A triple-cubic surface branch reduces after two successors

Given: A rational Gorenstein normal local surface in the permitted setting with square tangent conic whose nonzero controlling cubic is a scalar multiple of a cube.

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-double-plus-simple-cubic-rational-surface-branch-terminates. Assume AC and DC. Let $A$ be a rational Gorenstein normal local surface in the permitted canonical-module setting, with normal completion and generators $\mathfrak{m}=(x,y,z)$. If $z^2+ax{y}^2\in z{\mathfrak{m}}^2+(x,y{)}^4$ for a unit $a$, all its singular point-blowup branches terminate. (The double-plus-simple cubic surface branch terminates)

lem-nonsquare-tangent-conic-rational-surface-blowups-terminate. Assume AC and DC. For a nonregular rational normal local surface domain in the permitted class with invertible canonical module and normal completion, if its tangent-conic quadratic is not a scalar times a square, repeatedly blowing up its singular points terminates in a regular model. (Nonsquare tangent-conic surface singularities terminate under point blowups)

lem-rational-gorenstein-surface-tangent-conic-and-hilbert-function. Assume AC and DC. Let $A$ be a nonregular rational normal local surface domain in the permitted canonical-module setting, with ${\omega }_A\cong A$. Then its point blowup is normal with trivial canonical module. Its exceptional conormal $L$ has degree two and ${\dim }_{\kappa }{\mathfrak{m}}^n\mathrm{/}{\mathfrak{m}}^{n+1}=2n+1$. (The tangent conic of a rational Gorenstein surface singularity)

lem-square-tangent-conic-blowup-singularities-controlled-by-a-cubic. Assume AC and DC. For a rational Gorenstein normal local surface singularity with square tangent conic, choose $\mathfrak{m}=(x_1,x_2,z)$ and a relation $z^2=\sum a_{ijk}{x}_i{x}_j{x}_k$. There is a nonzero homogeneous cubic $H\in \kappa [X_1,X_2]$ whose zero scheme on the reduced exceptional line contains all singular successors. (A square-conic blowup has cubic-controlled singular successors)

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-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-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-local-quotient-by-parameter-is-regular. Assume the Axiom of Choice (The Axiom of Choice). Let $(R,\mathfrak{m},k)$ be regular local of dimension $d$, and let $x\in \mathfrak{m}\setminus {\mathfrak{m}}^2$. Then $R\mathrm{/}(x)$ is regular local, of dimension and embedding dimension $d-1$. (regular local quotient by parameter is regular)

lem-surface-regular-fibres-preserve-normality. Assume AC and DC. A flat map of Noetherian rings with regular fibres carries normality of the base to normality of the target. Consequently a normal essentially finite-type local ring over a field or complete equicharacteristic Noetherian local base has normal maximal-adic completion, which is a domain. (Surface regular fibres preserve normality)