Normalization

Normal varieties are regular in codimension 1.
- For curves, this implies nonsingular.
- For surfaces, this implies no singular curves, only isolated singular points.

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Idea: for affines, \(\operatorname{Spec}A\) is normal iff \(A\) is an integral domain.
Geometrically: for varieties over fields, \(X\) is normal iff any finite birational morphism is an isomorphism.
Definition: a scheme is normal iff all of its local rings are integral domains which are integrally closed.
Constructing the normalization: take an affine open cover \(\left\{{\operatorname{Spec}A_i}\right\}\rightrightarrows X\), let \(B_i = \operatorname{cl}^{\mathrm{int}} (A_i)\) be their integral closures, and glue.
Any reduced scheme has a unique normalization \(\tilde X\to X\) which is birational.
- If \(\dim X = 1\) then the normalization \(\tilde X\) is a regular scheme.
- If \(\dim X = 2\) then \(\tilde X\) has only isolated singularities.

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complete curves correspond to taking not just the prime places from the function field, but all places. Can take the projective closure to obtain a complete curve.

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Serre’s Criterion

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For curves, this yields a resolution of singularities: attachments/Pasted%20image%2020220905134622.png

Many surface singularities are normal. For example, every hypersurface singularity is \(S_2\), so that a hypersurface singularity is normal if and only if it is smooth in codimension one. Similarly, every quotient singularity is normal.

Examples

The simplest examples of non-normal varieties: must be dimension 1 or more, so take a curve. If degree 1 or 2, it will automatically be nonsingular and smooth implies normal in dimension 1. So take degree 3 or more, e.g. \(V(y^2=x^3+x^2)\) or \(V(y^2=x^3)\).