Trisecant lemma for nonequidimensional varieties

Let $X$ be an irreducible projective variety over an algebraically closed field of characteristic zero. For $r \geq3$, if every $(r-2)$-plane $\overline{x_1,\dots,x_{r-1}}$, where the $x_i$ are generic points, also meets $X$ in a point $x_r$ different from $x_1,\dots,x_{r-1}$, then $X$ is contained in a linear subspace $L$ such that $\operatorname{codim}_L X \leq r-2$. 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 $\mathbb P^r$, where $r \geq n+1$. 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 $\mathbb P^{n+1}$. 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 $\mathbb P^r$, where $r\geq n+1$, and let $Y$ be a proper subvariety of dimension $k\geq1$. Consider now $S$ being a component of maximal dimension of the closure of $\{l \in\mathbb G(1,r)\mid\exists p\in Y,\ q_1,q_2\in Z\setminus Y,q_1,q_2,p\in l\}$. 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.