The tangent conic of a rational Gorenstein surface singularity

Given: A nonregular rational normal local surface domain $(A,\mathfrak{m},\kappa )$ in the permitted canonical-module setting with ${\omega }_A\cong A$, its ordinary point blowup $f\text{\hspace{0.17em}⁣}:X\to \mathrm{Spec}A$, exceptional divisor $E$ and tautological ideal $I={\mathcal{O}}_X(1)$.

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)

def-hilbert-function-and-hilbert-series. Let $S={⨁}_{n\ge 0}{S}_n$ be a graded ring and $M={⨁}_{n\in \mathbb{Z}}{M}_n$ a graded $S$-module. Assume each homogeneous piece $M_n$ has finite length as an $S_0$-module and that $M_n=0$ for all sufficiently negative $n$. The Hilbert function of $M$ is (The Hilbert function and formal Hilbert series of a graded module with finite-length pieces)

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-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)

cor-degree-additive-proper-curve. Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field and let $C$ be a proper $k$-scheme (def-proper-morphism) whose underlying topological space has dimension at most one (def-dimension-noetherian-topological-space). For all invertible ${\mathcal{O}}_C$-modules $\mathcal{L}$ and $\mathcal{M}$ (def-invertible-sheaf): 1. (Degree is additive on invertible sheaves over a proper curve)

thm-serre-duality-for-coherent-sheaves-on-projective-cm-scheme. Assume AC. Let $k$ be a field and let $X$ be a projective, pure $d$-dimensional Cohen–Macaulay $k$-scheme. Let $D_X={\omega }_X[d]$ be its normalized dualizing complex, and let $t_X:H^d(X,{\omega }_X)\to k$ be its trace. (Serre duality for coherent sheaves on a projective Cohen–Macaulay scheme)

thm-cohomology-projective-space-twisting-sheaves. Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a commutative ring with $1$ (def-commutative-ring), let $n\ge 0$, let $d\in \mathbb{Z}$ and let $X={\mathbb{P}}_A^n\cong \mathrm{Proj}A[x_0,\dots ,x_n]$ be relative projective space (def-relative-projective-space-standard-charts, def-polynomial-ring-on-a-family-of-indeterminates), with twisting sheaf (Cohomology of O(d) on projective space)