Embedded strict-normal-crossings resolution of a reduced curve on a regular surface

By [F1] the curve $Z$ is an effective Cartier divisor on $S$, with invertible ideal sheaf ${\mathcal{O}}_S(-Z)$. By [F2] it has finitely many irreducible components $Z_1,\dots ,Z_r$, each integral of dimension one with finite normalization; if $r=0$ there is nothing to prove, so assume $r\ge 1$.

Apply [F5] to the original components: finitely many point blowups make their strict transforms pairwise disjoint regular curves. The centers supplied by that proof lie either at a singular point of one current integral curve or at an intersection of two distinct current curves. Each has ambient local dimension two. Indeed at a dimension-one point of a regular surface the ambient ring is a DVR; the defining prime of an integral curve through a closed point of that curve cannot be its maximal ideal, since the curve has local dimension one there, so the curve's local ring would equal the DVR and be regular. Nor can two distinct one-dimensional curves pass through such a point: their prime ideals would both be zero in this local domain, forcing the same irreducible component locally and hence globally. Thus every center in this sequence has dimension two. By [F3] every new exceptional divisor is a regular ${\mathbb{P}}^1$ over the center residue field. Later blowups preserve regularity of each already regular exceptional curve, because the induced point ideal on that curve is Cartier and [F4] identifies its strict transform with the curve. Consequently, at the end of this sequence all components of the total-transform support are regular, but their contacts need not yet have multiplicity one.

At every stage the inverse image of $Z$ is effective Cartier. Locally its ideal is generated by the pullback of a nonzero local equation on a regular irreducible component of the surface. Each blowup chart embeds into the function field of that component (Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains), so this pullback stays nonzero and is a nonzerodivisor in the regular local domain. Its ideal is thus the invertible ideal of an effective Cartier divisor (Total transform of a Cartier divisor). Its support consists of the strict transforms of the original components and precisely the exceptional curves whose centers lie on the preceding total-transform support; the sequence of step 2.1 has this property. These components are all regular by step 2.1. The same Cartier and regularity conclusions remain true during the further point blowups below.

Let $C_1,\dots ,C_s$ be the finitely many regular components of the current support. Every intersection of two distinct components is a finite set of closed points: it is a zero-dimensional closed subscheme of either Noetherian integral curve. The finite set of contact multiplicities therefore has a maximum $M$, unless the components are already disjoint. If $M>1$, blow up each of the finitely many points where this maximum occurs. These centers have local ambient dimension two by the intersection argument in step 2.1. For each pair through a center, [F4] strictly decreases its contact multiplicity at any remaining intersection above that center. Every strict transform stays regular, and its intersection with the new exceptional curve has multiplicity one by [F4]. Pairs away from the center are unchanged; pairs previously disjoint remain disjoint. Hence after this finite set of blowups every pairwise contact, including contacts with all new exceptional curves, has multiplicity less than $M$. Repeat; the positive integer maximum strictly decreases each time, so after finitely many blowups all pairwise contact multiplicities are one, or there are no contacts.

Suppose there is a closed point $p$ at which three or more components $C_1,\dots ,C_s$ ($s\ge 3$) of the current support meet. By step 4.1 all pairwise multiplicities $m_p(C_i\cap C_j)$ equal $1$, and all components are regular curves, so [F4] applies to each pair: blowing up $p$ makes the strict transforms $C_i^{\mathrm{\prime }}$ pairwise disjoint over $p$ and makes each $C_i^{\mathrm{\prime }}$ meet the new exceptional curve $E$ at its own point $q_i$ with $m_{q_i}(C_i^{\mathrm{\prime }}\cap E)=1$. Consequently every point of the new support lying over $p$ carries at most two components (one strict transform together with $E$, or $E$ alone), all components remain regular curves by [F3] and [F4], all pairwise multiplicities remain at most one by step 4.1, and no new point at which three components meet is created. Blowing up all points at which at least three components meet therefore strictly decreases their finite number, and iterating this finitely many times reaches a configuration in which at every closed point at most two components meet, and the pairwise contacts that exist have multiplicity one.

In the terminal configuration the total transform $Z_n$ of $Z$ is an effective Cartier divisor by step 3.1; its support has only regular irreducible components by steps 2.1 and 3.1; and at every closed point either no component passes, exactly one regular component passes, or exactly two regular components pass and meet transversally in the sense $m_p=1$ of Strict normal crossings divisor on a regular surface. No three components meet, by step 5.1. Hence the support is a strict normal crossings divisor in the sense of the definition, which is assertion 3; assertions 1 and 2 are steps 2.1 and 3.1. Every center used was a closed point of a regular surface, so by [F3] no smoothness over a ground field was used or asserted: the ambient surfaces and individual components are regular, the support is SNC, and no higher-dimensional statement is made.