Normality and fibre cohomology of a rational surface point blowup

Given: A rational permitted normal local surface domain $(A,\mathfrak{m},\kappa )$, its ordinary point blowup $X_0$, and the exceptional fibre $E$ with tautological conormal line $L={\mathcal{O}}_E(1)$.

cor-regular-quotient-cohen-macaulay-equivalence. Assume the Axiom of Choice (The Axiom of Choice). Under the hypotheses of lem-regular-quotient-preserves-depth-dimension-gap, $M$ is Cohen--Macaulay if and only if $M\mathrm{/}xM$ is Cohen--Macaulay. (Cohen--Macaulayness and a regular parameter quotient)

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-degree-invertible-sheaf-proper-dimension-one. Assume the Axiom of Choice, inherited from the Euler-characteristic supplier below (The Axiom of Choice). Let $k$ be a field (def-field) and let $C$ be a proper $k$-scheme (def-proper-morphism) whose underlying topological space is Noetherian of dimension at most one (def-dimension-noetherian-topological-space, def-locally-noetherian-and-noetherian-scheme). (Degree of an invertible sheaf on a proper one-dimensional scheme)

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-embedding-dimension-and-regular-local-ring. For a nonzero commutative Noetherian local ring $(R,\mathfrak{m},k)$, define $\mathrm{edim}R={\dim }_k(\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2)$. The ring is regular local when $\mathrm{edim}R=\dim R$. The cotangent space is intrinsic, and is finite-dimensional because $\mathfrak{m}$ is finitely generated. (embedding dimension and regular local ring)

def-rational-normal-surface-singularity-and-bounded-modification-h1. Assume AC and DC. A normal two-dimensional Noetherian local domain $A$ essentially of finite type over a field or complete equicharacteristic local base defines a rational singularity if $H^1(Y,{\mathcal{O}}_Y)=0$ for every normal integral proper modification $Y\to \mathrm{Spec}A$. Bounded modification H1 means these modules have uniformly bounded $A$-length. (Rational normal surface singularities and bounded modification cohomology)

lem-eventual-global-generation-coherent-twists. Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a Noetherian commutative ring (def-noetherian-ring-and-module) and let $X$ be a scheme projective over $A$ in the finite-dimensional H-projective convention (def-projective-morphism-pre-proj): the structure morphism $X\to \mathrm{Spec}A$ (def-affine-scheme-spectrum) factors over (Eventual generation of coherent projective twists)

lem-finite-over-projective-noetherian-affine-base-is-projective. Assume AC. Let $R$ be Noetherian, let $X$ be projective over $R$, and let $Y\to X$ be finite. Then $Y\to X$ admits a closed immersion into one relative projective space over $X$, and $Y$ is projective over $R$. Finite compositions of projective morphisms between such schemes are projective. (Finite schemes over projective schemes are projective over a Noetherian affine base)

lem-normal-domain-implies-s-two. Assume the Axiom of Choice (The Axiom of Choice). Every commutative Noetherian integrally closed domain satisfies $(S_2)$. (normal domain implies s two)

lem-rational-surface-exceptional-ideal-powers-and-sections. Assume AC and DC. Let $A$ be a rational permitted normal local surface domain and $X\to \mathrm{Spec}A$ a normal projective modification. A coherent globally generated sheaf $F$ on $X$ has $H^1(X,F)=0$. If the scheme-theoretic closed fibre is Cartier with ideal $I=\mathfrak{m}{\mathcal{O}}_X$, then $H^0(X,I^n)={\mathfrak{m}}^n$ and $H^1(X,I^n)=0$ for all $n\ge 0$. (Powers and sections of a rational surface exceptional ideal)

lem-surface-finite-type-normalization-finite. Assume AC and DC. Every integral finite-type algebra over a field or a complete equicharacteristic Noetherian local base has finite normalization, and so do its localizations. Integral schemes of finite type over these bases consequently have finite scheme normalization. (Surface finite type normalization finite)

thm-pullback-center-ideal-invertible. Assume the Axiom of Choice, inherited from the relative Proj construction (The Axiom of Choice). Let $\mathcal{I}$ be a quasi-coherent ideal sheaf of finite type on a scheme $X$ (def-quasi-coherent-ideal-sheaf), let $\pi \text{\hspace{0.17em}⁣}:{\mathrm{Bl}}_{\mathcal{I}}X\to X$ be its blowup and let $E={\pi }^{-1}(Z)$ be the exceptional subscheme, with the convention that $\mathcal{O}(1)$ (The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier)

thm-serre-vanishing. Assume the Axiom of Choice as inherited from the cited suppliers (The Axiom of Choice). Let $A$ be a Noetherian commutative ring with $1$, let $X$ be a scheme projective over $A$ in the finite-dimensional H-projective convention (def-projective-morphism-pre-proj): the structure morphism $X\to \mathrm{Spec}A$ factors as a closed immersion (Serre vanishing for coherent sheaves and ample twists)