Kleiman's theorem

In algebraic geometry, Kleiman's theorem, introduced by {{harvtxt|Kleiman|1974}}, concerns dimension and smoothness of scheme-theoretic intersection after some perturbation of factors in the intersection.

Precisely, it states:{{harvtxt|Fulton|1998|loc=Appendix B. 9.2.}} 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 $g V_1$ 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:{{harvtxt|Fulton|1998|loc=Example 11.4.5.}} 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 $g V_1 \times_X V_2$ 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 $p_0 : \Gamma_0 := (G \times V_{1, \text{sm}}) \times_X V_{2, \text{sm}} \to V_{1, \text{sm}} \times V_{2, \text{sm}}$ is a smooth morphism. It follows that a general fiber of $q_0 : \Gamma_0 \to G$ is smooth by generic smoothness. $\square$

Notes

References

{{citation|first=David|last=Eisenbud|author1-link=David Eisenbud|first2=Joe|last2=Harris|author2-link=Joe Harris (mathematician) |title=3264 and All That: A Second Course in Algebraic Geometry|publisher=Cambridge University Press|year=2016|isbn=978-1107602724}}
{{citation

| last = Kleiman | first = Steven L. | authorlink = Steven Kleiman

| journal = Compositio Mathematica

| mr = 0360616

| pages = 287–297

| title = The transversality of a general translate

| url = http://www.numdam.org/item/CM_1974__28_3_287_0/

| volume = 28

| year = 1974}}

{{Citation | title=Intersection Theory | publisher=Springer-Verlag | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-62046-4 | mr=1644323 | year=1998 | volume=2 | edition=2nd | first=William|last=Fulton|authorlink=William Fulton (mathematician)}}

Category:Theorems in algebraic geometry