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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.

The Zariski closure of a subset $S$ of an algebraic variety or a scheme is the smallest Zariski-closed subset containing $S$. For $S\subseteq k^n$, define the ideal $I(S)$ of all polynomials vanishing on $S$. ThenThis description follows because an intersection of polynomial zero sets is again the common zero set of their equations.^[ref]

Every infinite subset of the complex affine line is Zariski dense: a nonzero one-variable polynomial has only finitely many roots of a polynomial. Thus $S=\{1,1/2,1/3,\dots \}$ has Zariski closure all of $\mathbb{C}$, although its closure in the ordinary Euclidean topology is only $S\cup \{0\}$. In algebraic geometry, taking a closure retains all polynomial consequences, rather than all ordinary sequential limits.