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 ${\displaystyle V_{i}\to X,i=1,2}$ morphisms of varieties, G contains a nonempty open subset such that for each g in the set,

1. either ${\displaystyle gV_{1}\times _{X}V_{2}}$ is empty or has pure dimension ${\displaystyle \dim V_{1}+\dim V_{2}-\dim X}$, where ${\displaystyle gV_{1}}$ is ${\displaystyle V_{1}\to X{\overset {g}{\to }}X}$,
2. (Kleiman–Bertini theorem) If ${\displaystyle V_{i}}$ are smooth varieties and if the characteristic of the base field k is zero, then ${\displaystyle gV_{1}\times _{X}V_{2}}$ 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 ${\displaystyle f_{i}}$ for ${\displaystyle V_{i}\to X}$. Let ${\displaystyle h:G\times V_{1}\to X}$ be the composition that is ${\displaystyle (1_{G},f_{1}):G\times V_{1}\to G\times X}$ followed by the group action ${\displaystyle \sigma :G\times X\to X}$.

Let ${\displaystyle \Gamma =(G\times V_{1})\times _{X}V_{2}}$ be the fiber product of ${\displaystyle h}$ and ${\displaystyle f_{2}:V_{2}\to X}$; its set of closed points is

${\displaystyle \Gamma =\{(g,v,w)|g\in G,v\in V_{1},w\in V_{2},g\cdot f_{1}(v)=f_{2}(w)\}}$.

We want to compute the dimension of ${\displaystyle \Gamma }$. Let ${\displaystyle p:\Gamma \to V_{1}\times V_{2}}$ be the projection. It is surjective since ${\displaystyle G}$ acts transitively on X. Each fiber of p is a coset of stabilizers on X and so

${\displaystyle \dim \Gamma =\dim V_{1}+\dim V_{2}+\dim G-\dim X}$.

Consider the projection ${\displaystyle q:\Gamma \to G}$; the fiber of q over g is ${\displaystyle gV_{1}\times _{X}V_{2}}$ 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 ${\displaystyle p_{0}:\Gamma _{0}:=(G\times V_{1,{\text{sm}}})\times _{X}V_{2,{\text{sm}}}\to V_{1,{\text{sm}}}\times V_{2,{\text{sm}}}}$ is a smooth morphism. It follows that a general fiber of ${\displaystyle q_{0}:\Gamma _{0}\to G}$ is smooth by generic smoothness. ${\displaystyle \square }$

Notes

1. Fulton (1998, Appendix B. 9.2.)
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., 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323