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.

If X and Y are closed subvarieties of Aⁿ and A^m (so they are affine varieties), then a regular map ƒ:X→Y is the restriction of a polynomial map Aⁿ→A^m. Explicitly, it has the form

where the ${\displaystyle f_{i}}$'s are in the coordinate ring of X:

I the ideal defining X, so that the image ${\displaystyle f(X)}$ lies in Y; i.e., satisfying the defining equations of Y. (Two polynomials f and g define the same function on X if and only if f − g is in I.)

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