Trisecant lemma for nonequidimensional varieties

What is it about?

A classical trisecant lemma says that if $X\subset\Bbb P^r$ is a nondegenerate irreducible projective variety, then a general bisecant line $\langle x_1,x_2\rangle$, $x_1,x_2\in X$, $x_2\ne x_1$, meets $X$ in a point $x_3$, $x_3\ne x_i$, $i=1,2$, if and only if $X$ is a hypersurface of degree greater than two, i.e. $\dim{X}=r-1$, $\deg{X}\ge3$. In the paper under review the authors consider a generalization of this trisecant lemma to the case of reducible varieties with components of different dimensions. Let $Z\subset\Bbb P^r$, $\dim{Z}=n\le r-1$, be a possibly singular projective variety which may be reducible and may have components of different dimensions. Let $Y\subset Z$, $\dim{Y}=k\ge1$, be a proper subvariety, and let $S$ be a component of maximal dimension in $\overline{\{l\in\Bbb G(1,r)\mid\exists p\in Y, q_1,q_2\in Z\sbs Y,q_1,q_2,p\in l\}}$, where $\Bbb G(1,r)$ is the Grassmann variety of lines in $\Bbb P^r$ (thus $S$ is a component of trisecant lines of $Z$ meeting $Y$). Then $\dim{S}\le n+k$ with equality holding if and only if the union of all lines in $S$ has dimension $n+1$ (in this last case the union of lines in $S$ is a proper subvariety of $\Bbb P^r$ provided that $r>n+1$). Reviewed by Fyodor L. Zak

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