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 $V_{i}\to X,i=1,2$ 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 $gV_{1}$ is $V_{1}\to X{\overset {g}{\to }}X$ ,
(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:[2] 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 $gV_{1}\times _{X}V_{2}$ 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. $\square$

Notes

Fulton (1998, Appendix B. 9.2.)
Fulton (1998, Example 11.4.5.)

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

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[1] Fulton (1998, Appendix B. 9.2.)

[2] Fulton (1998, Example 11.4.5.)