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 $$P^r, where $r$³ n + 1. The variety of trisecant lines of $Z$, say $V$_1,3(Z), has dimension strictly less than $2n$, unless $Z$ is included in an $(n\; +\; 1)$-dimensional linear space and has degree at least $3$, in which case $dim\; V$_1,3(Z) = 2n. This also implies that if $dim\; V$_1,3(Z) = 2n, then $Z$ can be embedded in $$Pⁿ⁺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, q₁,q₂ Î Z\Y, q₁,q₂,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.

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Last modified: June 17, 2006