Normal scheme

Last updated January 26, 2024

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.

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 x² = y²(y + 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

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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

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

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

[1]

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