Secant variety (${\sigma }_r(X)$)

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.^[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.