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Normal scheme
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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.[citation needed] 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.[citation needed]
Normal varieties were introduced by Zariski.
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
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. For instance, in the case of the nodal cubic X in the figure, the local ring is not integrally closed in its field of fractions, since y/x is integral over A but is not in A. Therefore X is not normal at the point (0,0).
![{\displaystyle A=\left(k[x,y]/(y^{2}-x^{2}(x+1))\right)_{(x,y)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ecc3d8fce8672c3c60fb918f414d9b1df429cfe)
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 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.
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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.[citation needed] 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.[citation needed]
Normal varieties were introduced by Zariski.
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
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. For instance, in the case of the nodal cubic X in the figure, the local ring is not integrally closed in its field of fractions, since y/x is integral over A but is not in A. Therefore X is not normal at the point (0,0).
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 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.