The exceptional divisor is the projectivized normal cone

Let $U=\mathrm{Spec}A\subseteq X$ be affine with $\mathcal{I}{\mathrm{\mid }}_U=\tilde{I}$. Restricting the fibre product of [F1] to $U$ gives $E{\times }_{X}U=\mathrm{Spec}(A\mathrm{/}I){\times }_{\mathrm{Spec}A}{\mathrm{Bl}}_{I}U$, and ${\mathrm{Bl}}_{I}U={\mathrm{Proj}}_{A}R(I)$; applying [F2] to the morphism $\mathrm{Spec}(A\mathrm{/}I)\to \mathrm{Spec}A$ and the graded algebra $R(I)$ yields a canonical isomorphism ${\mathrm{Proj}}_{A}R(I){\times }_{\mathrm{Spec}A}\mathrm{Spec}(A\mathrm{/}I)\cong {\mathrm{Proj}}_{A\mathrm{/}I}(R(I){\otimes }_{A}A\mathrm{/}I)$.

In the standard chart $\mathrm{Spec}A[I\mathrm{/}a]$, $a\in I$, the ideal of $E$ is $\mathcal{I}{\mathcal{O}}_{\mathrm{Bl}}{\mathrm{\mid }}_{\mathrm{chart}}=IA[I\mathrm{/}a]=aA[I\mathrm{/}a]$ by [F1] and [F4], so $E$ meets the chart in $V(a)$, and the charts over a generating family of $I$ cover the blowup by [F4].

By [F3] the graded $A\mathrm{/}I$-algebra $R(I){\otimes }_{A}A\mathrm{/}I=R(I)\mathrm{/}IR(I)$ has degree-$n$ piece $I^n\mathrm{/}{I}^{n+1}$, so it is canonically ${\mathrm{gr}}_I(A)$; combining with step 1.1 gives a canonical isomorphism $E{\times }_{X}U\cong {\mathrm{Proj}}_{A\mathrm{/}I}{\mathrm{gr}}_I(A)$ over $\mathrm{Spec}(A\mathrm{/}I)$.

The isomorphisms of step 2.1 are canonical and compatible with restriction to smaller affine opens: for $A\to A_f$ both $\mathrm{Bl}$ and the associated graded construction localise, $I^n{A}_f\mathrm{/}{I}^{n+1}{A}_f=(I{A}_f{)}^n\mathrm{/}(I{A}_f{)}^{n+1}$, and the identifications of [F2] are natural; hence they glue over an affine cover of $X$ to a canonical isomorphism of $Z$-schemes $E\to {\mathrm{Proj}}_Z({\mathrm{gr}}_{\mathcal{I}}{\mathcal{O}}_X)$, the projectivized normal cone.

For a point $z\in Z$, applying [F2] to the morphism $\mathrm{Spec}\kappa (z)\to Z$ and the graded ${\mathcal{O}}_Z$-algebra ${\mathrm{gr}}_{\mathcal{I}}{\mathcal{O}}_X$ identifies the fibre of ${\mathrm{Proj}}_Z({\mathrm{gr}}_{\mathcal{I}}{\mathcal{O}}_X)$ over $z$, and hence the fibre of $E$ over $z$ by step 3.1, with ${\mathrm{Proj}}_{\kappa (z)}({\mathrm{gr}}_I(A){\otimes }_{A\mathrm{/}I}\kappa (z))$; for a closed point center, where $I=\mathfrak{m}$ is maximal, ${\mathrm{gr}}_{\mathfrak{m}}(A)={⨁}_{n\ge 0}{\mathfrak{m}}^n\mathrm{/}{\mathfrak{m}}^{n+1}$ is the associated graded ring of the local ring at the centre, so its Proj is the projectivized tangent cone.

Steps 2.1, 1.2 and 4.1 prove all the assertions: $E\cong {\mathrm{Proj}}_Z({\mathrm{gr}}_{\mathcal{I}}{\mathcal{O}}_X)$ over $Z$, the fibre description over points of $Z$, and the identification of $E$ with the divisor $V(a)$ in each standard chart.