In algebraic geometry, the secant variety $\backslash operatorname(V)$, 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' ...

$V\; \backslash subset\; \backslash mathbb^r$ is the Zariski closure
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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 $\backslash mathbb^r$:
:$\backslash operatorname(V)\; =\; \backslash bigcup\_\; \backslash overline$
(for $x\; =\; y$, the line $\backslash overline$ 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 $p\_3:\; (\backslash mathbb^r)^3\; \backslash to\; \backslash mathbb^r$ of the closure ''Z'' of the incidence variety
:$\backslash $.
Note that ''Z'' has dimension $2\; \backslash dim\; V\; +\; 1$ and so $\backslash operatorname(V)$ has dimension at most $2\; \backslash dim\; V\; +\; 1$.
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 $\backslash Sigma\_k$. The above secant variety is the first secant variety. Unless $\backslash Sigma\_k=\backslash mathbb^r$, it is always singular along $\backslash Sigma\_$, but may have other singular points.
If $V$ has dimension ''d'', the dimension of $\backslash Sigma\_k$ 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 asmooth
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 $\backslash mathbb^3$ as follows. Let $C\; \backslash subset\; \backslash mathbb^r$ 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 $\backslash mathbb^r$ that is not on ''S'' and so we have the projection $\backslash pi\_p$ from ''p'' to a hyperplane ''H'', which gives the embedding $\backslash pi\_p:\; C\; \backslash hookrightarrow\; H\; \backslash simeq\; \backslash mathbb^$. Now repeat.
If $S\; \backslash subset\; \backslash mathbb^5$ is a surface that does not lie in a hyperplane and if $\backslash operatorname(S)\; \backslash ne\; \backslash mathbb^5$, 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