Canonical pullback is surjective after blowing up a rational singular point

Given: A nonregular rational normal local surface domain $(A,\mathfrak{m},\kappa )$ in the permitted regular-base dualizing setting, its ordinary point blowup $f\text{\hspace{0.17em}⁣}:X\to \mathrm{Spec}A$, the exceptional divisor $E$ and the 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)

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-cm-projective-curve-canonical-positive-twist-vanishing-generation. Assume AC and DC. Let $E$ be a projective pure CM curve over a field $\kappa$, with $H^0(E,O_E)=\kappa$ and canonical module ${\omega }_E$. If $L$ is globally generated and nontrivial, $H^1(E,{\omega }_E\otimes L)=0$. If $L$ is very ample with ${\mathrm{deg}}_{\kappa }L\ge 2$, then ${\omega }_E\otimes L$ is globally generated. These statements allow nonreduced $E$. (Positive canonical twists on projective Cohen–Macaulay curves)

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)

lem-normal-surface-trace-cokernel-dualizes-h1-and-bounds-it. Assume AC and DC. Let $R$ be regular local of dimension two and $A$ a finite normal local $R$-domain in the permitted class. For a projective normal modification $X$, put $M=H^1(X,{\mathcal{O}}_X)$. (Trace cokernels detect and bound normal surface H1)

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)

thm-nakayama-lemma. Assume the Axiom of Choice. Let $R$ be a commutative ring, let $I⊴R$ satisfy $I\subseteq J(R)$, and let $M$ be a finitely generated left $R$-module. If $IM=M$, then $M=0$. (Assuming the Axiom of Choice, Nakayama's lemma)

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)