between the processes of lowering of the maximum multiplicity of $X$ and $X'$ by blowing up equimultiple centers.

This is a joint work with Ana Bravo and Orlando Villamayor.