A Note on the Strict Transformation of an Effective Cartier Divisor

Abstract

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.