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 .
Geometric and algebraic interpretations of normality.
A morphism of varieties is finite if the inverse image of every point is finite and the morphism is proper. A morphism of varieties
is birational if it restricts to an isomorphism between dense open subsets. So, for example, the cuspidal cubic curve "X" in the affine plane "A"2 defined by "x"2 = "y"3 is not normal, because there is a finite birational morphism "A"1 → "X"
(namely, "t" maps to ("t"3, "t"2)) which is not an isomorphism. By contrast, the affine line "A"1 is normal: it cannot be simplified any further by finite birational morphisms.
A normal complex variety "X" has the property, when viewed as a stratified space using the classical topology, that every link is connected. Equivalently, every complex point "x" has arbitrarily small neighborhoods "U" such that "U" minus
the singular set of "X" is connected. For example, it follows that the nodal cubic curve "X" in the figure, defined by "x"2 = "y"2("y" + 1), is not normal. This also follows from the definition of normality, since there is a finite birational morphism from "A"1 to "X" which is not an isomorphism; it sends two points of "A"1 to the same point in "X".
More generally, a scheme "X" is normal if each of its local rings
is an integrally closed domain. That is, each of these rings is an integral domain "R", and every ring "S" with "R" ⊆ "S" ⊆ Frac("R") such that "S" is finitely generated as an "R"-module is equal to "R". (Here Frac("R") denotes the field of fractions of "R".) This is a direct translation, in terms of local rings, of the geometric condition that every finite birational morphism to "X" is an isomorphism.
An older notion is that a subvariety "X" of projective space is linearly normal if the linear system giving the embedding is complete. Equivalently, "X" ⊆ Pn is not the linear projection of an embedding "X" ⊆ Pn+1 (unless "X" is contained
in a hyperplane Pn). This is the meaning of "normal" in the phrases rational normal curve and rational normal scroll.
Every regular scheme is normal. Conversely, showed that every normal variety is regular outside a subset of codimension at least 2, and a similar result is true for schemes. So, for example, every normal curve is regular.
The normalization.
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 "A"i. Let "B"i be the integral closure of "A"i in its fraction field. Then the normalization of "X" is defined by gluing together the affine schemes
Spec "B"i.
Examples.
If the initial scheme is not irreducible, the normalization is defined to be the disjoint union of the normalizations of the irreducible components.
Normalization of a cusp.
Consider the affine curveformula_1with the cusp singularity at the origin. Its normalization can be given by the mapformula_2induced from the algebra mapformula_3
Normalization of axes in affine plane.
For example,formula_4is not an irreducible scheme since it has two components. Its normalization is given by the scheme morphismformula_5induced from the two quotient mapsformula_6
formula_7
Normalization of reducible projective variety.
Similarly, for homogeneous irreducible polynomials formula_8 in a UFD, the normalization offormula_9is given by the morphismformula_10