In algebraic geometry, Kleiman's theorem, introduced by , 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
Acting is an activity in which a story is told by means of its enactment by an actor or actress who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode.
Acting involves a broad r ...
transitively on an algebraic variety ''X'' over an algebraically closed field ''k'' and
morphisms of varieties, ''G'' contains a nonempty open subset such that for each ''g'' in the set,
# either
is empty or has pure dimension
, where
is
,
# (Kleiman–
Bertini theorem In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broadest ...
) If
are smooth varieties and if the characteristic of the base field ''k'' is zero, then
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
for
. Let
be the composition that is
followed by the
group action .
Let
be the
fiber product of
and
; its set of closed points is
:
.
We want to compute the dimension of
. Let
be the projection. It is surjective since
acts transitively on ''X''. Each fiber of ''p'' is a coset of stabilizers on ''X'' and so
:
.
Consider the
projection ; the fiber of ''q'' over ''g'' is
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
is a
smooth morphism. It follows that a general fiber of
is smooth by
generic smoothness In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smoo ...
.
Notes
References
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{{algebraic-geometry-stub
Algebraic geometry