Kleiman's theorem

In algebraic geometry, Kleiman's theorem, introduced in (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 $gV_{1}\times _{X}V_{2}$ is empty or has pure dimension $\dim V_{1}+\dim V_{2}-\dim X$, where $gV_{1}$ is $V_{1}\to X{\overset {g}{\to }}X$,
2. (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: after some perturbation of cycles on X, their intersection has expected dimension.

References

• Eisenbud, David; Joe, Harris (2016), 3264 and All That: A Second Course in Algebraic Geometry, C. U.P., ISBN 978-1107602724
• Steven Kleiman, "The transversality of a generic translate," Math. 28 (1974), 287–297.
• William Fulton. (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