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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''
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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 In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broadest ...
) 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 In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smoo ...
. \square


Notes


References

* * * {{algebraic-geometry-stub Algebraic geometry