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,
# 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 $V_1 \to X \overset\to X$,
# (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 = \$.
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 $p_0 : \Gamma_0 := (G \times V_) \times_X V_ \to V_ \times V_$ is a smooth morphism. It follows that a general fiber of $q_0 : \Gamma_0 \to G$ is smooth by generic smoothness. $\square$

Notes

References

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{{algebraic-geometry-stub

Algebraic geometry