Morphism of algebraic varieties

<p>In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, a morphism between <a href="/facts/Algebraic_varieties/Vk7f97vn">algebraic varieties</a> is a function between the varieties that is given locally by <a href="/facts/Polynomial/Lzak8VVx">polynomials</a>. It is also called a regular map. A morphism from an algebraic variety to the <a href="/facts/Affine_line/tlvI3oX1">affine line</a> is also called a regular function.
A regular map whose inverse is also regular is called biregular, and the biregular maps are the <a href="/facts/Isomorphism/pj8KgkxU">isomorphisms</a> of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on <a href="/facts/Projective_variety/aB2XdB7B">projective varieties</a> – the concepts of <a href="/facts/Rational_map/kqbMfnKn">rational</a> and <a href="/facts/Birational/hsWqRhJ1">birational</a> maps are widely used as well; they are  <a href="/facts/Partial_function/0vJMw0Nm">partial functions</a> that are defined locally by <a href="/facts/Rational_fraction/rgzXYjyr">rational fractions</a> instead of polynomials.
</p><p>An algebraic variety has naturally the structure of a <a href="/facts/Locally_ringed_space/JyeaTMtX">locally ringed space</a>; a morphism between algebraic varieties is precisely a morphism of the underlying locally ringed spaces.
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Morphism of algebraic varieties open-in-new

Morphism of algebraic varieties