Normal scheme

In algebraic geometry, an algebraic variety or scheme X is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety X (understood to be irreducible) is normal if and only if the ring O(X) of regular functions on X is an integrally closed domain. A variety X over a field is normal if and only if every finite birational morphism from any variety Y to X is an isomorphism. Normal varieties were introduced by .

Any reduced scheme X has a unique normalization: a normal scheme Y with an integral birational morphism Y → X. (For X a variety over a field, the morphism Y → X is finite, which is stronger than "integral".) The normalization of a scheme of dimension 1 is regular, and the normalization of a scheme of dimension 2 has only isolated singularities. Normalization is not usually used for resolution of singularities for schemes of higher dimension. To define the normalization, first suppose that X is an irreducible reduced scheme X. Every affine open subset of X has the form Spec R with R an integral domain. Write X as a union of affine open subsets Spec Ai. Let Bi be the integral closure of Ai in its fraction field. Then the normalization of X is defined by gluing together the affine schemes
Spec Bi. If the initial scheme is not irreducible, the normalization is defined to be the disjoint union of the normalizations of the irreducible components.

Consider the affine curvewith the cusp singularity at the origin. Its normalization can be given by the mapinduced from the algebra map

For example,is not an irreducible scheme since it has two components. Its normalization is given by the scheme morphisminduced from the two quotient maps