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In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties such that (i) For each y in Y, the fiber is an orbit of G.(ii) The topology of Y is the quotient topology: a subset is open if and only if is open.(iii) For any open subset , is an isomorphism. (Here, k is the base field.)

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  • Geometric quotient (en)
  • Geometriskt kvot (sv)
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  • Inom algebraisk geometri, en del av matematiken, är ett geometriskt kvot av en algebraisk varietet X med verkan av en G en så att (i) För varje y i Y är fibern en bana av G.(ii) Topologin av Y är : en delmängd är öppen om och bara om är öppen.(iii) För varje öppen delmängd är en isomorfi. (Här är k baskroppen.) (sv)
  • In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties such that (i) For each y in Y, the fiber is an orbit of G.(ii) The topology of Y is the quotient topology: a subset is open if and only if is open.(iii) For any open subset , is an isomorphism. (Here, k is the base field.) (en)
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  • In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties such that (i) For each y in Y, the fiber is an orbit of G.(ii) The topology of Y is the quotient topology: a subset is open if and only if is open.(iii) For any open subset , is an isomorphism. (Here, k is the base field.) The notion appears in geometric invariant theory. (i), (ii) say that Y is an orbit space of X in topology. (iii) may also be phrased as an isomorphism of sheaves . In particular, if X is irreducible, then so is Y and : rational functions on Y may be viewed as invariant rational functions on X (i.e., of X). For example, if H is a closed subgroup of G, then is a geometric quotient. A GIT quotient may or may not be a geometric quotient: but both are categorical quotients, which is unique; in other words, one cannot have both types of quotients (without them being the same). (en)
  • Inom algebraisk geometri, en del av matematiken, är ett geometriskt kvot av en algebraisk varietet X med verkan av en G en så att (i) För varje y i Y är fibern en bana av G.(ii) Topologin av Y är : en delmängd är öppen om och bara om är öppen.(iii) För varje öppen delmängd är en isomorfi. (Här är k baskroppen.) (sv)
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