Trisecant lemma for nonequidimensional varieties

Abstract

Let X be an irreducible projective variety over an algebraically closed field of characteristic zero. For ≥ 3, if every (r−2)-plane \(\overline {x_1 , \ldots ,x_{r - 1} } \), where the x i are generic points, also meets X in a point x r different from x 1,..., x r−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 ≥ 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 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 ≥ n + 1, and let Y be a proper subvariety of dimension k ≥ 1. Consider now S being a component of maximal dimension of the closure of \(\{ l \in \mathbb{G}(1,r)|\exists p \in Y, q_1 , q_2 \in Z\backslash Y, q_1 , q_2 ,p \in l\} \). 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 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.