Regular centers have projective-bundle exceptional divisors

Let $\mathrm{Spec}A\subseteq X$ be an affine chart on which $\mathcal{I}$ is generated by an $A$-regular sequence $f_1,\dots ,f_c$ and put $J=(f_1,\dots ,f_c)$, so that $J\mathrm{/}{J}^2$ is free on the classes of the $f_i$ with ${\mathrm{Sym}}_{A\mathrm{/}J}(J\mathrm{/}{J}^2)\cong A\mathrm{/}J[X_1,\dots ,X_c]$. By [F2] the canonical graded homomorphism $A\mathrm{/}J[X_1,\dots ,X_c]\to {\mathrm{gr}}_{J}A$, $X_i\mapsto f_i+J^2$, is an isomorphism, so it is a canonical isomorphism ${\mathrm{Sym}}_{A\mathrm{/}J}(J\mathrm{/}{J}^2)\to {\mathrm{gr}}_{J}A$; for the empty sequence this reads $\mathrm{Sym}(0)=A\mathrm{/}J={\mathrm{gr}}_{0}A$.

Over this chart [F1] identifies the exceptional divisor $E$ with ${\mathrm{Proj}}_Z({\mathrm{gr}}_{\mathcal{I}}{\mathcal{O}}_X)$, that is, with ${\mathrm{Proj}}_{\mathrm{Spec}A\mathrm{/}J}({\mathrm{gr}}_{J}A)$, canonically in $Z$; combining with step 1.1 gives a canonical isomorphism $E\cong {\mathrm{Proj}}_{A\mathrm{/}J}\mathrm{Sym}(J\mathrm{/}{J}^2)={\mathbb{P}}_{\mathrm{Spec}A\mathrm{/}J}(J\mathrm{/}{J}^2)$ over the chart, the projective bundle in the quotient convention recorded in [F6]. In the degenerate case $J=0$ both sides of this display are empty and the identification is the empty isomorphism.

These neighborhoods cover $Z$. Over $X\setminus Z$ the inverse-image exceptional subscheme is empty, so no regular-sequence presentation of the unit ideal is required there. The chartwise isomorphisms of step 2.1 are canonical: on an overlap of two charts each is induced by the canonical graded multiplication map $\mathrm{Sym}(\mathcal{I}\mathrm{/}{\mathcal{I}}^2)\to {\mathrm{gr}}_{\mathcal{I}}{\mathcal{O}}_X$, together with the canonical isomorphism of [F1], which involves no choices; hence they agree on overlaps and glue to a single canonical isomorphism of $Z$-schemes $E\to {\mathbb{P}}_Z(\mathcal{I}\mathrm{/}{\mathcal{I}}^2)={\mathrm{Proj}}_Z\mathrm{Sym}(\mathcal{I}\mathrm{/}{\mathcal{I}}^2)$, the projective bundle of [F6] in the quotient convention.

For the second claim let $p$ have the stated two-dimensional regular local ring on $S$ and let $\mathcal{I}$ be the ideal sheaf of $Z=\{p\}$, so that $\mathcal{I}\mathrm{/}{\mathcal{I}}^2={\mathfrak{m}}_p\mathrm{/}{\mathfrak{m}}_p^2$. The point immersion is regular: its regular parameters form a regular sequence at $p$, and this property and generation of the point ideal extend to a neighborhood by killing the finite coherent quotients and multiplication kernels whose stalks vanish at $p$. Thus step 3.1 gives a canonical isomorphism $E\to {\mathbb{P}}_Z({\mathfrak{m}}_p\mathrm{/}{\mathfrak{m}}_p^2)$ of schemes over $\mathrm{Spec}\kappa (p)$. By hypothesis ${\mathcal{O}}_{S,p}$ is a regular local ring of dimension two with residue field $\kappa (p)$, so [F4] gives $\mathrm{edim}{\mathcal{O}}_{S,p}={\dim }_{\kappa (p)}({\mathfrak{m}}_p\mathrm{/}{\mathfrak{m}}_p^2)=\dim {\mathcal{O}}_{S,p}=2$; thus ${\mathfrak{m}}_p\mathrm{/}{\mathfrak{m}}_p^2$ is a free $\kappa (p)$-module of rank two.

Let $u,v$ be a $\kappa (p)$-basis of ${\mathfrak{m}}_p\mathrm{/}{\mathfrak{m}}_p^2$; it has exactly two elements by step 4.1, and by [F5] the lifted elements $u,v$ minimally generate ${\mathfrak{m}}_p$, so their initial classes generate ${\mathrm{gr}}_{{\mathfrak{m}}_p}{\mathcal{O}}_{S,p}$ in degree one. Applying [F3] to the two-dimensional regular local ring ${\mathcal{O}}_{S,p}$ with this cotangent basis gives a graded $\kappa (p)$-algebra isomorphism $\kappa (p)[X,Y]\to {\mathrm{gr}}_{{\mathfrak{m}}_p}{\mathcal{O}}_{S,p}$ with $X\mapsto u$, $Y\mapsto v$.

Finally ${\mathbb{P}}_Z({\mathfrak{m}}_p\mathrm{/}{\mathfrak{m}}_p^2)={\mathrm{Proj}}_{\kappa (p)}\mathrm{Sym}({\mathfrak{m}}_p\mathrm{/}{\mathfrak{m}}_p^2)$ for the free rank-two module ${\mathfrak{m}}_p\mathrm{/}{\mathfrak{m}}_p^2$ is by [F6] exactly ${\mathrm{Proj}}_{\kappa (p)}\kappa (p)[X,Y]={\mathbb{P}}_{\kappa (p)}^1$, and step 5.1 identifies this graded algebra with ${\mathrm{gr}}_{{\mathfrak{m}}_p}{\mathcal{O}}_{S,p}$; hence, with the canonical isomorphism $E\to {\mathbb{P}}_Z(\mathcal{I}\mathrm{/}{\mathcal{I}}^2)$ of step 3.1, the exceptional divisor satisfies $E\cong {\mathbb{P}}_{\kappa (p)}^1$ and $\mathcal{I}\mathrm{/}{\mathcal{I}}^2$ is free of rank two over $\kappa (p)$, as claimed.