Example
Consider the most basic example of a blowup: \(Y = \mathbb A^2_{xy}\), \(X = \pt = \{0\}\), \(\widetilde Y = \operatorname{Bl}_XY\xrightarrow{\pi}Y\)the blowup of \(Y\) at \(X\) and \(\widetilde X = \pi^{-1}(X)\) the exceptional divisor. This is