FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2006, VOLUME 12, NUMBER 2, PAGES 71-87

**Trisecant lemma for nonequidimensional varieties**

J. Y. Kaminski

A. Kanel-Belov

M. Teicher

Abstract

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Let $X$ be an
irreducible projective variety over an algebraically closed field of
characteristic zero.
For $r$³ 3, if every
$(r$-
2)-plane $\backslash overline\; \{x$_{1},...,
x_{r−1}}, where the
$x$_{i}
are generic points, also meets $X$ in
a point $x$_{r} different
from $x$_{1},..., x_{r-1}, then
$X$ is contained
in a linear subspace $L$ such that $codim$_{L} X ≤ r − 2.
In this paper, our purpose is to present another derivation of this
result for $r\; =\; 3$
and then to introduce a generalization to nonequidimensional
varieties.
For the sake of clarity, we shall reformulate our problem as follows.
Let $Z$ be an
equidimensional variety (maybe singular and/or reducible) of
dimension $n$, other than a linear
space, embedded into $$**P**^{r}, where
$r$³
n + 1.
The variety of trisecant lines of $Z$, say $V$_{1,3}(Z), has
dimension strictly less than $2n$, unless $Z$ is included in an
$(n\; +\; 1)$-dimensional linear
space and has degree at least $3$, in which case
$dim\; V$_{1,3}(Z) = 2n.
This also implies that if $dim\; V$_{1,3}(Z) = 2n,
then $Z$ can be
embedded in $$**P**^{n+1}.
Then we inquire the more general case, where $Z$ is not required to be
equidimensional.
In that case, let $Z$ be a possibly
singular variety of dimension $n$, which may be neither
irreducible nor equidimensional, embedded into $$**P**^{r}, where
$r$³
n + 1, and let $Y$ be a proper
subvariety of dimension $k$³ 1.
Consider now $S$
being a component of maximal dimension of the closure of
$\{l$Î
**G**(1,r) | $ p Î Y, q_{1},q_{2} Î Z\Y, q_{1},q_{2},p Î l}.
We show that $S$
has dimension strictly less than $n\; +\; k$, unless the union of
lines in $S$
has dimension $n\; +\; 1$, in which case
$dim\; S\; =\; n\; +\; k$.
In the latter case, if the dimension of the space is strictly greater
than $n\; +\; 1$, then
the union of lines in $S$ cannot cover the whole
space.
This is the main result of our paper.
We also introduce some examples showing that our bound is strict.

Location: http://mech.math.msu.su/~fpm/eng/k06/k062/k06204h.htm

Last modified: June 17, 2006