A point blowup drops pairwise intersection multiplicity by at least one

Put $A={\mathcal{O}}_{X,p}$ with maximal ideal $\mathfrak{m}$, let $I=I_{Y,p}$ and $J=I_{Z,p}$ be the stalk ideals, so that $A\mathrm{/}I={\mathcal{O}}_{Y,p}$ is a one-dimensional regular local ring and hence a discrete valuation ring with valuation $v$ [F1]. Choose $x_1\in \mathfrak{m}$ whose image ${\bar{x}}_1\in A\mathrm{/}I$ is a uniformizer; then $\mathfrak{m}=I+(x_1)$, because both sides are ideals contained in $\mathfrak{m}$ and their images in the discrete valuation ring $A\mathrm{/}I$ are equal to its maximal ideal. The image $J(A\mathrm{/}I)$ is a nonzero ideal of the valuation ring, so $J(A\mathrm{/}I)=({\bar{x}}_1^N)$ for a unique integer $N\ge 1$, and $N=\min \{v(\bar{g}):g\in J\}$; pick $f\in J$ with $v(\bar{f})=N$. By [F2], $N={\mathrm{\ell }}_{A\mathrm{/}I}(A\mathrm{/}(I+J))=m_p(Y\cap Z)$. All claims are local over $p$ and its unique preimage, and blowups commute with the flat base change $\mathrm{Spec}A\to X$ (Flat base change for blowups, and failure without flatness), and localizing the chart saturation $(JB):x_1^{\mathrm{\infty }}$ commutes with taking this schematic closure: a fraction lies in the localized saturation exactly when some power of $x_1$ carries it into the localized ideal. Thus we compute in the local model ${\mathrm{Bl}}_{\mathfrak{m}}(\mathrm{Spec}A)$.

The center ideal on $Y$ is ${\mathcal{I}}_p{\mathcal{O}}_Y$, whose stalk at $p$ is $\mathfrak{m}(A\mathrm{/}I)=({\bar{x}}_1)$; it is not the zero ideal $I{\mathcal{O}}_Y$ defining $Y$ itself. It is the unit ideal away from $p$, and its generator at $p$ is a nonzerodivisor. Coherence allows the stalk generator to generate on a neighbourhood of $p$, so this ideal sheaf is invertible on $Y$. The strict transform of $Y$ is its blowup in this induced center ideal (Strict transforms of closed subschemes are blowups of the subscheme), and the blowup of an invertible ideal is an isomorphism (Blowing up an effective Cartier divisor does nothing). Hence $Y^{\mathrm{\prime }}\to Y$ is an isomorphism and has exactly one point $q$ above $p$, with local ring $A\mathrm{/}I$.

The $x_1$-chart of the local blowup is $B=A[\mathfrak{m}\mathrm{/}{x}_1]$. Mapping $y\mathrm{/}{x}_1^n\in B$ (with $y\in {\mathfrak{m}}^n$) to the unique element $\bar{a}\in A\mathrm{/}I$ with $\bar{y}=\bar{a}{\bar{x}}_1^n$ gives a well-defined $A$-algebra homomorphism $\psi :B\to A\mathrm{/}I$: existence holds because ${\mathfrak{m}}^n\subseteq I+(x_1^n)$, uniqueness because ${\bar{x}}_1$ is a nonzerodivisor of the domain $A\mathrm{/}I$, and compatibility with the relations of the localisation holds because $x_1^k(x_1^{m}y-x_1^n{y}^{\mathrm{\prime }})=0$ implies ${\bar{x}}_1^{k+m+n}(\psi (y\mathrm{/}{x}_1^n)-\psi (y^{\mathrm{\prime }}\mathrm{/}{x}_1^m))=0$. The composite of $\psi$ with the chart morphism $\mathrm{Spec}B\to \mathrm{Spec}A$ is the closed immersion $\mathrm{Spec}(A\mathrm{/}I)\hookrightarrow \mathrm{Spec}A$; since the pullback of $\mathfrak{m}$ to $\mathrm{Spec}(A\mathrm{/}I)$ is the invertible ideal $({\bar{x}}_1)$, the universal property of the blowup gives a unique lift $\mathrm{Spec}(A\mathrm{/}I)\to {\mathrm{Bl}}_{\mathfrak{m}}(\mathrm{Spec}A)$, and this lift is the strict transform $Y^{\mathrm{\prime }}$ in the chart (Universal property of the blowup, Affine blowup standard charts and overlaps, Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains, Strict transform of a closed subscheme).

In the chart the exceptional subscheme is $V(x_1)$: the pulled back center ideal $\mathfrak{m}\mathcal{O}$ is generated by the nonzerodivisor $x_1$ and $E=V(\mathfrak{m}\mathcal{O})$ (The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier, Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains). Hence the scheme-theoretic intersection $Y^{\mathrm{\prime }}\cap E$ is computed in the chart by $A\mathrm{/}I{\otimes }_{B}B\mathrm{/}(x_1)=(A\mathrm{/}I)\mathrm{/}({\bar{x}}_1(A\mathrm{/}I))=A\mathrm{/}(I+(x_1))=A\mathrm{/}\mathfrak{m}=\kappa (p)$; so $Y^{\mathrm{\prime }}\cap E$ is a single reduced point $q$ with residue field $\kappa (p)$ and $m_q(Y^{\mathrm{\prime }}\cap E)={\mathrm{\ell }}_{{\mathcal{O}}_{X^{\mathrm{\prime }},q}}(\kappa (q))=1$, proving claim 2.

The strict transform of $Z$ in the chart is cut out by the saturation $J^{\mathrm{\prime }}=(JB):x_1^{\mathrm{\infty }}$, the closure of $V(JB)$ on the open chart complement of $E$ (Strict transform of a closed subscheme); it contains $f\mathrm{/}{x}_1$, because $x_1\cdot (f\mathrm{/}{x}_1)=f\in JB$. Let $K=\psi (J^{\mathrm{\prime }})\subseteq A\mathrm{/}I$ be the image, an ideal of the discrete valuation ring $A\mathrm{/}I$ containing $\psi (f\mathrm{/}{x}_1)=\bar{f}\mathrm{/}{\bar{x}}_1$, which has valuation $N-1$. If $q\in Z^{\mathrm{\prime }}$, then the closed subscheme $Y^{\mathrm{\prime }}\cap Z^{\mathrm{\prime }}$ of $Y^{\mathrm{\prime }}=\mathrm{Spec}(A\mathrm{/}I)$ contains the closed point $q$, so $K$ is a proper ideal contained in the maximal ideal $({\bar{x}}_1)$ and therefore $K=({\bar{x}}_1^j)$ for some $j\ge 1$; from ${\bar{x}}_1^{N-1}\in K$ we get $j\le N-1$, and hence $m_q(Y^{\mathrm{\prime }}\cap Z^{\mathrm{\prime }})={\mathrm{\ell }}_{A\mathrm{/}I}(A\mathrm{/}I\mathrm{/}K)=j\le N-1<N=m_p(Y\cap Z)$ [F1], proving claim 3.

Finally, if $m_p(Y\cap Z)=N=1$, then the element $\bar{f}\mathrm{/}{\bar{x}}_1$ of step 4.2 is a unit, so $K$ is the unit ideal whenever $q\in Z^{\mathrm{\prime }}$, a contradiction; hence $q\mathrm{\notin }{Z}^{\mathrm{\prime }}$, and since $q$ is the only point of $Y^{\mathrm{\prime }}$ over $p$ by step 2.1, the strict transforms $Y^{\mathrm{\prime }}$ and $Z^{\mathrm{\prime }}$ are disjoint over $p$. This completes claims 1-3 and the additional assertions.