“Kleiman's theorem”的意思、由来-开放百科全书

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Kleiman's theorem

释义

References

In algebraic geometry, Kleiman's theorem, introduced in {{harv|Kleiman|1974}}, concerns dimension and smoothness of scheme-theoretic intersection after some perturbation of factors in the intersection.

Precisely, it states:^[1] given a connected algebraic group G acting transitively on an algebraic variety X over an algebraically closed field k and morphisms of varieties, G contains a nonempty open subset such that for each g in the set,

either is empty or has pure dimension , where is ,
(Kleiman–Bertini theorem) If are smooth varieties and if the characteristic of the base field k is zero, then is smooth.

Statement 1 establishes a version of Chow's moving lemma:^[2] after some perturbation of cycles on X, their intersection has expected dimension.

References

1. ^{{harvnb|Fulton|loc=Appendix B. 9.2.}}
2. ^{{harvnb|Fulton|loc=Example 11.4.5.}}

{{citation|first=David|last=Eisenbud|first2=Harris|last2=Joe|title=3264 and All That: A Second Course in Algebraic Geometry|publisher=C. U.P.|year=2016|isbn=978-1107602724}}
Steven Kleiman, "The transversality of a generic translate," Math. 28 (1974), 287–297.
{{Citation | title=Intersection theory | publisher=Springer-Verlag | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-62046-4 | mr=1644323 | year=1998 | volume=2 | edition=2nd | author=William Fulton.}}

1 : Algebraic geometry

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