Strict transform of a closed subscheme

Assume the Axiom of Choice, inherited from the relative Proj construction used by the blowup (The Axiom of Choice). Let $X$ be a scheme, let $\mathcal{I}$ be a quasi-coherent ideal sheaf of finite type (Quasi-coherent ideal sheaves), 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), and let $Y\hookrightarrow X$ be a closed subscheme (Closed immersions of schemes) with scheme-theoretic inverse image $Y{\times }_X{\mathrm{Bl}}_{\mathcal{I}}X$. Write $U:=(Y{\times }_X{\mathrm{Bl}}_{\mathcal{I}}X)\setminus E$ for the open subscheme obtained by deleting $E$ from this inverse image, with its open immersion $j\text{\hspace{0.17em}⁣}:U\to Y{\times }_X{\mathrm{Bl}}_{\mathcal{I}}X$.

The strict transform (or proper transform) of $Y$ is the scheme-theoretic closure of $U$ in $Y{\times }_X{\mathrm{Bl}}_{\mathcal{I}}X$, that is, the scheme-theoretic image $Y^{\mathrm{\prime }}:=\bar{U}$ of $j$ (Scheme-theoretic image). This image exists for all the data above: the kernel ideal sheaf $\mathcal{K}=\ker ({\mathcal{O}}_{Y{\times }_X\mathrm{Bl}}\to j_{\ast }{\mathcal{O}}_U)$ is quasi-coherent by the chart calculation below. The corresponding closed subscheme exists by Quasi-coherent ideals and closed subschemes, complete route and is the smallest closed subscheme through which $j$ factors. Its structural map $Y^{\mathrm{\prime }}\to Y$ is the restriction of the projection.

Chartwise description. The strict transform is determined by its restriction to the standard affine charts of the blowup. Let $U_0=\mathrm{Spec}A\subseteq X$ be an affine open and let $a\in \mathrm{\Gamma }(U_0,\mathcal{I})$; on the chart $\mathrm{Spec}A[\mathcal{I}\mathrm{/}a]\subseteq {\mathrm{Bl}}_{\mathcal{I}}X$ the exceptional subscheme is cut out by $a$ by Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains and Exceptional subscheme of a blowup, the inverse image of $Y$ is cut out by the ideal $JA[\mathcal{I}\mathrm{/}a]$, where $J=\mathrm{\Gamma }(U_0,{\mathcal{I}}_Y)$, and the strict transform meets the chart in the closed subscheme defined by the saturation $(JA[\mathcal{I}\mathrm{/}a]:a^{\mathrm{\infty }})=\{f\in A[\mathcal{I}\mathrm{/}a]:a^{n}f\in JA[\mathcal{I}\mathrm{/}a]\text{ for some }n\ge 0\},$ the largest ideal of the chart ring containing $JA[\mathcal{I}\mathrm{/}a]$ whose localisation at $a$ equals the localisation of $JA[\mathcal{I}\mathrm{/}a]$. To verify both existence and this description, put $C=A[\mathcal{I}\mathrm{/}a]$ and $H=JC$. The inverse image of $Y$ on this chart is $\mathrm{Spec}(C\mathrm{/}H)$, and its intersection with $U$ is $D(a)$ (A principal localization identifies its spectrum with a distinguished open). The kernel of $C\to (C\mathrm{/}H{)}_a$ is precisely $(H:a^{\mathrm{\infty }})$: a class becomes zero after localization exactly when some power of $a$ annihilates it. For every $g\in C$, localization of this kernel at $g$ is the kernel of $C_g\to (C\mathrm{/}H{)}_{ag}$, by Localisation of modules is exact. Thus $\mathcal{K}$ on $\mathrm{Spec}(C\mathrm{/}H)$ is the associated sheaf of $(H:a^{\mathrm{\infty }})\mathrm{/}H$, so it is quasi-coherent without any Noetherian hypothesis. These kernels agree on chart overlaps, since they all consist of sections whose restriction to $U$ is zero; equivalently the ratio transition of Affine blowup standard charts and overlaps makes $a$ and $b$ unit multiples. Consequently the chart subschemes glue to the closed subscheme cut out by $\mathcal{K}$. Its ideal restricts to zero on $U$, so $j$ factors through it. Any other closed subscheme through which $j$ factors has ideal contained in $\mathcal{K}$, and therefore contains this one. This proves that the glued saturation construction is the scheme-theoretic closure in all cases, including an empty deleted open, whose closure is empty.

Reducedness. If $Y$ is reduced, then $Y^{\mathrm{\prime }}$ is reduced. Indeed, on a chart as above the ring of the strict transform is $A[\mathcal{I}\mathrm{/}a]\mathrm{/}(JA[\mathcal{I}\mathrm{/}a]:a^{\mathrm{\infty }})$, and the saturation is by construction the kernel of the localisation $A[\mathcal{I}\mathrm{/}a]\mathrm{/}(JA[\mathcal{I}\mathrm{/}a])\to A[\mathcal{I}\mathrm{/}a][1\mathrm{/}a]\mathrm{/}(J)$, so this ring embeds into $(A[\mathcal{I}\mathrm{/}a]\mathrm{/}(J))[1\mathrm{/}a]=(A\mathrm{/}J)[1\mathrm{/}a]$, which is reduced when $Y$ is; a subring of a reduced ring is reduced, and reducedness is local, so $Y^{\mathrm{\prime }}$ is reduced.

Iteration. The construction applies verbatim to a blowup of ${\mathrm{Bl}}_{\mathcal{I}}X$ along any quasi-coherent ideal sheaf of finite type on it: for a closed subscheme $W\hookrightarrow {\mathrm{Bl}}_{\mathcal{I}}X$, its inverse image under a further blowup and the deletion of that blowup's exceptional subscheme define the strict transform of $W$, and the chartwise saturation description is unchanged. In particular a strict transform of a strict transform may be formed along a further blowup.

• The strict transform depends on the scheme structure of the center, not just its underlying closed set. Multiplication of its ideal by an invertible ideal can preserve the blowup canonically while changing the center, exceptional locus and deleted open; it need not preserve strict transforms. For example, on ${\mathbb{A}}_k^2$ the ideals $\mathcal{O}$ and $(x)$ both have identity blowup. The strict transform of $V(x)$ is $V(x)$ for the first center and empty for the second.
• The chartwise saturation description is the one used in computations: the strict transform of a hypersurface with local equation $f$ in the chart of $a$ is cut out by the saturation $(f:a^{\mathrm{\infty }})$, which removes the components supported inside the exceptional divisor; no closure operation is visible beyond this saturation.
• No reducedness, regularity or normality of $X$ or $Y$ is assumed; the reducedness conclusion above is a statement about the strict transform, not about $Y{\times }_X{\mathrm{Bl}}_{\mathcal{I}}X$, which need not be reduced even for reduced $Y$.