Blowing up (algebraic geometry, ${\mathrm{Bl}}_{Z}X$)

The projective morphism ${\mathrm{Proj}}_X{⨁}_{j\ge 0}{\mathcal{I}}^j\to X$ associated to the powers of the ideal sheaf of a closed centre. Off that centre it is an isomorphism. Blowing up a smooth point of an $n$-dimensional smooth variety replaces the point by a projective space ${\mathbb{P}}^{n-1}$, the algebraic exceptional divisor.

For a blowup of an algebraic variety $\pi :\tilde{X}\to X$ with centre $Z$, the strict transform of an algebraic subvariety of a subvariety $Y$ not contained in $Z$ is the closure of ${\pi }^{-1}(Y\setminus Z)$. It excludes components supported entirely in the algebraic exceptional divisor.

The divisor contracted by a blowup of an algebraic variety to its centre. For the blowup of a smooth point on an $n$-dimensional smooth variety it is ${\mathbb{P}}^{n-1}$. Being a closed projective subvariety of positive dimension, it can obstruct affineness of a fibre containing it.