A note on the categorical resolution 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. This paper provides a sufficient condition for $B{l}_{X}Y$ to be 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 $D^b\left(B{l}_{X}Y\right)$ which is a weak categorical crepant resolution of Y, in the sense of Kuznetsov [2].