# Divisorial contractions and singularities

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.

• f only contacts the strict transform of D? Nov 15 '20 at 6:47
• Exactly, $f$ only contracts the strict transform of $D$ and it is birational on its complement. Nov 16 '20 at 13:34