Secant variety

In algebraic geometry, the secant variety

Sect ⁡ (

{\displaystyle \operatorname {Sect} (V)}

, or the variety of chords, of a projective variety

is the Zariski closure of the union of all secant lines (chords) to V in

is the tangent line.)

It is also the image under the projection

of the closure Z of the incidence variety Note that Z has dimension

Sect ⁡ (

{\displaystyle \operatorname {Sect} (V)}

secant variety is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on

The above secant variety is the first secant variety.

, but may have other singular points.

A useful tool for computing the dimension of a secant variety is Terracini's lemma.

A secant variety can be used to show the fact that a smooth projective curve can be embedded into the projective 3-space

be a smooth curve.

Since the dimension of the secant variety S to C has dimension at most 3, if

from p to a hyperplane H, which gives the embedding

is a surface that does not lie in a hyperplane and if

Sect ⁡ (

, then S is a Veronese surface.

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