Blowups of finite type ideals are locally H-projective, and proper

Let $U=\mathrm{Spec}A\subseteq X$ be affine and let $f_0,\dots ,f_r\in A$ generate $I=\mathrm{\Gamma }(U,\mathcal{I})$. The graded $A$-algebra homomorphism $\phi \text{\hspace{0.17em}⁣}:A[x_0,\dots ,x_r]\to R(I)$, $x_i\mapsto f_{i}t$, is surjective, because in degree $n$ the images $f^{\alpha }{t}^n$ of the monomials $x^{\alpha }$ of degree $n$ generate $I^n{t}^n$; hence by [F1] it presents $\mathrm{Proj}R(I)={\mathrm{Bl}}_{I}U\cong {\pi }^{-1}(U)$ as the closed subscheme $V_{+}(\ker \phi )\hookrightarrow {\mathbb{P}}_U^r$ over $U$, which is assertion 1.

If $\mathcal{I}$ is generated by global sections $f_0,\dots ,f_r$, then the evaluation morphism ${\mathcal{O}}_X^{r+1}\to \mathcal{I}$ is surjective by [F4], and consequently the induced morphism of graded ${\mathcal{O}}_X$-algebras $\psi \text{\hspace{0.17em}⁣}:{\mathcal{O}}_X[x_0,\dots ,x_r]\to \mathcal{R}(\mathcal{I})$, $x_i\mapsto f_{i}t$, is surjective in every degree, because in degree $n$ its image is the subsheaf generated by the products of $n$ of the global sections, which is ${\mathcal{I}}^n$ by hypothesis.

Assume the global generation of step 1.2. On every affine open $U\subseteq X$ the restriction of $\psi$ is the surjection $\phi$ of step 1.1 for the restricted generators, so by step 1.1 the morphisms ${\pi }^{-1}(U)\to {\mathbb{P}}_U^r\subseteq {\mathbb{P}}_X^r$ are closed immersions over $U$; these affine-local morphisms agree on overlaps because they are induced by the restrictions of the single graded morphism $\psi$ and the identifications of [F2] are canonical, so they glue to a morphism ${\mathrm{Bl}}_{\mathcal{I}}X\to {\mathbb{P}}_X^r$ over $X$.

The glued morphism of step 2.1 is a closed immersion: surjectivity of the structure map of sheaves is checked on stalks, and a subset of ${\mathbb{P}}_X^r$ whose traces on the members of an open cover are closed is closed, so both conditions of [F6] are affine-local and hold because each restriction is a closed immersion; hence ${\mathrm{Bl}}_{\mathcal{I}}X$ is a closed subscheme of ${\mathbb{P}}_X^r$ over $X$, and $\pi$ is H-projective by Projective morphisms before Proj, which is assertion 2.

For every affine open $U\subseteq X$ the restriction ${\pi }^{-1}(U)\to U$ is a closed subscheme of ${\mathbb{P}}_U^r$ over $U$ by step 1.1, hence H-projective and therefore proper by [F5]; since properness is local on an open cover of the base by [F5], the morphism $\pi$ itself is proper, which is assertion 3, and it holds whether or not $\mathcal{I}$ is globally generated.