NDLI: Trisecant lemma for nonequidimensional varieties

Content Provider	Springer Nature Link
Author	Kaminski, Jeremy Yirmeyahu Kanel Belov, Alexei Teicher, Mina
Copyright Year	2008
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 ℙ$^{r}$, 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 ℙ$^{ n + 1}$. 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 ℙ$^{r}$, 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 $$\{ l \in \mathbb{G}(1,r)\|\exists p \in Y, q_1 , q_2 \in Z\backslash Y, q_1 , q_2 ,p \in l\} $$ . 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.
Starting Page	1087
Ending Page	1097
Page Count	11
File Format	PDF
ISSN	10723374
Journal	Journal of Mathematical Sciences
Volume Number	149
Issue Number	2
e-ISSN	15738795
Language	English
Publisher	Springer US
Publisher Date	2008-01-01
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mathematics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Statistics and Probability

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