Kleiman's theorem
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 [math]\displaystyle{ V_i \to X, i = 1, 2 }[/math] morphisms of varieties, G contains a nonempty open subset such that for each g in the set,
- either [math]\displaystyle{ gV_1 \times_X V_2 }[/math] is empty or has pure dimension [math]\displaystyle{ \dim V_1 + \dim V_2 - \dim X }[/math], where [math]\displaystyle{ g V_1 }[/math] is [math]\displaystyle{ V_1 \to X \overset{g}\to X }[/math],
- (Kleiman–Bertini theorem) If [math]\displaystyle{ V_i }[/math] are smooth varieties and if the characteristic of the base field k is zero, then [math]\displaystyle{ gV_1 \times_X V_2 }[/math] is smooth.
Statement 1 establishes a version of Chow's moving lemma:[2] after some perturbation of cycles on X, their intersection has expected dimension.
Sketch of proof
We write [math]\displaystyle{ f_i }[/math] for [math]\displaystyle{ V_i \to X }[/math]. Let [math]\displaystyle{ h: G \times V_1 \to X }[/math] be the composition that is [math]\displaystyle{ (1_G, f_1): G \times V_1 \to G \times X }[/math] followed by the group action [math]\displaystyle{ \sigma: G \times X \to X }[/math].
Let [math]\displaystyle{ \Gamma = (G \times V_1) \times_X V_2 }[/math] be the fiber product of [math]\displaystyle{ h }[/math] and [math]\displaystyle{ f_2: V_2 \to X }[/math]; its set of closed points is
- [math]\displaystyle{ \Gamma = \{ (g, v, w) | g \in G, v \in V_1, w \in V_2, g \cdot f_1(v) = f_2(w) \} }[/math].
We want to compute the dimension of [math]\displaystyle{ \Gamma }[/math]. Let [math]\displaystyle{ p: \Gamma \to V_1 \times V_2 }[/math] be the projection. It is surjective since [math]\displaystyle{ G }[/math] acts transitively on X. Each fiber of p is a coset of stabilizers on X and so
- [math]\displaystyle{ \dim \Gamma = \dim V_1 + \dim V_2 + \dim G - \dim X }[/math].
Consider the projection [math]\displaystyle{ q: \Gamma \to G }[/math]; the fiber of q over g is [math]\displaystyle{ g V_1 \times_X V_2 }[/math] 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 [math]\displaystyle{ 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}} }[/math] is a smooth morphism. It follows that a general fiber of [math]\displaystyle{ q_0 : \Gamma_0 \to G }[/math] is smooth by generic smoothness. [math]\displaystyle{ \square }[/math]
Notes
References
- Eisenbud, David; Harris, Joe (2016), 3264 and All That: A Second Course in Algebraic Geometry, Cambridge University Press, ISBN 978-1107602724
- "The transversality of a general translate", Compositio Mathematica 28: 287–297, 1974, http://www.numdam.org/item/CM_1974__28_3_287_0/
- Fulton, William (1998), Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4
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