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.

À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Graph Chatbot

Chattez avec Graph Search

Posez n’importe quelle question sur les cours, conférences, exercices, recherches, actualités, etc. de l’EPFL ou essayez les exemples de questions ci-dessous.

AVERTISSEMENT : Le chatbot Graph n'est pas programmé pour fournir des réponses explicites ou catégoriques à vos questions. Il transforme plutôt vos questions en demandes API qui sont distribuées aux différents services informatiques officiellement administrés par l'EPFL. Son but est uniquement de collecter et de recommander des références pertinentes à des contenus que vous pouvez explorer pour vous aider à répondre à vos questions.