Trisecant Lemma for Non Equidimensional Varietiesстатья

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1. Полный текст 0712.3878v1.pdf 204,9 КБ 16 апреля 2016 [AlexeiBelov]

[1] Kaminski J. Y., Kanel-Belov A., Teicher M. Trisecant lemma for non equidimensional varieties // ArXiv e-prints. — 2007. — no. 0712.3878. The classic trisecant lemma states that if X is an integral curve of 3 then the variety of trisecants has dimension one, unless the curve is planar and has degree at least 3, in which case the variety of trisecants has dimension 2. In this paper, our purpose is first to present another derivation of this result and then to introduce a generalization to non-equidimensional varities. For the sake of clarity, we shall reformulate our first 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, r≥n+1. The variety of trisecant lines of Z, say V1,3(Z), has dimension strictly less than 2n, unless Z is included in a (n+1)−dimensional linear space and has degree at least 3, in which case dim(V1,3(Z))=2n. 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, that may be neither irreducible nor equidimensional, embedded into r, where r≥n+1, and Y a proper subvariety of dimension k≥1. Consider now S being a component of maximal dimension of the closure of {l ∈(1,r) ∃pY, q_1, q_2 ∈Z \Y, q_1,q_2,pl}. 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 stricly greater then n+1, the union of lines in S cannot cover the whole space. This is the main result of our work. We also introduce some examples showing than our bound is strict.

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