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 Zariski (1939, section III).

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 A² defined by x² = y³ is not normal, because there is a finite birational morphism A¹ → X (namely, t maps to (t³, t²)) which is not an isomorphism. By contrast, the affine line A¹ 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 y² = x²(x + 1), is not normal. This also follows from the definition of normality, since there is a finite birational morphism from A¹ to X which is not an isomorphism; it sends two points of A¹ to the same point in X.

More generally, a scheme X is normal if each of its local rings

O_X,x

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, Zariski (1939, theorem 11) showed that every normal variety is regular outside a subset of codimension at least 2, and a similar result is true for schemes.^[1] 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".^[2]) 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.

If the initial scheme is not irreducible, the normalization is defined to be the disjoint union of the normalizations of the irreducible components.

Examples

Normalization of a cusp

Consider the affine curve

$C={\text{Spec}}\left({\frac {k[x,y]}{y^{2}-x^{5}}}\right)$

with the cusp singularity at the origin. Its normalization can be given by the map

${\text{Spec}}(k[t])\to C$

induced from the algebra map

$x\mapsto t^{2},y\mapsto t^{5}$

Normalization of axes in affine plane

For example,

$X={\text{Spec}}(\mathbb {C} [x,y]/(xy))$

is not an irreducible scheme since it has two components. Its normalization is given by the scheme morphism

${\text{Spec}}(\mathbb {C} [x,y]/(x)\times \mathbb {C} [x,y]/(y))\to {\text{Spec}}(\mathbb {C} [x,y]/(xy))$

induced from the two quotient maps

$\mathbb {C} [x,y]/(xy)\to \mathbb {C} [x,y]/(x,xy)=\mathbb {C} [x,y]/(x)$

$\mathbb {C} [x,y]/(xy)\to \mathbb {C} [x,y]/(y,xy)=\mathbb {C} [x,y]/(y)$

Normalization of reducible projective variety

Similarly, for homogeneous irreducible polynomials $f_{1},\ldots ,f_{k}$ in a UFD, the normalization of

${\text{Proj}}\left({\frac {k[x_{0},\ldots ,x_{n}]}{(f_{1}\cdots f_{k},g)}}\right)$

is given by the morphism

${\text{Proj}}\left(\prod {\frac {k[x_{0}\ldots ,x_{n}]}{(f_{i},g)}}\right)\to {\text{Proj}}\left({\frac {k[x_{0},\ldots ,x_{n}]}{(f_{1}\cdots f_{k},g)}}\right)$

Notes

^ Eisenbud, D. Commutative Algebra (1995). Springer, Berlin. Theorem 11.5
^ Eisenbud, D. Commutative Algebra (1995). Springer, Berlin. Corollary 13.13

References

Eisenbud, David (1995), Commutative algebra. With a view toward algebraic geometry., Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-5350-1, ISBN 978-0-387-94268-1, MR 1322960
Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, p. 91
Zariski, Oscar (1939), "Some Results in the Arithmetic Theory of Algebraic Varieties.", Amer. J. Math., 61 (2): 249–294, doi:10.2307/2371499, JSTOR 2371499, MR 1507376

[1] Eisenbud, D. Commutative Algebra (1995). Springer, Berlin. Theorem 11.5

[2] Eisenbud, D. Commutative Algebra (1995). Springer, Berlin. Corollary 13.13

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