Secant variety

In algebraic geometry, the secant variety $$\operatorname{Sect}(V)$$, or the variety of chords, of a projective variety $$V \subset \mathbb{P}^r$$ is the Zariski closure of the union of all secant lines (chords) to V in $$\mathbb{P}^r$$:
 * $$\operatorname{Sect}(V) = \bigcup_{x, y \in V} \overline{xy}$$

(for $$x = y$$, the line $$\overline{xy}$$ is the tangent line.) It is also the image under the projection $$p_3: (\mathbb{P}^r)^3 \to \mathbb{P}^r$$ of the closure Z of the incidence variety
 * $$\{ (x, y, r) | x \wedge y \wedge r = 0 \}$$.

Note that Z has dimension $$2 \dim V + 1$$ and so $$\operatorname{Sect}(V)$$ has dimension at most $$2 \dim V + 1$$.

More generally, the $$k^{th}$$ secant variety is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on $$V$$. It may be denoted by $$\Sigma_k$$. The above secant variety is the first secant variety. Unless $$\Sigma_k=\mathbb{P}^r$$, it is always singular along $$\Sigma_{k-1}$$, but may have other singular points.

If $$V$$ has dimension d, the dimension of $$\Sigma_k$$ is at most $$kd+d+k$$. A useful tool for computing the dimension of a secant variety is Terracini's lemma.

Examples
A secant variety can be used to show the fact that a smooth projective curve can be embedded into the projective 3-space $$\mathbb{P}^3$$ as follows. Let $$C \subset \mathbb{P}^r$$ be a smooth curve. Since the dimension of the secant variety S to C has dimension at most 3, if $$r > 3$$, then there is a point p on $$\mathbb{P}^r$$ that is not on S and so we have the projection $$\pi_p$$ from p to a hyperplane H, which gives the embedding $$\pi_p: C \hookrightarrow H \simeq \mathbb{P}^{r-1}$$. Now repeat.

If $$S \subset \mathbb{P}^5$$ is a surface that does not lie in a hyperplane and if $$\operatorname{Sect}(S) \ne \mathbb{P}^5$$, then S is a Veronese surface.