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 they are isomorphisms in the category of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the weaker condition of a rational map and birational maps are frequently used as well.

Definition

If X and Y are closed subvarieties of An and Am, then a regular map ƒ: X→Y is the restriction of a polynomial map An→Am. 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 ƒ:X→Y between two varieties is regular at a point x if there is a neighbourhood U of x and a neighbourhood V of ƒ such that ƒ ⊂ V and the restricted function ƒ:U→V is regular as a function on some affine charts of U and V. Then ƒ is called regular, if it is regular at all points of X.
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 ƒ: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 ƒ is the inclusion, then ƒ# is the restriction of regular functions on Y to X. See #Examples below for more examples.

Regular functions

In the particular case that Y equals A1 the regular map ƒ:X→A1 is called a regular function, 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 ƒ:X→A1 is regular at a point x if, in some open affine neighborhood of x, it is a rational function that is regular at x; i.e., there are regular functions g, h near x such that f = g/h and h does not vanish at x. 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 X = Spec A and Y = Spec B 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

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 ƒ:X→Y of algebraic varieties is said to be a 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 ƒ ⊂ V and then is injective. Thus, the dominant map ƒ induces an injection on the level of function fields:
where the 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.

Morphisms to a projective space

Let
be a morphism from a projective variety to a projective space. Let x be a point of X. Then some i-th homogeneous coordinate of f is nonzero; say, i = 0 for simplicity. Then, by continuity, there is an open affine neighborhood U of x such that
is a morphism, where yi are the homogeneous coordinates. Note the target space is the affine space Am through the identification. Thus, by definition, the restriction f |U is given by
where gi's are regular functions on U. Since X is projective, each gi is a fraction of homogeneous elements of the same degree in the homogeneous coordinate ring k of X. We can arrange the fractions so that they all have the same homogeneous denominator say f0. Then we can write gi = fi/f0 for some homogeneous elements fi's in k. Hence, going back to the homogeneous coordinates,
for all x in U and by continuity for all x in X as long as the fi's do not vanish at x simultaneously. If they vanish simultaneously at a point x of X, then, by the above procedure, one can pick a different set of fi's that do not vanish at x simultaneously
In fact, the above description is valid for any quasi-projective variety X, an open subvariety of a projective variety ; the difference being that fi's are in the homogeneous coordinate ring of.
Note: The above does not say a morphism from a projective variety to a projective space is given by a single set of polynomials. For example, let X be the conic in P2. Then two maps and agree on the open subset of X and so defines a morphism.

Fibers of a morphism

The important fact is:
In Mumford's red book, the theorem is proved by means of Noether's normalization lemma. For an algebraic approach where the generic freeness plays a main role and the notion of "universally catenary ring" is a key in the proof, see Eisenbud, Ch. 14 of "Commutative algebra with a view toward algebraic geometry." In fact, the proof there shows that if f is flat, then the dimension equality in 2. of the theorem holds in general.

Degree of a finite morphism

Let f: X → Y be a finite surjective morphism between algebraic varieties over a field k. Then, by definition, the degree of f is the degree of the finite field extension of the function field k over f*k. By generic freeness, there is some nonempty open subset U in Y such that the restriction of the structure sheaf OX to is free as OYU-module. The degree of f is then also the rank of this free module.
If f is étale and if X, Y are complete, then for any coherent sheaf F on Y, writing χ for the Euler characteristic,
In general, if f is a finite surjective morphism, if X, Y are complete and F a coherent sheaf on Y, then from the Leray spectral sequence, one gets:
In particular, if F is a tensor power of a line bundle, then and since the support of has positive codimension if q is positive, comparing the leading terms, one has:
If f is étale and k is algebraically closed, then each geometric fiber f−1 consists exactly of deg points.