## Лемма о 3-секущих для многообразий с компонентами различной размерностистатья

Статья опубликована в журнале из списка RSCI Web of Science

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Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 20 апреля 2016 г.
• Авторы:
• Журнал: Фундаментальная и прикладная математика
• Том: 12
• Номер: 2
• Год издания: 2006
• Издательство: Интуит
• Местоположение издательства: М.
• Первая страница: 71
• Последняя страница: 87
• DOI: 10.1007/s10958-008-0047-7
• Аннотация: 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 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 $\PP^r$, $r \geq n+1$. The variety of trisecant lines of $Z$, say $V_{1,3}(Z)$, has dimension strictly less than $2n$, unless $Z$ is included in a $(n+1)-$dimensional linear space and has degree at least 3, in which case $\dim(V_{1,3}(Z)) = 2n$. 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 $\PP^r$, where $r \geq n+1$, and $Y$ a proper subvariety of dimension $k \geq 1$. Consider now $S$ being a component of maximal dimension of the closure of $\{l \in \G(1,r) \vtl \exists p \in Y, q_1, q_2 \in Z \backslash Y, q_1,q_2,p \in l\}$. We show that $S$ has 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.
• Добавил в систему: Белов Алексей Яковлевич

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1. Полный текст fpm935.pdf 242,5 КБ 17 апреля 2016 [AlexeiBelov]

 [1] Каминский Д. Й., Канель-Белов А. Я., Тайхер М. Лемма о 3-секущих для многообразий с компонентами различной размерности // Фундаментальная и прикладная математика. — 2006. — Т. 12, № 2. — С. 71–87. The classic trisecant lemma states that if $X$ is an integral curve of $PP3$ 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 $PPr$, $r geq n+1$. The variety of trisecant lines of $Z$, say $V_{1,3}(Z)$, has dimension strictly less than $2n$, unless $Z$ is included in a $(n+1)-$dimensional linear space and has degree at least 3, in which case $dim(V_{1,3}(Z)) = 2n$. 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 $PPr$, where $r geq n+1$, and $Y$ a proper subvariety of dimension $k geq 1$. Consider now $S$ being a component of maximal dimension of the closure of ${l in G(1,r) vtl exists p in Y, q_1, q_2 in Z backslash Y, q_1,q_2,p in l}$. We show that $S$ has 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. [ DOI ]