  We study the local geometry of the pullback of a variety via a finite
holomorphic map. In particular, we are looking for properties of $V =
F^{-1}(W)$ such that if $V$ has the property $A$, then $W$ must have the
property $A$. We show that $A$ can be the property of normality or
prefactoriality. We also show that $A$ can be the property of smoothness, under
extra assumptions.
