Kleiman's theorem

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 $$gV_1 \times_X V_2$$ is empty or has pure dimension $$\dim V_1 + \dim V_2 - \dim X$$, where $$g V_1$$ is $$V_1 \to X \overset{g}\to X$$,
 * 2) (Kleiman–Bertini theorem) If $$V_i$$ are smooth varieties and if the characteristic of the base field k is zero, then $$gV_1 \times_X V_2$$ 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 $$h: G \times V_1 \to X$$ be the composition that is $$(1_G, f_1): G \times V_1 \to G \times X$$ followed by the group action $$\sigma: G \times X \to X$$.

Let $$\Gamma = (G \times V_1) \times_X V_2 $$ be the fiber product of $$h$$ and $$f_2: V_2 \to X$$; its set of closed points is
 * $$\Gamma = \{ (g, v, w) | g \in G, v \in V_1, w \in V_2, g \cdot f_1(v) = f_2(w) \}$$.

We want to compute the dimension of $$\Gamma$$. Let $$p: \Gamma \to V_1 \times V_2$$ 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 $$g V_1 \times_X V_2$$ 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 $$p_0 : \Gamma_0 := (G \times V_{1, \text{sm}}) \times_X V_{2, \text{sm}} \to V_{1, \text{sm}} \times V_{2, \text{sm}}$$ is a smooth morphism. It follows that a general fiber of $$q_0 : \Gamma_0 \to G$$ is smooth by generic smoothness. $$\square$$