The Stacks project

Definition 31.33.1. With $Z \subset S$ and $f : X \to S$ as above.

  1. Given a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the strict transform of $\mathcal{F}$ with respect to the blowup of $S$ in $Z$ is the quotient $\mathcal{F}'$ of $\text{pr}_ X^*\mathcal{F}$ by the submodule of sections supported on $\text{pr}_{S'}^{-1}E$.

  2. The strict transform of $X$ is the closed subscheme $X' \subset X \times _ S S'$ cut out by the quasi-coherent ideal of sections of $\mathcal{O}_{X \times _ S S'}$ supported on $\text{pr}_{S'}^{-1}E$.

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