# Morphism of varieties

In algebraic geometry, a regular map between affine varieties is a mapping which is given by polynomials. To be explicit, suppose X and Y are subvarieties (or algebraic subsets) of An and Am respectively. A regular map f from X to Y has the form $f = (f_1, \dots, f_m)$ where the $f_i$ are in the coordinate ring $k[x_1, \dots, x_n]/I$, I the ideal defining X, so that the image $f(X)$ lies in Y; i.e., satisfying the defining equations of Y.[1]

More generally, a map ƒ:XY between two varieties is regular at a point x if there is a neighbourhood U of x and a neighbourhood V of ƒ(x) such that the restricted function ƒ:UV is regular as a function on the coordinate patches of U and V. Then ƒ is called regular, if it is regular at all points of X.

In the particular case that Y equals A1 the map ƒ:XA1 is called a regular function, and correspond to scalar functions in differential geometry. In other words, a scalar function is regular at a point x if, in a neighborhood of x, it is a rational function (i.e., a fraction of polynomials) such that the denominator does not vanish at x.[2] The ring of regular functions (that is the coordinate ring or more abstractly the ring of global sections of the structure sheaf) is a fundamental object in affine algebraic geometry. The only regular function on a connected projective variety is constant (this can be viewed as an algebraic analogue of Liouville's theorem in complex analysis); thus, in the projective case, one usually considers the global sections of a line bundle (or divisor) instead.

Regular maps are, by definition, morphisms in the category of algebraic varieties. In particular, a regular map between affine varieties corresponds contravariantly in one-to-one to a ring homomorphism between the coordinate rings.

## Isomorphism

A regular map whose inverse is also regular is called biregular, and are isomorphisms in the category of algebraic varieties. A morphism between algebraic varieties that is a homeomorphism between the underlying topological spaces need not be an isomorphism (a counterexample is given by a Frobenius morphism $t \mapsto t^p$.) On the other hand, if f is bijective birational and the target space of f is a normal variety, then f is biregular. (cf. Zariski's main theorem.)

## Official definition

An (abstract) algebraic variety is defined to be a particular kind of a locally ringed space (see for example projective variety for a ringed structure of a projective variety). When this definition is used, a morphism of varieties is a morphism of the locally ringed spaces underlying the varieties (so for example it is continuous by definition).

## Relation to rational functions

Taking the function field k(V) 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 (this is a consequence of the completeness of projective varieties). That is, unless F is actually constant, we have to attribute to F the value ∞ at some points of V. Now in some sense F is no worse behaved at those points than anywhere else: ∞ is just the chosen point at infinity on the projective line, and by using a Möbius transformation we can move it anywhere we wish. But it is in some way inadequate to the needs of geometry to use only the affine line as target for functions, since we shall end up only with constants.

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

## Properties

A morphism between varieties is continuous with respect to Zariski topologies on the source and the target.

If f is a morphism between varieties, then the image of f contains an open dense subset of its closure. (cf. constructible set.)

On a normal variety, a rational function is regular if and only if it has no poles of codimension one.[3] This is an algebraic analog of Hartogs' extension theorem. There is also a relative version of this fact; see [1].

A regular map between complex algebraic varieties is a holomorphic map. (There is actually a slight technical difference: a regular map is a meromorphic map whose singular points are removable, but the distinction is usually ignored in practice.) In particular, a regular map into the complex numbers is just a usual holomorphic function (complex-analytic function).

## Fibers of a morphism

The important fact is:[4]

Theorem — Let f: XY be a dominating (i.e., having dense image) morphism of algebraic varieties, and let r = dim X - dim Y. Then

1. For every irreducible closed subset W of Y and every irreducible component Z of f-1(W) dominating W,
$\dim Z \ge \dim W + r.$
2. There exists a nonempty open subset U in Y such that (a) $U \subset f(X)$ and (b) for every irreducible closed subset W of Y intersecting U and every irreducible component Z of f-1(W) intersecting f-1(U),
$\dim Z = \dim W + r.$

Corollary — Let f: XY be a morphism of algebraic varieties. For each x in X, define

$e(x) = \max \{ \dim Z | Z$ an irreducible component of $f^{-1}(f(x))$ containing $x \}.$

Then e is upper-semicontinuous; i.e., for each integer n, the set

$X_n = \{ x \in X | e(x) \ge n \}$

is closed.

Corollary (Chevalley)[5] — Let f: XY be a morphism of algebraic varieties. For each integer n, let

$C_n = \{ y \in Y | \dim f^{-1}(y) = n \}.$

Then $C_n$ are constructible and $C_r$ contains an open dense subset of Y.

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 (not just generically).

## Degree of a finite morphism

Let f: XY be a finite morphism between algebraic varieties over a field k. Then the degree of f is the degree of the finite field extension of the function field k(X) over f*k(Y). By generic freeness, there is some nonempty open subset U in Y such that the restriction of the structure sheaf OX to f−1(U) is free as OY|U-module. The degree of f is then 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,

$\chi(f^* F) = \deg(f) \chi (F).$[6]

(The Riemann–Hurwitz formula for a ramified covering shows the "étale" here cannot be omitted.)

If f is étale and k is algebraically closed, then each geometric fiber f−1(y) consists exactly of deg(f) points.