In algebraic geometry, the secant variety
, 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 ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
is the
Zariski closure
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is ...
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
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
.) It is also the image under the projection
of the closure ''Z'' of the
incidence variety
Incidence may refer to:
Economics
* Benefit incidence, the availability of a benefit
* Expenditure incidence, the effect of government expenditure upon the distribution of private incomes
* Fiscal incidence, the economic impact of government ta ...
:
.
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
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables ...
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
Projection, projections or projective may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphic ...
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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