In algebraic geometry, the secant variety
, or the variety of chords, of a
projective variety is the
Zariski closure of the union of all
secant line
Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to:
* a secant line, in geometry
* the secant variety, in algebraic geometry
* secant (trigonometry) (Latin: secans), the multiplicative inverse (or recipr ...
s (chords) to ''V'' in
:
:
(for
, the line
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
and so
has dimension at most
.
More generally, the
secant variety is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on
. It may be denoted by
. The above secant variety is the first secant variety. Unless
, it is always singular along
, but may have other singular points.
If
has dimension ''d'', the dimension of
is at most
.
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 algebrai ...
projective curve can be embedded into the projective 3-space
as follows. Let
be a smooth curve. Since the dimension of the secant variety ''S'' to ''C'' has dimension at most 3, if
, then there is a point ''p'' on
that is not on ''S'' and so we have the
projection from ''p'' to a hyperplane ''H'', which gives the embedding
. Now repeat.
If
is a surface that does not lie in a hyperplane and if
, 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 Giu ...
.
References
*
*
* Joe Harris, ''Algebraic Geometry, A First Course'', (1992) Springer-Verlag, New York.
Algebraic geometry
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