H1 of a normal surface modification injects off its special fibre

Given: A normal two-dimensional Noetherian local domain $A$ essentially of finite type over a field or complete equicharacteristic local base, a projective normal modification $X\to \mathrm{Spec}A$, and $U$ the inverse image of the punctured spectrum.

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-normal-surface-modification-and-normalized-point-blowup. Normal schemes. A locally Noetherian scheme is normal if every local ring ${\mathcal{O}}_{X,x}$ is an integrally closed domain (def-normal-noetherian-ring). This is a local condition on the local rings and is checked on an affine open cover; it does not require the global section ring to be a domain. The empty scheme is normal vacuously. (def-normal-surface-modification-and-normalized-point-blowup)

lem-local-normal-surface-modification-dimension-and-projective-cohomology. Assume AC and DC. Let $(A,\mathfrak{m})$ be a normal Noetherian local domain of dimension two and $f:X\to \mathrm{Spec}A$ an integral modification. Then $X$ has dimension two, all closed points have local dimension two, $f$ is an isomorphism off the closed point, $f_{\ast }{\mathcal{O}}_X={\mathcal{O}}_{\mathrm{Spec}A}$, and its special fibre has dimension at most one. (Dimension and cohomology of local normal surface modifications)

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)

lem-normal-surface-fibre-divisor-conormal-degree-positive. Assume AC and DC. Let $A$ be a normal Noetherian local domain of dimension two and $f:X\to \mathrm{Spec}A$ a normal integral modification. For any nonempty effective Cartier divisor $Z$ supported in the special fibre, some integral component $C$ of $Z$ satisfies ${\mathrm{deg}}_C({\mathcal{O}}_X(-Z){\mathrm{\mid }}_C)>0$. In particular its conormal bundle is not trivial. (Positive conormal degree for a fibre divisor on a normal surface)

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)

thm-proper-pushforward-coherent. Assume the Axiom of Choice and the Axiom of Dependent Choice, inherited from the affine localization theorem, the Čech comparison and the dévissage lemma cited below (The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). (Coherent higher direct images under proper morphisms)