(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 $\left(r$- 2)-plane $\overline \left\{x$1,..., xr−1}, where the $x$i are generic points, also meets $X$ in a point $x$r different from $x$1,..., xr-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 Pr, 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 $\left(n + 1\right)$-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 Pr, 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 $\left\{l$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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