Rational normal surfaces reduce to an invertible canonical module

Given: A rational normal local surface domain $A$ in the permitted regular-base dualizing setting, its finite torsion-free rank-one canonical module ${\omega }_A$, and projective normal modifications of $A$.

def-axiom-of-choice. The Axiom of Choice (AC) is the following statement. > Every family of nonempty sets has a choice function > (def-choice-function). Written out: for every set $\mathcal{F}$ all of whose members are nonempty, there exists a function $g$ with domain $\mathcal{F}$ satisfying $g(S)\in S$ for all $S\in \mathcal{F}$. (The Axiom of Choice)

def-dependent-choice. Let $X$ be a set and let $R\subseteq X\times X$ be a binary relation on $X$. Call $R$ entire on $X$ when $\text{for every }x\in X\text{ there is }y\in X\text{ with }xRy\mathrm{.}$ The Axiom of Dependent Choice, written $\mathrm{D}\mathrm{C}$, is the following statement. (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)

lem-normal-projective-surface-dualizing-module-over-regular-local-base. Assume AC and DC. Let $R$ be a regular Noetherian local ring of dimension two, let $A$ be a finite normal local $R$-domain of dimension two, with $R\hookrightarrow A$ local, and let $X$ be a normal integral scheme of dimension two projective over $R$, with a proper birational map $f:X\to \mathrm{Spec}A$. Put ${\omega }_A={\mathrm{Hom}}_R(A,R)$. (Dualizing modules and trace pairing for normal projective surface modifications)

lem-rank-one-torsion-free-surface-module-principalized-by-an-ideal-blowup. Assume AC and DC. For a finite torsion-free rank-one module $M$ over a Noetherian domain $A$, there is a nonzero ideal $J$ and a blowup $b:Y={\mathrm{Bl}}_J\mathrm{Spec}A\to \mathrm{Spec}A$ such that $b^{\ast }M$ modulo torsion is invertible; the same holds on every integral model dominating $Y$. (A rank-one surface module is principalized by an ideal blowup)

lem-normalized-point-blowups-dominate-local-normal-surface-modifications. Assume AC and DC. Let $A$ be a normal two-dimensional Noetherian local domain essentially of finite type over a field or a complete equicharacteristic Noetherian local ring. Let $S$ be a normal integral modification of $\mathrm{Spec}A$, and let $Y\to S$ be an integral modification with normal $Y$. (Normalized point blowups dominate local normal surface modifications)

lem-rational-normal-surface-point-blowup-normal-and-fibre-cohomology. Assume AC and DC. For a rational permitted normal local surface domain $(A,\mathfrak{m},\kappa )$, its ordinary point blowup $X$ is normal. Its exceptional fibre $E$ is a projective pure CM curve, its tautological conormal line $L={\mathcal{O}}_E(1)$ is very ample, and $H^1(E,L^n)=0$, $H^0(E,L^n)={\mathfrak{m}}^n\mathrm{/}{\mathfrak{m}}^{n+1}$ for $n\ge 0$. (Normality and fibre cohomology of a rational surface point blowup)

lem-rational-singular-point-blowup-canonical-pullback-surjective. Assume AC and DC. For a nonregular rational normal local surface domain in the permitted regular-base dualizing setting, let $f:X\to \mathrm{Spec}A$ be its ordinary point blowup, $E$ its exceptional divisor and $I=O_X(1)$. Then $H^1(X,{\omega }_X\otimes I^n)=0$ for $n\ge 0$ and the canonical evaluation $f^{\ast }{\omega }_A\to {\omega }_X$ is surjective. (Canonical pullback is surjective after blowing up a rational singular point)

lem-rational-surface-local-rings-propagate-by-point-sequence-spreading. Assume AC and DC. If a permitted normal local surface domain $A$ is rational and $A\subset B$ is a normal two-dimensional local domain with the same fraction field, essentially of finite type over $A$, then $B$ is rational. (Rationality propagates to birational local surface rings)

lem-regular-base-dualizing-traces-compose-on-rational-modifications. Assume AC and DC. Let $R$ be regular local of dimension two, $A$ finite normal local over $R$, and let $g:X^{\mathrm{\prime }}\to X$ be a morphism of projective normal modifications over $A$. Their regular-base dualizing complexes are independent of the chosen projective embeddings up to the unique isomorphism preserving their duality pairings. (Dualizing traces compose and become isomorphisms on rational modifications)

lem-regular-base-surface-cartier-curve-canonical-adjunction. Assume AC and DC. For a normal projective surface modification $X$ over a finite normal local domain $A$ of a regular two-dimensional local ring $R$, let $E$ be a Cartier closed fibre with residue field $\kappa$ and conormal $L=O_X(-E){\mathrm{\mid }}_E$. (Canonical adjunction for a Cartier fibre curve)

lem-blowup-of-closed-point-of-regular-surface-is-regular. Assume the Axiom of Choice (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 (def-embedding-dimension-and-regular-local-ring). (Point blowups of regular surfaces stay regular, with rational exceptional curves at two-dimensional local rings)