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I have a smooth $6$-fold $X\subset\mathbb{P}^n$ and a divisor $D\subset X$ cut out by a quadratic polynomial. I know that $D$ in singular along a smooth $3$-fold $Y\subset X$, and that if $Z$ is the blow-up of $X$ along $Y$ then the strict transform $\widetilde{D}\subset Z$ of $D$ is smooth and there is a birational morphism $f:Z\rightarrow W$ contracting $\widetilde{D}$.

Is this information enough to determine the dimension of $f(\widetilde{D})$ and the type of the singularity of $W$ along $f(\widetilde{D})$ (terminal/canonical, multiplicity of the singularity)?

Thank you very much.

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  • $\begingroup$ f only contacts the strict transform of D? $\endgroup$
    – Karl Schwede
    Commented Nov 15, 2020 at 6:47
  • $\begingroup$ Exactly, $f$ only contracts the strict transform of $D$ and it is birational on its complement. $\endgroup$
    – user125056
    Commented Nov 16, 2020 at 13:34

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