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 {{harvs|txt|last=Zariski|authorlink=Oscar Zariski|year=1939|loc=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.

Image:Newtonsche Knoten.png

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, {{harvtxt|Zariski|1939|loc=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.Eisenbud, D. Commutative Algebra (1995). Springer, Berlin. Theorem 11.5 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".Eisenbud, D. Commutative Algebra (1995). Springer, Berlin. Corollary 13.13) 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

References

{{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra. With a view toward algebraic geometry. | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94268-1 | mr=1322960 | year=1995 | volume=150 | doi=10.1007/978-1-4612-5350-1}}
{{Hartshorne AG}}, p. 91
{{citation|mr=1507376|last=Zariski|first= Oscar|title=Some Results in the Arithmetic Theory of Algebraic Varieties.

|journal=Amer. J. Math.|volume= 61 |year=1939|issue= 2|pages= 249–294|jstor=2371499|doi=10.2307/2371499}}

Category:Scheme theory

Category:Algebraic geometry