Blowups of integral projective complex schemes are projective

$X$ carries the closed immersion $i:X\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^N$ given by its projectivity, and $L_0=i^{\ast }{\mathcal{O}}_{{\mathbb{P}}_{\mathbb{C}}^N}(1)$ is invertible. The standard charts $U_0,\dots ,U_N$ are affine and cover ${\mathbb{P}}_{\mathbb{C}}^N$, so the finitely many intersections $X\cap U_j=X_{s_j}$, where $s_j=i^{\ast }(x_j)\in \mathrm{\Gamma }(X,L_0)$, are affine: each is a closed subscheme of the affine chart $U_j$, hence $\mathrm{Spec}(A_j)$ for a quotient $A_j$ of the polynomial ring over $\mathbb{C}$ [lem-closed-immersion-affine-quotient-and-base-change]. Every $A_j$ is Noetherian [lem-field-is-noetherian, cor-finite-variable-polynomial-ring-noetherian, thm-noetherian-ring-quotients-and-localisations], so this finite affine cover exhibits the scheme $X$ as locally Noetherian and quasi-compact [def-locally-noetherian-and-noetherian-scheme, def-quasi-compact-and-quasi-separated-scheme]. The coordinate section $x_j$ has nonvanishing locus $U_j$ and pullback along a closed immersion of an invertible sheaf has as nonvanishing locus the preimage of the original one [def-very-ample-invertible-sheaf-relative, def-closed-immersion-schemes], so the charts $X_{s_j}=X\cap U_j$ are affine and cover $X$; the ampleness criterion of [def-ample-invertible-sheaf] therefore makes $L_0$ ample on $X$.

By [F2] the quasi-coherent finite-type ideal sheaf $\mathcal{I}$ is coherent on the locally Noetherian $X$, and since $X$ is projective over the Noetherian ring $\mathbb{C}$ and $L_0$ is ample there is $d_0$ such that $\mathcal{I}\otimes L_0^{\otimes d}$ is globally generated for every $d\ge d_0$. Fix $d\ge \max (1,d_0)$, put $L=L_0^{\otimes d}$ and $\mathcal{F}=\mathcal{I}\otimes L\subseteq L$, and choose finitely many global sections $s_0,\dots ,s_n\in \mathrm{\Gamma }(X,\mathcal{F})$ generating $\mathcal{F}$. To justify finiteness, global generation supplies finitely many such sections generating each finite-type stalk; their coherent cokernel vanishes on a neighbourhood of that point (on an affine chart, each of its finitely many generators is killed after localizing away from that point). Quasi-compactness of $X$ supplies finitely many of these neighbourhoods, and the union of their finite section tuples generates $\mathcal{F}$ everywhere. Since $\mathcal{I}\mathrm{\ne }0$ and $L$ is invertible, $\mathcal{F}\mathrm{\ne }0$, so not all $s_i$ are zero and $\mathcal{F}={\sum }_i{\mathcal{O}}_X{s}_i$ as subsheaves of $L$.

Apply [F1] to the integral finite-type $\mathbb{C}$-scheme $X$, the invertible sheaf $L$ and the meromorphic sections $s_0,\dots ,s_n$: the fractional blowup $B={\mathrm{Proj}}_X{⨁}_{k\ge 0}(\mathcal{F}\otimes L^{-1}{)}^k$ is integral and $(\pi ,\psi ):B\to X{\times }_{\mathbb{C}}{\mathbb{P}}_{\mathbb{C}}^n$ is the schematic closure of the graph of the induced rational map, hence a closed immersion.

The canonical isomorphisms $\mathcal{F}\otimes L^{-1}=(\mathcal{I}\otimes L)\otimes L^{-1}\cong \mathcal{I}\otimes (L\otimes L^{-1})\cong \mathcal{I}$ of ${\mathcal{O}}_X$-modules [def-invertible-sheaf] identify the graded ${\mathcal{O}}_X$-algebras ${⨁}_{k\ge 0}(\mathcal{F}\otimes L^{-1}{)}^k$ and ${⨁}_{k\ge 0}{\mathcal{I}}^k$, hence induce an $X$-isomorphism $B\cong {\mathrm{Proj}}_X{⨁}_{k\ge 0}{\mathcal{I}}^k={\mathrm{Bl}}_{\mathcal{I}}X$ [def-blowup-scheme-along-ideal, def-blowup-fractional-ideal]. Composing with the closed immersion of step 3.1, the blowup ${\mathrm{Bl}}_{\mathcal{I}}X$ is a closed subscheme of $X{\times }_{\mathbb{C}}{\mathbb{P}}_{\mathbb{C}}^n$ over $X$.

Base change of the closed immersion $X\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^N$ along ${\mathbb{P}}_{\mathbb{C}}^n\to \mathrm{Spec}\mathbb{C}$ is a closed immersion $X{\times }_{\mathbb{C}}{\mathbb{P}}_{\mathbb{C}}^n\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^N{\times }_{\mathbb{C}}{\mathbb{P}}_{\mathbb{C}}^n$ [F4], and the Segre embedding of [F5] is a closed immersion ${\mathbb{P}}_{\mathbb{C}}^N{\times }_{\mathbb{C}}{\mathbb{P}}_{\mathbb{C}}^n\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^M$ with $M=(N+1)(n+1)-1$. A composite of closed immersions is again a closed immersion [F4], so ${\mathrm{Bl}}_{\mathcal{I}}X$ is a closed subscheme of ${\mathbb{P}}_{\mathbb{C}}^M$; all exhibited maps are morphisms over $\mathbb{C}$, so the structure morphism ${\mathrm{Bl}}_{\mathcal{I}}X\to \mathrm{Spec}\mathbb{C}$ factors as this closed immersion followed by the projection, and ${\mathrm{Bl}}_{\mathcal{I}}X$ is projective over $\mathbb{C}$ in the sense of [def-projective-morphism-pre-proj].

For the composition clause argue by induction on the number of factors: let $X_0$ be integral projective over $\mathbb{C}$ and $X_{k+1}={\mathrm{Bl}}_{{\mathcal{I}}_k}{X}_k$ for nonzero finite-type quasi-coherent ideal sheaves ${\mathcal{I}}_k\subseteq {\mathcal{O}}_{X_k}$. If $X_k$ is integral projective then steps 1.1-5.1 applied to $(X_k,{\mathcal{I}}_k)$ show that $X_{k+1}$ is projective over $\mathbb{C}$, and [F6] shows that it stays integral, so the induction runs through every finite composition.

Boundaries and Choice accounting. An integral scheme is nonempty and $\mathcal{I}\mathrm{\ne }0$ by hypothesis, so the empty and zero cases are excluded; the chosen twist satisfies $d\ge 1$; the endpoint $n=0$ of a single generator is allowed, the Segre factor then being ${\mathbb{P}}_{\mathbb{C}}^0$ with $M=N$; and $N=0$ is covered by the same argument with the single chart $U_0$. The Axiom of Choice [F7] is inherited exactly from [F1] and [F2]; no further selection is made.