Kleiman's theorem explained

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

Precisely, it states: given a connected algebraic group G acting transitively on an algebraic variety X over an algebraically closed field k and

V _{i} \to X, i = 1 , 2 morphisms of varieties, G contains a nonempty open subset such that for each g in the set,

1. either

gV _{1} x _{X} V _{2} is empty or has pure dimension \dim V _{1} + \dim V _{2} - \dim X , where g V _{1} is V _{1} \to X \overset{g}\to X ,

1. (Kleiman–Bertini theorem) If

V _{i} are smooth varieties and if the characteristic of the base field k is zero, then gV _{1} x _{X} V _{2} is smooth.

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

Sketch of proof

We write

f _{i} for V _{i} \to X . Let h: G x V _{1} \to X be the composition that is ( 1 _{G,} f _{1):} G x V _{1} \to G x X followed by the group action \sigma: G x X \to X .

Let

\Gamma = (G x V _{1)} x _{X} V _{2} be the fiber product of h and f _{2:} V _{2} \to X ; its set of closed points is\Gamma = \{ (g, v, w) | g \in G, v \in V _{1,} w \in V _{2,} g ⋅ f _{1(v)} = f _{2(w)} \} .We want to compute the dimension of \Gamma . Let p: \Gamma \to V _{1} x V _{2} be the projection. It is surjective since G acts transitively on X. Each fiber of p is a coset of stabilizers on X and so\dim \Gamma = \dim V _{1} + \dim V _{2} + \dim G - \dim X .Consider the projection q: \Gamma \to G ; the fiber of q over g is g V _{1} x _{X} V _{2} 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

p _{0} : \Gamma _{0} : = (G x V _{1,} ) x _{X} V _{2,} \to V _{1,} x V _{2,} is a smooth morphism. It follows that a general fiber of q _{0} : \Gamma _{0} \to G is smooth by generic smoothness. \square