Blowing up replaces the center by its projectivized normal directions

A blowup is not the deletion of the center. Let $\mathcal{I}$ be a quasi-coherent ideal sheaf of finite type on a scheme $X$ with zero scheme $Z=V(\mathcal{I})$, and let $\pi \text{\hspace{0.17em}⁣}:{\mathrm{Bl}}_{\mathcal{I}}X\to X$ be the blowup of Blowup of a scheme along an ideal sheaf, with exceptional subscheme $E={\pi }^{-1}(Z)$ (Exceptional subscheme of a blowup). Then:

• $\pi$ is proper over $X$ (Blowups of finite type ideals are locally H-projective, and proper); no properness of the base over a field or global H-projective embedding is required;
• $\pi$ is an isomorphism over the open complement $X\setminus Z$ (The blowup is an isomorphism off the center). It can also be an isomorphism over the center: an effective Cartier center has identity blowup, as is seen on its single principal regular chart;
• the pullback ideal $\mathcal{I}{\mathcal{O}}_{\mathrm{Bl}}$ is invertible (The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier), so after the blowup the center is an effective Cartier divisor and further blowups along it do nothing;
• the center is replaced, not deleted: $E$ is identified with the projectivized normal cone ${\mathrm{Proj}}_Z({\mathrm{gr}}_{\mathcal{I}}{\mathcal{O}}_X)$ (The exceptional divisor is the projectivized normal cone), so the points of $E$ lying over $z\in Z$ record the normal directions to $Z$ at $z$. For a closed point center on a regular finite-type surface of pure dimension two over a field this is a projective line over the residue field, ${\mathbb{P}}_{\kappa (p)}^1$, and the blowup stays a regular surface (Point blowups of regular surfaces stay regular, with rational exceptional fibre over the residue field).

Surjectivity is part of the same picture but carries a hypothesis. If the center is empty ($\mathcal{I}={\mathcal{O}}_X$) then $\pi$ is an isomorphism; if $\mathcal{I}$ vanishes identically on an open set, then the blowup has empty fibres over that set, so no unconditional surjectivity holds. In the integral finite-type situations used on this page, a nonzero center ideal has a nonempty dense complement. The proper image of the blowup is closed and contains that complement, hence is all of $X$. Thus every fiber is nonempty, including the projectivized normal-cone fibers over the center.

Finally, strict transforms record how subvarieties approach the center: a closed subscheme $W\subseteq X$ has strict transform (Strict transform of a closed subscheme) cut out on the standard affine charts by the saturation of the pullback of its ideal by a local equation of the center, so parts supported entirely in the exceptional locus are removed. A subscheme contained in the center has empty strict transform; a single blowup need not separate all remaining branches.