## Abstract

Let X be an irreducible projective variety over an algebraically closed field of characteristic zero. For r ≥ 3, if every (r-2)-plane x _{1},...,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 ℘^{r}, 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 ℘^{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 ℘^{r}, 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 ∈ script G sign(1,r)|∃ p ∈ Y, q_{1}, q_{2} ∈ Z\Y, q _{1}, q_{2},p ∈ 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.

Original language | English |
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Pages (from-to) | 1087-1097 |

Number of pages | 11 |

Journal | Journal of Mathematical Sciences |

Volume | 149 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2008 |