Separation of finitely many curve components by point blowups

Apply [F1] successively to the strict transforms of $Y_1,\dots ,Y_r$. These applications remain legitimate: a blowup centered away from another component leaves it unchanged; at a center on that component, its strict transform is its intrinsic point blowup (Strict transforms of closed subschemes are blowups of the subscheme), which is finite and retains the same finite normalization (The blowup of a one-dimensional integral Noetherian scheme at a closed point is finite, The finite normalization of a curve factors through the blowup of a closed point). Thus every not-yet-regularized component still satisfies the finite-normalization hypothesis of [F1]. During each later sequence, every already regular component stays regular by [F2]. Since there are finitely many components and each intrinsic regularization is finite, after finitely many ambient point blowups all strict transforms are regular integral curves. Rename these curves and the current ambient scheme $Y_1,\dots ,Y_r$ and $X$.

(The multiplicity maximum) By [F4] each intersection $Y_i\cap Y_j$ with $i\mathrm{\ne }j$ is a finite set of closed points, so the set of numbers $m_p(Y_i\cap Y_j)$, over all pairs and all intersection points, is finite. If it is empty, the $Y_i$ are already pairwise disjoint and we are done. Otherwise let $n\ge 1$ be its maximum.

(Phase $n\ge 2$: lowering the maximum) Suppose $n\ge 2$ and let $p_1,\dots ,p_s$ be the finitely many points at which the maximum $n$ is attained. Blow up these points one after another (each is a closed point of the current ambient scheme, and the strict transforms are updated). For each pair $(i,j)$ meeting at a blown-up point $p_t$, [F3] shows that every multiplicity of $Y_i^{\mathrm{\prime }}\cap Y_j^{\mathrm{\prime }}$ over $p_t$ is strictly smaller than $n$; new intersections arise only with the exceptional curves of the blowups and have multiplicity $1$; and the local data at points that are not blown up are unchanged. Hence after these finitely many blowups either no pairwise intersection remains, in which case the curves are already disjoint and the process stops, or the maximum of the remaining pairwise multiplicities is strictly smaller than $n$. All strict transforms remain regular integral curves by [F2], and their pairwise intersections remain finite. Repeating this phase at most $n-1$ times, always lowering the current maximum, the process either stops with pairwise disjoint curves or reaches the case in which the maximum is $1$. Every phase consists of finitely many blowups, so the total number of blowups is finite.

(Phase $n=1$: separating the components) Assume the maximum of all pairwise multiplicities is $1$ and let $p_1,\dots ,p_s$ be the finitely many points lying in at least two of the curves. Blow up these points one after another. For each pair $(i,j)$ meeting at a point $p_t$ with $m_{p_t}(Y_i\cap Y_j)=1$, the final clause of [F3] shows that $Y_i^{\mathrm{\prime }}$ and $Y_j^{\mathrm{\prime }}$ are disjoint over $p_t$; after the finitely many blowups of this phase, every pair of strict transforms meets over none of the points $p_t$. Since intersections can only occur at the $p_t$ (the strict transforms agree with the original curves away from the blown-up points), the final strict transforms $Y_i^{\mathrm{\prime }}$ are pairwise disjoint, and they remain regular curves by [F2].

Combining steps 1.1, 3.1 and 3.2: first make all components regular, then lower the maximum pairwise multiplicity by finitely many point blowups until either no intersection remains or the maximum is $1$; in the latter case separate the remaining multiplicity-$1$ contacts by finitely many further point blowups. The composite is a finite sequence of blowups of $X$ at closed points whose final strict transforms are pairwise disjoint regular curves, which is the assertion.