Let $Y$ be an effective Cartier divisor of a smooth variety $Z$. Let $X_{ i}$, $i\in \{ 1,\cdots,n\}$ be a set of pairwise disjoint smooth subvarieties in $Y$ such that their union contains the singular locus of $Y$. In this paper, we give a sufficient condition such that $Bl_{ X}Y$ is smooth, where $X$ is the disjoint union of all $X_{ i}$. Moreover, we prove that assuming a dimensional condition, there is an admissible subcategory of $D^{ b}(Bl_{ X}Y)$ which is a weak categorical crepant resolution of $Y$ in the sense of Kuznetsov.
Categories: math.AG