Normal scheme modifications and normalized point blowups

Assume the Axiom of Choice as inherited from the blowup and scheme-construction suppliers.

Normal schemes. A locally Noetherian scheme is normal if every local ring ${\mathcal{O}}_{X,x}$ is an integrally closed domain (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.

Normalization of an integral scheme. Let $S$ be an integral scheme with function field $K=K(S)$ (Integral schemes), and let $\mathrm{Spec}A\subseteq S$ be a nonempty affine open subset. Write $\bar{A}$ for the integral closure of $A$ in $K$, that is, the subring of elements of $K$ that are integral over $A$. The normalization of $S$ is the $S$-scheme

$\nu \text{\hspace{0.17em}⁣}:S^{\nu }⟶S$

obtained by gluing the affine schemes $\mathrm{Spec}\bar{A}$ over the nonempty affine open subsets $\mathrm{Spec}A\subseteq S$, inside the common function field $K$. This is well defined: by Integrality and integral closure commute with localisation the formation of $\bar{A}$ commutes with localization, so for a principal open $D(f)\subseteq \mathrm{Spec}A$ the canonical map $(\bar{A}{)}_f\to \overline{A_f}$ is an isomorphism onto the integral closure of $A_f$ in $K$; the affine integral-closure algebras therefore identify on overlaps inside $K$ and satisfy the cocycle identity, and Glue relative spectra of affine-local algebras constructs the glued scheme and its canonical morphism to $S$. The scheme $S^{\nu }$ is integral, $\nu$ is affine, and $K(S^{\nu })=K$; the normalization is characterized up to unique isomorphism over $S$ by this construction. Finiteness of the normalization — that $S^{\nu }\to S$ is a finite morphism — is an additional assertion about $S$ and is never part of the definition; it is proved separately on this page for the surfaces that occur below.

Modifications. Let $S$ and $X$ be integral schemes. A modification of $S$ is a proper morphism $f\text{\hspace{0.17em}⁣}:X\to S$ (Proper morphisms) that induces an isomorphism $K(S)\to K(X)$ of function fields, so that $f$ is birational. A regular resolution of $S$ is a modification $f\text{\hspace{0.17em}⁣}:X\to S$ whose source $X$ is regular (all local rings regular). The modulus here is a property of the morphism $f$, not of the source alone: the same scheme $X$ may occur as source of modifications of several different integral schemes.

Point blowups and their normalization. Let $S$ be a locally Noetherian integral scheme (Locally Noetherian and Noetherian schemes) and let $s\in S$ be a closed point whose ideal sheaf ${\mathcal{I}}_s\subseteq {\mathcal{O}}_S$ is coherent and nonzero. The point blowup of $S$ at $s$ is the blowup ${\mathrm{Bl}}_s(S)\to S$ of $S$ along ${\mathcal{I}}_s$ (Blowup of a scheme along an ideal sheaf). It is locally projective, hence proper, over $S$; a single global projective-space embedding is asserted only when the additional global-generation or projective-base hypotheses are available. The normalized point blowup of $S$ at $s$ is the composite

${\mathrm{Bl}}_s(S{)}^{\nu }⟶{\mathrm{Bl}}_s(S)⟶S,$

that is, the point blowup followed by normalization of its source. It is a morphism of integral schemes. It is proper when the normalization morphism ${\mathrm{Bl}}_s(S{)}^{\nu }\to {\mathrm{Bl}}_s(S)$ is finite; no properness assertion is made when that finiteness has not been established. The finite-normalization results below verify this condition in the surface classes used in the resolution arguments.

Resolution by normalized point blowups. A resolution of $S$ by normalized point blowups is a morphism $X=S_n\to S_{n-1}\to \cdots \to S_0=S^{\nu }\to S$ obtained as follows: start with the normalization $S_0:=S^{\nu }$ of $S$; choose closed points $s_i\in S_i$ for $i=0,\dots ,n-1$, each with nonzero coherent point ideal, and let $S_{i+1}$ be the normalized point blowup of $S_i$ at $s_i$; require that each normalization in the sequence, including the initial normalization $S^{\nu }\to S$, be finite, and that the terminal scheme $S_n$ be regular. Each arrow $S_{i+1}\to S_i$ is then a modification, the composite is proper, and the terminal scheme is a regular resolution of $S$. If $S$ is already normal, the initial normalization $S^{\nu }\to S$ is an isomorphism, so the definition then starts effectively at $S$ itself; if $S$ is regular, the empty sequence $n=0$ with $S_0=S^{\nu }=S$ exhibits the identity as a resolution by normalized point blowups.

• The definition does not assert that a resolution by normalized point blowups exists for a given integral scheme $S$: it names the shape of the object. Existence for surfaces is proved later on this page, under the hypotheses stated there.
• Blowing up a regular point on a regular surface is generally not an isomorphism: its exceptional fibre is a projective line. The blowup remains regular, so its normalization is the identity and the map is still a proper birational modification. On a regular one-dimensional integral scheme, a closed-point ideal is invertible and its blowup is an isomorphism: at the point its stalk is the principal maximal ideal of a DVR, and away from the point it is the unit ideal. This need not hold on a singular curve. The resolution arguments below choose singular centres when they need to alter the regularity.
• The phrase modification is used here only for integral schemes, so that function fields are defined and the birationality condition makes sense. No separatedness hypothesis beyond properness is imposed.
• The normalization is defined by gluing over the nonempty affine opens of an integral scheme; on the empty scheme there is no function field and no normalization is defined.