Morphism of algebraic varieties


In algebraic geometry,[] a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function.
A regular map whose inverse is also regular is called biregular, and the biregular maps are the isomorphisms of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the concepts of rational and birational maps are widely used as well; they are partial functions that are defined locally by rational fractions instead of polynomials.
An algebraic variety has naturally the structure of a locally ringed space; a morphism between algebraic varieties is precisely a morphism of the underlying locally ringed spaces.

Definition

If X and Y are closed subvarieties of and , then a regular map is the restriction of a polynomial map. Explicitly, it has the form:
where the s are in the coordinate ring of X:
where I is the ideal defining X. The image f lies in Y, and hence satisfies the defining equations of Y. That is, a regular map is the same as the restriction of a polynomial map whose components satisfy the defining equations of.
More generally, a map f : X→''Y between two varieties is regular at a point x'' if there is a neighbourhood U of x and a neighbourhood V of f such that f ⊂ V and the restricted function f : U→''V is regular as a function on some affine charts of U'' and V. Then f is called regular, if it is regular at all points of X.
  • Note: It is not immediately obvious that the two definitions coincide: if X and Y are affine varieties, then a map f : X→''Y is regular in the first sense if and only if it is so in the second sense. Also, it is not immediately clear whether regularity depends on a choice of affine charts This kind of a consistency issue, however, disappears if one adopts the formal definition. Formally, an algebraic variety is defined to be a particular kind of a locally ringed space. When this definition is used, a morphism of varieties is just a morphism of locally ringed spaces.
The composition of regular maps is again regular; thus, algebraic varieties form the category of algebraic varieties where the morphisms are the regular maps.
Regular maps between affine varieties correspond contravariantly in one-to-one to algebra homomorphisms between the coordinate rings: if f'' : X→''Y is a morphism of affine varieties, then it defines the algebra homomorphism
where are the coordinate rings of X'' and Y; it is well-defined since is a polynomial in elements of. Conversely, if is an algebra homomorphism, then it induces the morphism
given by: writing
where are the images of 's. Note as well as In particular, f is an isomorphism of affine varieties if and only if f# is an isomorphism of the coordinate rings.
For example, if X is a closed subvariety of an affine variety Y and f is the inclusion, then f# is the restriction of regular functions on Y to X. See #Examples below for more examples.

Regular functions

In the particular case that equals the regular maps are called regular functions, and are algebraic analogs of smooth functions studied in differential geometry. The ring of regular functions is a fundamental object in affine algebraic geometry. The only regular function on a projective variety is constant.
A scalar function is regular at a point if, in some open affine neighborhood of, it is a rational function that is regular at ; i.e., there are regular functions, near such that and does not vanish at. Caution: the condition is for some pair not for all pairs ; see [|Examples].
If X is a quasi-projective variety; i.e., an open subvariety of a projective variety, then the function field k is the same as that of the closure of X and thus a rational function on X is of the form g/''h for some homogeneous elements g'', h of the same degree in the homogeneous coordinate ring of . Then a rational function f on X is regular at a point x if and only if there are some homogeneous elements g, h of the same degree in such that f = g/''h and h'' does not vanish at x. This characterization is sometimes taken as the definition of a regular function.

Comparison with a morphism of schemes

If and are affine schemes, then each ring homomorphism determines a morphism
by taking the pre-images of prime ideals. All morphisms between affine schemes are of this type and gluing such morphisms gives a morphism of schemes in general.
Now, if X, Y are affine varieties; i.e., A, B are integral domains that are finitely generated algebras over an algebraically closed field k, then, working with only the closed points, the above coincides with the definition given at #Definition.
This fact means that the category of affine varieties can be identified with a full subcategory of affine schemes over k. Since morphisms of varieties are obtained by gluing morphisms of affine varieties in the same way morphisms of schemes are obtained by gluing morphisms of affine schemes, it follows that the category of varieties is a full subcategory of the category of schemes over k.
For more details, see .

Examples

  • The regular functions on are exactly the polynomials in variables and the regular functions on are exactly the constants.
  • Let be the affine curve. Then is a morphism; it is bijective with the inverse. Since is also a morphism, is an isomorphism of varieties.
  • Let be the affine curve. Then is a morphism. It corresponds to the ring homomorphism which is seen to be injective.
  • Continuing the preceding example, let U = A1 − . Since U is the complement of the hyperplane t = 1, U is affine. The restriction is bijective. But the corresponding ring homomorphism is the inclusion, which is not an isomorphism and so the restriction f |U is not an isomorphism.
  • Let X be the affine curve x2 + y2 = 1 and let Then f is a rational function on X. It is regular at despite the expression since, as a rational function on X, f can also be written as.
  • Let. Then X is an algebraic variety since it is an open subset of a variety. If f is a regular function on X, then f is regular on and so is in. Similarly, it is in. Thus, we can write: where g, h are polynomials in k. But this implies g is divisible by xn and so f is in fact a polynomial. Hence, the ring of regular functions on X is just k.
  • Suppose by identifying the points with the points x on A1 and ∞ =. There is an automorphism σ of P1 given by σ = ; in particular, σ exchanges 0 and ∞. If f is a rational function on P1, then and f is regular at ∞ if and only if f is regular at zero.
  • Taking the function field k of an irreducible algebraic curve V, the functions F in the function field may all be realised as morphisms from V to the projective line over k. The image will either be a single point, or the whole projective line. That is, unless F is actually constant, we have to attribute to F the value ∞ at some points of V.
  • For any algebraic varieties X, Y, the projection is a morphism of varieties. If X and Y are affine, then the corresponding ring homomorphism is where.

    Properties

A morphism between varieties is continuous with respect to Zariski topologies on the source and the target.
The image of a morphism of varieties need not be open nor closed. However, one can still say: if f is a morphism between varieties, then the image of f contains an open dense subset of its closure.
A morphism f:''X→Y'' of algebraic varieties is said to be dominant if it has dense image. For such an f, if V is a nonempty open affine subset of Y, then there is a nonempty open affine subset U of X such that f ⊂ V and then is injective. Thus, the dominant map f induces an injection on the level of function fields:
where the direct limit runs over all nonempty open affine subsets of Y. Conversely, every inclusion of fields is induced by a dominant rational map from X to Y. Hence, the above construction determines a contravariant-equivalence between the category of algebraic varieties over a field k and dominant rational maps between them and the category of finitely generated field extension of k.
If X is a smooth complete curve and if f is a rational map from X to a projective space Pm, then f is a regular map X → Pm. In particular, when X is a smooth complete curve, any rational function on X may be viewed as a morphism X → P1 and, conversely, such a morphism as a rational function on X.
On a normal variety, a rational function is regular if and only if it has no poles of codimension one. This is an algebraic analog of Hartogs' extension theorem. There is also a relative version of this fact; see .
A morphism between algebraic varieties that is a homeomorphism between the underlying topological spaces need not be an isomorphism On the other hand, if f is bijective birational and the target space of f is a normal variety, then f is biregular.
A regular map between complex algebraic varieties is a holomorphic map. In particular, a regular map into the complex numbers is just a usual holomorphic function.