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In algebraic geometry, the secant variety $\operatorname\left(V\right)$, or the variety of chords, of a
projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective spaces, projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n' ...
$V \subset \mathbb^r$ is the
Zariski closure , birth_date = , birth_place = Kobrin, Russian Empire The Russian Empire, . was a historical empire that extended across Eurasia Eurasia () is the largest continental area on Earth, comprising all of Europe and Asia. Pri ...
of the union of all
secant line In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
s (chords) to ''V'' in $\mathbb^r$: :$\operatorname\left(V\right) = \bigcup_ \overline$ (for $x = y$, the line $\overline$ is the
tangent line In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

.) It is also the image under the projection $p_3: \left(\mathbb^r\right)^3 \to \mathbb^r$ of the closure ''Z'' of the incidence variety :$\$. Note that ''Z'' has dimension $2 \dim V + 1$ and so $\operatorname\left(V\right)$ 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 Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...
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 $\operatorname\left(S\right) \ne \mathbb^5$, then ''S'' is a
Veronese surface In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Gius ...
.

References

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