Multi-secant lemma

What is it about?

The classical trisecant lemma states that if $X$ is an integral curve of $\Bbb{P}^3$ then the variety of trisecants has dimension 1, unless the curve is planar and has degree at least 3, in which case the variety of trisecants has dimension 2. The papers [B. Ådlandsvik, Math. Scand. 61 (1987), no. 2, 213–222; MR0947474; J. Reine Angew. Math. 392 (1988), 16–26; MR0965054; H. Flenner, L. O'Carroll and W. Vogel, Joins and intersections, Springer Monogr. Math., Springer, Berlin, 1999; MR1724388] contain a generalization of this result to the following one: Let $X$ be an irreducible projective variety over an algebraically closed field of characteristic 0; for $r\geq 3$, if every $(r-2)$-plane that intersects $X$ in $r-1$ generic points also meets $X$ in another $r$-th point then $X$ is contained in a linear subspace $L$ with ${\rm codim}_LX\leq r-2$ [see also L. Chiantini and C. Ciliberto, Trans. Amer. Math. Soc. 354 (2002), no. 1, 151–178; MR1859030]. The authors of the paper under review study the case of lines that intersect $X$ (equidimensional) in $m$ points such that $m\leq \dim X +1$. They introduce the notion of strong connectivity in which two $m$-secants $l_1$ and $l_2$ can be joined by a limit sequence $\{(p_i,u_i)\}_{i=1, \ldots , n}$ where $u_1=l_1$, $u_n=l_n$ and each line $u_i$ is an $m$-secant passing through $p_i\in X$. With this notion they prove the following result. Consider an equidimensional variety $X$, of dimension $d$. For $m\leq d+1$, if the variety of $m$-secants satisfies the following assumptions: (1) through every point in $X$ there passes at least one $m$-secant, (2) the variety of $m$-secants is strongly connected, (3) every $m$-secant is also an $(m+1)$-secant, then the variety $X$ can be embedded in $\Bbb{P}^{d+1}$. They also give examples that show the importance of the hypotheses of the theorem. Reviewed by Alessandra Bernardi

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