Secant Variety

In algebraic geometry, the secant variety $\operatorname(V)$, or the variety of chords, of a projective variety $V \subset \mathbb^r$ is the Zariski closure of the union of all secant lines (chords) to ''V'' in $\mathbb^r$:
:$\operatorname(V) = \bigcup_ \overline$
(for $x = y$, the line $\overline$ is the tangent line.) It is also the image under the projection $p_3: (\mathbb^r)^3 \to \mathbb^r$ of the closure ''Z'' of the incidence variety
:$\$.
Note that ''Z'' has dimension $2 \dim V + 1$ and so $\operatorname(V)$ has dimension at most $2 \dim V + 1$.

More generally, the $k^$ 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 $\Sigma_k=\mathbb^r$, it is always singular along $\Sigma_$, 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 $\mathbb^3$ as follows. Let $C \subset \mathbb^r$ 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 $\mathbb^r$ that is not on ''S'' and so we have the projection $\pi_p$ from ''p'' to a hyperplane ''H'', which gives the embedding $\pi_p: C \hookrightarrow H \simeq \mathbb^$. Now repeat.

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

References

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* Joe Harris, ''Algebraic Geometry, A First Course'', (1992) Springer-Verlag, New York.

Algebraic geometry

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