  The classic trisecant lemma states that if $X$ is an integral curve of
$\PP^3$ then the variety of trisecants has dimension one, unless the curve is
planar and has degree at least 3, in which case the variety of trisecants has
dimension 2. In this paper, our purpose is first to present another derivation
of this result and then to introduce a generalization to non-equidimensional
varities. For the sake of clarity, we shall reformulate our first problem as
follows. Let $Z$ be an equidimensional variety (maybe singular and/or
reducible) of dimension $n$, other than a linear space, embedded into $\PP^r$,
$r \geq n+1$. The variety of trisecant lines of $Z$, say $V_{1,3}(Z)$, has
dimension strictly less than $2n$, unless $Z$ is included in a
$(n+1)-$dimensional linear space and has degree at least 3, in which case
$\dim(V_{1,3}(Z)) = 2n$. 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 $n$, that may be neither irreducible nor
equidimensional, embedded into $\PP^r$, where $r \geq n+1$, and $Y$ a proper
subvariety of dimension $k \geq 1$. Consider now $S$ being a component of
maximal dimension of the closure of $\{l \in \G(1,r) \vtl \exists p \in Y, q_1,
q_2 \in Z \backslash Y, q_1,q_2,p \in l\}$. 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 stricly greater then $n+1$, the union of lines in $S$ cannot cover the whole
space. This is the main result of our work. We also introduce some examples
showing than our bound is strict.
