= Secant variety
{title2=$\sigma_r(X)$}
For a <projective variety> $X\subseteq\mathbb P^N$, its <secant variety> $\sigma_2(X)$ is the <Zariski closure> of the union of <projective lines> joining distinct points of $X$. More generally $\sigma_r(X)$ is the closure of the spans of $r$ points. Taking the closure includes limiting configurations; it is essential to the definition.<Algebraic Geometry Fall 2024 Supplementary Notes and Exercises 9 3 24 by J. M. Landsberg>[section 6, joins and secant varieties]{ref}
For a nonsingular <conic> in the <Complex projective plane>, secants fill a dense subset of the plane and their closure is the entire plane. As the two points merge, a secant line tends to a <tangent line>. For <tensor rank> problems, sums of $r$ decomposable <tensors> give secant constructions; the closure also permits limits of such sums. This is why secant geometry naturally describes <border rank>, which can be smaller than actual <tensor rank>.
Back to article page
