In algebraic geometry, Kleiman's theorem, introduced by 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,

  1. either is empty or has pure dimension , where is ,
  2. (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.

Sketch of proof

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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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  1. ^ Fulton (1998, Appendix B. 9.2.)
  2. ^ Fulton (1998, Example 11.4.5.)

References

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  • Eisenbud, David; Harris, Joe (2016), 3264 and All That: A Second Course in Algebraic Geometry, Cambridge University Press, ISBN 978-1107602724
  • Kleiman, Steven L. (1974), "The transversality of a general translate", Compositio Mathematica, 28: 287–297, MR 0360616
  • Fulton, William (1998), Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323