Geometric quotient

In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties $\pi: X \to Y$ such that

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 $\mathcal{O}_Y \simeq (\pi_*\mathcal{O}_X)^G$ . In particular, if X is irreducible, then so is Y and $k(Y) = k(X)^G$ : rational functions on Y may be viewed as invariant rational functions on X (i.e., rational-invariants of X).

单词	Geometric quotient
释义	Geometric quotient 英语百科 Geometric quotient In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties $\pi: X \to Y$ such that 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 $\mathcal{O}_Y \simeq (\pi_*\mathcal{O}_X)^G$ . In particular, if X is irreducible, then so is Y and $k(Y) = k(X)^G$ : rational functions on Y may be viewed as invariant rational functions on X (i.e., rational-invariants of X).
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