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Concept
Morphism of algebraic varieties
Summary
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.
If X and Y are closed subvarieties of and (so they are affine varieties), 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 (note: two polynomials f and g define the same function on X if and only if f − g is in I). The image f(X) 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(x) such that f(U) ⊂ 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 (it does not.
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