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, Statement 1 establishes a version of Chow's moving lemma:[2] after some perturbation of cycles on X, their intersection has expected dimension.
We write
be the composition that is
followed by the group action
σ :
be the fiber product of
; its set of closed points is We want to compute the dimension of
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.
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