Strict normal crossings divisor on a regular surface

Assume the Axiom of Choice for the local factoriality and the equivalence argument below (The Axiom of Choice). Let $S$ be a regular surface: a locally Noetherian scheme of pure dimension two all of whose local rings are regular (A local ring is a nonzero commutative ring with a unique maximal ideal, embedding dimension and regular local ring). Let $D\subseteq S$ be a reduced curve, that is, a reduced closed subscheme of pure dimension one; equivalently, a reduced effective Cartier divisor on $S$ of pure dimension one (Effective cartier divisor, Cartier divisor).

Then $D$ is a strict normal crossings divisor (an SNC divisor) on $S$ if at every closed point $p$ of $S$ one of the following holds:

(a) no irreducible component of $D$ passes through $p$;

(b) exactly one irreducible component of $D$ passes through $p$, and it is regular at $p$;

(c) exactly two irreducible components $Y_1,Y_2$ of $D$ pass through $p$, both are regular at $p$, and they meet transversally at $p$, that is, $m_p(Y_1\cap Y_2)=1$ (Intersection multiplicity of closed subschemes at a point).

Equivalently (Stacks, Lemma 41.21.2, in the locally Noetherian case): $D$ is reduced, every irreducible component of $D$ is a regular scheme, every nonempty scheme-theoretic intersection of two distinct irreducible components of $D$ is a regular scheme of dimension zero, and no three distinct irreducible components of $D$ meet at a point. The equivalent local equation form is: at every point $p\in S$ the stalk ideal $I_{D,p}\subseteq {\mathcal{O}}_{S,p}$ is principal, with a generator of the form $x_1\cdots x_r$ where $r\ge 0$ and $x_1,\dots ,x_r$ extend to a regular system of parameters of ${\mathcal{O}}_{S,p}$ (for a closed point of the surface, $0\le r\le 2$; $r=0$ means the unit ideal).

No smoothness over a ground field is part of the definition; over a non-perfect field the components are regular but need not be smooth. The empty curve $D=\mathrm{\varnothing }$ and a single regular component are SNC divisors by (a) and (b).

[R1] The pointwise conditions and the regular-intersection criterion agree. At a closed point $p\in D$, the local ring of an integral curve component through $p$ has dimension one. Its defining prime in $A={\mathcal{O}}_{S,p}$ is nonzero, since that component has dimension one and is not an irreducible component of the pure two-dimensional surface. The chain from the zero prime of the regular local domain $A$ through this prime to its maximal ideal has length two. Thus $A$ has dimension two, and is a unique factorization domain (Regular local rings are unique factorization domains). The irreducible components of $D$ through $p$ correspond to the height-one prime ideals ${\mathfrak{p}}_1,\dots ,{\mathfrak{p}}_r$ containing the radical ideal $I_{D,p}$, and $I_{D,p}={\mathfrak{p}}_1\cap \cdots \cap {\mathfrak{p}}_r$ is generated by the product of prime generators $x_1\cdots x_r$ of those primes, a nonzerodivisor; so near $p$ the curve has the local equation $x_1\cdots x_r$. A component $Y_i$ is regular at $p$ exactly when its local equation $x_i$ extends to a regular system of parameters of $A$ (regular local quotient by parameter is regular), which for one element of a two-dimensional regular local ring means that $x_i$ is not in ${\mathfrak{m}}_p^2$. Conversely, if $A\mathrm{/}(x_i)$ is regular of dimension one, its cotangent space has dimension one; if $x_i$ lay in ${\mathfrak{m}}_p^2$, the quotient cotangent space would still be the two-dimensional ${\mathfrak{m}}_p\mathrm{/}{\mathfrak{m}}_p^2$, a contradiction. For two distinct components $Y_1,Y_2$ the scheme-theoretic intersection near $p$ is cut out by $(x_1,x_2)$; it is a regular scheme of dimension zero precisely when $(x_1,x_2)={\mathfrak{m}}_p$, that is, when $x_1,x_2$ form a regular system of parameters, and this is exactly $m_p(Y_1\cap Y_2)=1$. Three distinct components cannot meet at $p$ when the criterion holds, because ${\bigcap }_{j\in J}{D}_j$ for a three-element set $J$ would have to be regular of codimension three inside the surface $S$ and therefore empty. Conversely, the pointwise conditions give the criterion: components are regular, every local intersection of two distinct components is a reduced point, and no three components meet. This is the surface case of the finite-subset form of Stacks Lemma 41.21.2.

[R2] Why the two descriptions of the ambient object agree. For a reduced closed subscheme $D\subseteq S$ of pure dimension one, being an effective Cartier divisor is a local condition: at $p$ the ideal $I_{D,p}$ is generated by one nonzerodivisor. If $p\mathrm{\notin }D$ it is the unit ideal; if $p$ is the generic point of a component, ${\mathcal{O}}_{S,p}$ is a discrete valuation ring and $I_{D,p}$ is principal; if $p$ is a closed point, [R1] exhibits the principal generator $x_1\cdots x_r$. Thus every reduced curve on a regular surface is a reduced effective Cartier divisor, and conversely a reduced effective Cartier divisor of pure dimension one is a reduced curve.

[R3] Closed points suffice. On a scheme of dimension at most two, every non-closed point of $D$ is the generic point of an irreducible component, where the local ring of $D$ is a field and no two distinct components meet; so regularity of the components and all pairwise contacts are detected at closed points. This is why the definition tests only closed points.