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: 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 $g{V}_1{\times }_X{V}_2$ is empty or has pure dimension $\dim V_1+\dim V_2-\dim X$, where $g{V}_1$ is $V_1\to X\stackrel{g}{\to }X$,
2. (Kleiman–Bertini theorem) If $V_i$ are smooth varieties and if the characteristic of the base field k is zero, then $g{V}_1{\times }_X{V}_2$ 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 $\mathrm{\Gamma }=(G\times V_1){\times }_X{V}_2$ be the fiber product of $h$ and $f_2:V_2\to X$; its set of closed points is

$\mathrm{\Gamma }=\{(g,v,w)|g\in G,v\in V_1,w\in V_2,g\cdot f_1(v)=f_2(w)\}$.

We want to compute the dimension of $\mathrm{\Gamma }$. Let $p:\mathrm{\Gamma }\to V_1\times V_2$ 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

$\dim \mathrm{\Gamma }=\dim V_1+\dim V_2+\dim G-\dim X$.

Consider the projection $q:\mathrm{\Gamma }\to G$; 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 $p_0:{\mathrm{\Gamma }}_0:=(G\times V_{1,\text{sm}}){\times }_X{V}_{2,\text{sm}}\to V_{1,\text{sm}}\times V_{2,\text{sm}}$ is a smooth morphism. It follows that a general fiber of $q_0:{\mathrm{\Gamma }}_0\to G$ is smooth by generic smoothness. $◻$

References

• Eisenbud, David; Harris, Joe (2016), 3264 and All That: A Second Course in Algebraic Geometry, Cambridge University Press, ISBN 978-1107602724
• Kleiman, Steven L. (1974), "The transversality of a general translate", Compositio Mathematica, 28: 287–297, MR 0360616
• Fulton, William (1998), Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323