A consequence of Gruson-Peskine $k$-secant lemma is the following : if $2N -3n+1>0$ then the trisecants of $X$ do not fill the ambiant space. In particular, if you know that the secant variety of $X$ fills the ambiant space, then you have a numerical condition that guarantees that a general secant is not a trisecant.

A consequence of Gruson-Peskine $k$-secant lemma is the following : if $2N -3n-1>0$ then the trisecants of $X$ do not fill the ambiant space. In particular, if you know that the secant variety of $X$ fills the ambiant space, then you have a numerical condition that guarantees that a general secant is not a trisecant.

A consequence of Gruson-Peskine $k$-secant lemma is the following : if $2N -3n+1>0$ then the trisecants of $X$ do not fill the ambiant space. In particular, if you know that the secant cariety of $X$ fills the ambiant space,then you have a numerical condition that guarantees that a general secant is not a trisecant.

A consequence of Gruson-Peskine $k$-secant lemma is the following : if $2N -3n+1>0$ then the trisecants of $X$ do not fill the ambiant space. In particular, if you know that the secant variety of $X$ fills the ambiant space, then you have a numerical condition that guarantees that a general secant is not a trisecant.