Blowup of a complex manifold at a point

The blowup replaces a point of a complex $n$-manifold by the projective space of complex tangent directions ${\mathbb{CP}}^{n-1}$. A biholomorphism carrying one center to another lifts to a biholomorphism of their blowups.

The exceptional divisor of the blowup of a complex manifold at a point $p$ is the fiber over $p$. In complex dimension $n$ it is naturally the projective space $\mathbb{P}(T_{p}X)\cong {\mathbb{CP}}^{n-1}$ of tangent directions at the center.

The strict transform of a subvariety $Y$ under a blowup $\sigma :\tilde{X}\to X$ is the closure of ${\sigma }^{-1}(Y\setminus Z)$, where $Z$ is the center. Its total transform also contains the exceptional divisor with the multiplicity with which $Y$ passes through $Z$.