I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2006, VOLUME 12, NUMBER 2, PAGES 71-87
Trisecant lemma for nonequidimensional varieties
J. Y. Kaminski
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Let be an
irreducible projective variety over an algebraically closed field of
For , if every
-plane , where the
are generic points, also meets in
a point different
from , then
in a linear subspace such that .
In this paper, our purpose is to present another derivation of this
and then to introduce a generalization to nonequidimensional
For the sake of clarity, we shall reformulate our problem as follows.
Let be an
equidimensional variety (maybe singular and/or reducible) of
dimension , other than a linear
space, embedded into , where
The variety of trisecant lines of , say , has
dimension strictly less than , unless is included in an
space and has degree at least , in which case
This also implies that if ,
then can be
embedded in .
Then we inquire the more general case, where is not required to be
In that case, let be a possibly
singular variety of dimension , which may be neither
irreducible nor equidimensional, embedded into , where
, and let be a proper
subvariety of dimension .
being a component of maximal dimension of the closure of
We show that
has dimension strictly less than , unless the union of
has dimension , in which case
In the latter case, if the dimension of the space is strictly greater
than , then
the union of lines in cannot cover the whole
This is the main result of our paper.
We also introduce some examples showing that our bound is strict.
Last modified: June 17, 2006