Trisecant Lemma for Non Equidimensional Varieties - Mathematics > Algebraic GeometryReport as inadecuate




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Abstract: The classic trisecant lemma states that if $X$ is an integral curve of$\PP^3$ then the variety of trisecants has dimension one, unless the curve isplanar and has degree at least 3, in which case the variety of trisecants hasdimension 2. In this paper, our purpose is first to present another derivationof this result and then to introduce a generalization to non-equidimensionalvarities. For the sake of clarity, we shall reformulate our first problem asfollows. Let $Z$ be an equidimensional variety maybe singular and-orreducible of dimension $n$, other than a linear space, embedded into $\PP^r$,$r \geq n+1$. The variety of trisecant lines of $Z$, say $V {1,3}Z$, hasdimension strictly less than $2n$, unless $Z$ is included in a$n+1-$dimensional linear space and has degree at least 3, in which case$\dimV {1,3}Z = 2n$. Then we inquire the more general case, where $Z$ isnot required to be equidimensional. In that case, let $Z$ be a possiblysingular variety of dimension $n$, that may be neither irreducible norequidimensional, embedded into $\PP^r$, where $r \geq n+1$, and $Y$ a propersubvariety of dimension $k \geq 1$. Consider now $S$ being a component ofmaximal dimension of the closure of $\{l \in \G1,r \vtl \exists p \in Y, q 1,q 2 \in Z \backslash Y, q 1,q 2,p \in l\}$. We show that $S$ has dimensionstrictly less than $n+k$, unless the union of lines in $S$ has dimension $n+1$,in which case $dimS = n+k$. In the latter case, if the dimension of the spaceis stricly greater then $n+1$, the union of lines in $S$ cannot cover the wholespace. This is the main result of our work. We also introduce some examplesshowing than our bound is strict.



Author: J.Y. Kaminski, A. Kanel-Belov, M. Teicher

Source: https://arxiv.org/







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