Kleiman's theorem
In algebraic geometry, Kleiman's theorem, introduced by {{harvtxt|Kleiman|1974}}, concerns dimension and smoothness of scheme-theoretic intersection after some perturbation of factors in the intersection.
Precisely, it states:{{harvtxt|Fulton|1998|loc=Appendix B. 9.2.}} 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:{{harvtxt|Fulton|1998|loc=Example 11.4.5.}} after some perturbation of cycles on X, their intersection has expected dimension.
Sketch of proof
We write for . Let be the composition that is followed by the group action .
Let be the fiber product of and ; its set of closed points is
:.
We want to compute the dimension of . Let be the projection. It is surjective since acts transitively on X. Each fiber of p is a coset of stabilizers on X and so
:.
Consider the projection ; the fiber of q over g is and has the expected dimension unless empty. This completes the proof of Statement 1.
For Statement 2, since G acts transitively on X and the smooth locus of X is nonempty (by characteristic zero), X itself is smooth. Since G is smooth, each geometric fiber of p is smooth and thus is a smooth morphism. It follows that a general fiber of is smooth by generic smoothness.
Notes
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References
- {{citation|first=David|last=Eisenbud|author1-link=David Eisenbud|first2=Joe|last2=Harris|author2-link=Joe Harris (mathematician) |title=3264 and All That: A Second Course in Algebraic Geometry|publisher=Cambridge University Press|year=2016|isbn=978-1107602724}}
- {{citation
| last = Kleiman | first = Steven L. | authorlink = Steven Kleiman
| journal = Compositio Mathematica
| mr = 0360616
| pages = 287–297
| title = The transversality of a general translate
| url = http://www.numdam.org/item/CM_1974__28_3_287_0/
| volume = 28
| year = 1974}}
- {{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 | first=William|last=Fulton|authorlink=William Fulton (mathematician)}}
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