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A form in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c0210501.png" /> variables, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c0210502.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c0210503.png" /> a closed algebraic subvariety of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c0210504.png" />-dimensional projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c0210505.png" />, which is determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c0210506.png" /> uniquely up to a constant factor and itself uniquely determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c0210507.png" />. The exact definition is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c0210508.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c0210509.png" />-dimensional projective space of all hyperplanes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105010.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105011.png" /> be the subset of the variety
+
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c0210501.png
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$#A+1 = 105 n = 0
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$#C+1 = 105 : ~/encyclopedia/old_files/data/C021/C.0201050 Cayley form
 +
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if TeX found to be correct.
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105012.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105013.png" /> factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105014.png" />) consisting of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105015.png" />-tuples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105017.png" /> is a point in the intersection of the hyperplanes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105018.png" /> and
+
A form in  $  ( n + 1) ( N + 1) $
 +
variables, where  $  n = \mathop{\rm dim}  X $,
 +
with  $  X $
 +
a closed algebraic subvariety of an  $  N $-
 +
dimensional projective space  $  \mathbf P  ^ {N} $,  
 +
which is determined by  $  X $
 +
uniquely up to a constant factor and itself uniquely determines  $  X $.  
 +
The exact definition is as follows. Let  $  \widetilde{\mathbf P}  {}  ^ {N} $
 +
be the  $  N $-
 +
dimensional projective space of all hyperplanes in $  \mathbf P  ^ {N} $,
 +
and let  $  \Gamma $
 +
be the subset of the variety
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105019.png" /></td> </tr></table>
+
$$
 +
{\widetilde{\mathbf P}  } {}  ^ {N}
 +
\times \dots \times
 +
{\widetilde{\mathbf P}  } {}  ^ {N}
 +
\times X
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105020.png" /> factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105021.png" /> each time) is the natural projection. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105022.png" /> is an irreducible subvariety of codimension 1 in
+
( $  n + 1 $
 +
factors $  \widetilde{\mathbf P}  {}  ^ {N} $)
 +
consisting of all  $  ( n + 2) $-
 +
tuples  $  ( \pi _ {1} \dots \pi _ {n + 1 }  , x) $,
 +
where  $  x \in X $
 +
is a point in the intersection of the hyperplanes  $  \pi _ {1} \dots \pi _ {n + 1 }  $
 +
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105023.png" /></td> </tr></table>
+
$$
 +
\phi : \widetilde{\mathbf P}  {}  ^ {N} \times \dots \times {\widetilde{\mathbf P}  } {}  ^ {N}
 +
\times X  \rightarrow \
 +
{\widetilde{\mathbf P}  } {}  ^ {N} \times \dots \times {\widetilde{\mathbf P}  } {}  ^ {N}
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105024.png" /> factors) and so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105025.png" /> is the zero variety for some form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105026.png" /> on
+
( $  n + 1 $
 +
factors $  \widetilde{\mathbf P}  {}  ^ {N} $
 +
each time) is the natural projection. Then  $  \phi ( \Gamma ) $
 +
is an irreducible subvariety of codimension 1 in
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105027.png" /></td> </tr></table>
+
$$
 +
{\widetilde{\mathbf P}  } {}  ^ {N} \times \dots \times {\widetilde{\mathbf P}  } {}  ^ {N}
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105028.png" /> factors).
+
( $  n + 1 $
 +
factors) and so  $  \phi ( \Gamma ) $
 +
is the zero variety for some form  $  F _ {X} $
 +
on
  
It may always be assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105029.png" /> has no multiple factors, and this condition defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105030.png" /> uniquely up to multiplication by a constant, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105031.png" />. Conversely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105032.png" /> uniquely determines the set of all possible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105033.png" />-tuples of hyperplanes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105034.png" /> that intersect at points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105035.png" />, and so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105036.png" /> uniquely determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105037.png" />. The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105038.png" /> is called the Cayley form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105039.png" />.
+
$$
 +
{\widetilde{\mathbf P}  } {}  ^ {N} \times \dots \times {\widetilde{\mathbf P}  } {}  ^ {N}
 +
$$
  
The Cayley form is also frequently known as the Chow form, or the associated form, of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105040.png" />. The idea of defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105041.png" /> by the complex of linear subspaces of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105042.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105043.png" /> that intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105044.png" /> goes back to A. Cayley [[#References|[5]]], who applied it to the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105046.png" />. The coefficients of the Cayley form are known as the Chow coordinates of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105047.png" />.
+
( $  n + 1 $
 +
factors).
  
The Cayley form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105048.png" /> is homogeneous in each of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105049.png" /> coordinate systems of the space
+
It may always be assumed that  $  F _ {X} $
 +
has no multiple factors, and this condition defines  $  F _ {X} $
 +
uniquely up to multiplication by a constant, given  $  X $.
 +
Conversely,  $  F _ {X} $
 +
uniquely determines the set of all possible  $  ( n + 1) $-
 +
tuples of hyperplanes in  $  \mathbf P  ^ {N} $
 +
that intersect at points of  $  X $,
 +
and so  $  F _ {X} $
 +
uniquely determines  $  X $.
 +
The form  $  F _ {X} $
 +
is called the Cayley form of $  X $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105050.png" /></td> </tr></table>
+
The Cayley form is also frequently known as the Chow form, or the associated form, of the variety  $  X $.
 +
The idea of defining  $  X $
 +
by the complex of linear subspaces of dimension  $  N - n - 1 $
 +
in  $  \mathbf P  ^ {N} $
 +
that intersect  $  X $
 +
goes back to A. Cayley [[#References|[5]]], who applied it to the case  $  n = 1 $,
 +
$  N = 3 $.  
 +
The coefficients of the Cayley form are known as the Chow coordinates of the variety  $  X $.
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105051.png" /> factors) (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105052.png" />-th coordinate system is the coordinate system of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105053.png" />-th factor in this space). The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105054.png" /> has the same degree of homogeneity with respect to each system; this common degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105055.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105056.png" /> and is called the degree of the subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105057.png" />. It admits a geometrical interpretation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105058.png" /> is the maximum number of points of intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105059.png" /> with all possible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105060.png" />-dimensional linear spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105061.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105062.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105063.png" /> is a finite set (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105064.png" /> is the number of points in the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105065.png" /> with a "general" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105066.png" />-dimensional linear subspace).
+
The Cayley form $  F _ {X} $
 +
is homogeneous in each of the  $ n + 1 $
 +
coordinate systems of the space
  
The set of all forms (up to multiplication by a non-zero constant) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105067.png" /> groups of variables consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105068.png" /> variables and having degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105069.png" /> in each group, forms a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105070.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105071.png" />. The Cayley form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105072.png" /> may be identified with a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105073.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105074.png" /> of all points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105075.png" /> which are Cayley forms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105076.png" />-dimensional closed subvarieties of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105077.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105078.png" /> is a quasi-projective variety; it parametrizes the family of all such subvarieties in such a way that the family is algebraic relative to the parametrization. In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105079.png" /> need not be closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105080.png" />.
+
$$
 +
{\widetilde{\mathbf P}  } {}  ^ {N} \times \dots \times {\widetilde{\mathbf P}  } {}  ^ {N}
 +
$$
  
The construction of the Cayley form carries over in a natural way to closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105081.png" />-dimensional cycles in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105082.png" />, i.e. to formal linear combinations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105083.png" /> of closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105084.png" />-dimensional subvarieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105085.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105086.png" /> with integer coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105087.png" />. In fact, one puts
+
( $  n + 1 $
 +
factors) (the  $  i $-
 +
th coordinate system is the coordinate system of the  $  i $-
 +
th factor in this space). The form  $  F _ {X} $
 +
has the same degree of homogeneity with respect to each system; this common degree  $  d $
 +
is denoted by  $  \mathop{\rm deg}  X $
 +
and is called the degree of the subvariety  $  X $.
 +
It admits a geometrical interpretation: $  d $
 +
is the maximum number of points of intersection of  $  X $
 +
with all possible  $  ( N - n) $-
 +
dimensional linear spaces  $  L $
 +
in $  \mathbf P  ^ {N} $
 +
such that  $  X \cap L $
 +
is a finite set (i.e. $  d $
 +
is the number of points in the intersection of $  X $
 +
with a "general" $  ( N - n) $-
 +
dimensional linear subspace).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105088.png" /></td> </tr></table>
+
The set of all forms (up to multiplication by a non-zero constant) in  $  n + 1 $
 +
groups of variables consisting of  $  N + 1 $
 +
variables and having degree  $  d $
 +
in each group, forms a projective space  $  \mathbf P ^ {\nu _ {N, n, d }  } $
 +
of dimension  $  \nu _ {N, n, d }  $.
 +
The Cayley form  $  F _ {X} $
 +
may be identified with a point of  $  \mathbf P ^ {\nu _ {N, n, d }  } $.
 +
The set  $  C _ {N, n, d }  $
 +
of all points in  $  \mathbf P ^ {\nu _ {N, n, d }  } $
 +
which are Cayley forms of  $  n $-
 +
dimensional closed subvarieties of degree  $  d $
 +
in  $  \mathbf P  ^ {N} $
 +
is a quasi-projective variety; it parametrizes the family of all such subvarieties in such a way that the family is algebraic relative to the parametrization. In general,  $  C _ {N, n, d }  $
 +
need not be closed in  $  \mathbf P ^ {\nu _ {N, n, d }  } $.
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105089.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105090.png" /> of all Cayley forms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105091.png" />-dimensional cycles of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105092.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105093.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105094.png" />.
+
The construction of the Cayley form carries over in a natural way to closed  $  n $-
 +
dimensional cycles in $  \mathbf P  ^ {N} $,
 +
i.e. to formal linear combinations  $  m _ {1} X _ {1} + \dots + m _ {s} X _ {s} $
 +
of closed $  n $-
 +
dimensional subvarieties  $  X _ {1} \dots X _ {s} $
 +
in $  \mathbf P  ^ {N} $
 +
with integer coefficients  $  m _ {i} > 0 $.  
 +
In fact, one puts
  
The investigation of Cayley forms and the properties of the varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105096.png" /> constitute an important aspect in the problem of classifying subvarieties and cycles in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105097.png" />. The first stage in this classification is to study the decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105098.png" /> into irreducible components. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c02105099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c021050100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c021050101.png" /> (curves of degree 2 in three-dimensional space), the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c021050102.png" /> decomposes into two irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c021050103.png" />-dimensional components. The first corresponds to planar curves of the second order, the second to pairs of straight lines. The birational classification of the varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c021050104.png" /> is an important problem (in all known examples these varieties are rational).
+
$$
 +
\mathop{\rm deg}  X  = \
 +
\sum _ {i = 1 } ^ { s }
 +
m _ {i}  \mathop{\rm deg}  X _ {i}  $$
  
====References====
+
and  $  F _ {X} = F _ {X _ {1}  } ^ {m _ {1} } \dots F _ {X _ {s} } ^ {m _ {s} } $.  
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich,   "Basic algebraic geometry" , Springer (1977) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> W.V.D. Hodge,  D. Pedoe,  "Methods of algebraic geometry" , '''2''' , Cambridge Univ. Press (1952)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P. Samuel,  "Méthodes d'algèbre abstraite en géométrie algébrique" , Springer (1955)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> W.-L. Chow,  B.L. van der Waerden,  "Zur algebraische Geometrie IX" ''Math. Ann.'' , '''113''' (1937) pp. 692–704</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Cayley,   "On a new analytical representation of curves in space" , ''Collected mathematical papers'' , '''4''' , Cambridge Univ. Press (1891)  pp. 446–455  (Quart. J. Pure Appl. Math. 3 (1860), 225–236)</TD></TR></table>
+
The set  $  {C _ {N, n, d } } bar $
 +
of all Cayley forms of $ n $-
 +
dimensional cycles of degree $ d $
 +
in $ \mathbf P ^ {N} $
 +
is closed in $ \mathbf P ^ {\nu _ {N, n, d } } $.
  
 +
The investigation of Cayley forms and the properties of the varieties  $  C _ {N, n, d }  $
 +
and  $  {C _ {N, n, d }  } bar $
 +
constitute an important aspect in the problem of classifying subvarieties and cycles in  $  \mathbf P  ^ {n} $.
 +
The first stage in this classification is to study the decomposition of  $  {C _ {N, n, d }  } bar $
 +
into irreducible components. For example, if  $  N = 3 $,
 +
$  n = 1 $,
 +
$  d = 2 $(
 +
curves of degree 2 in three-dimensional space), the variety  $  {C _ {3, 1, 2 }  } bar $
 +
decomposes into two irreducible  $  8 $-
 +
dimensional components. The first corresponds to planar curves of the second order, the second to pairs of straight lines. The birational classification of the varieties  $  C _ {N, n, d }  $
 +
is an important problem (in all known examples these varieties are rational).
  
 +
====References====
 +
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''2''' , Cambridge Univ. Press (1952) {{MR|0048065}} {{ZBL|0048.14502}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P. Samuel, "Méthodes d'algèbre abstraite en géométrie algébrique" , Springer (1955) {{MR|0072531}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> W.-L. Chow, B.L. van der Waerden, "Zur algebraische Geometrie IX" ''Math. Ann.'' , '''113''' (1937) pp. 692–704 {{MR|}} {{ZBL|0016.04004}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A. Cayley, "On a new analytical representation of curves in space" , ''Collected mathematical papers'' , '''4''' , Cambridge Univ. Press (1891) pp. 446–455 (Quart. J. Pure Appl. Math. 3 (1860), 225–236)</TD></TR></table>
  
 
====Comments====
 
====Comments====
The varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021050/c021050105.png" /> are special cases of Hilbert schemes (cf. [[Hilbert scheme|Hilbert scheme]]).
+
The varieties $  C _ {N,n,d} $
 +
are special cases of Hilbert schemes (cf. [[Hilbert scheme|Hilbert scheme]]).

Latest revision as of 16:43, 4 June 2020


A form in $ ( n + 1) ( N + 1) $ variables, where $ n = \mathop{\rm dim} X $, with $ X $ a closed algebraic subvariety of an $ N $- dimensional projective space $ \mathbf P ^ {N} $, which is determined by $ X $ uniquely up to a constant factor and itself uniquely determines $ X $. The exact definition is as follows. Let $ \widetilde{\mathbf P} {} ^ {N} $ be the $ N $- dimensional projective space of all hyperplanes in $ \mathbf P ^ {N} $, and let $ \Gamma $ be the subset of the variety

$$ {\widetilde{\mathbf P} } {} ^ {N} \times \dots \times {\widetilde{\mathbf P} } {} ^ {N} \times X $$

( $ n + 1 $ factors $ \widetilde{\mathbf P} {} ^ {N} $) consisting of all $ ( n + 2) $- tuples $ ( \pi _ {1} \dots \pi _ {n + 1 } , x) $, where $ x \in X $ is a point in the intersection of the hyperplanes $ \pi _ {1} \dots \pi _ {n + 1 } $ and

$$ \phi : \widetilde{\mathbf P} {} ^ {N} \times \dots \times {\widetilde{\mathbf P} } {} ^ {N} \times X \rightarrow \ {\widetilde{\mathbf P} } {} ^ {N} \times \dots \times {\widetilde{\mathbf P} } {} ^ {N} $$

( $ n + 1 $ factors $ \widetilde{\mathbf P} {} ^ {N} $ each time) is the natural projection. Then $ \phi ( \Gamma ) $ is an irreducible subvariety of codimension 1 in

$$ {\widetilde{\mathbf P} } {} ^ {N} \times \dots \times {\widetilde{\mathbf P} } {} ^ {N} $$

( $ n + 1 $ factors) and so $ \phi ( \Gamma ) $ is the zero variety for some form $ F _ {X} $ on

$$ {\widetilde{\mathbf P} } {} ^ {N} \times \dots \times {\widetilde{\mathbf P} } {} ^ {N} $$

( $ n + 1 $ factors).

It may always be assumed that $ F _ {X} $ has no multiple factors, and this condition defines $ F _ {X} $ uniquely up to multiplication by a constant, given $ X $. Conversely, $ F _ {X} $ uniquely determines the set of all possible $ ( n + 1) $- tuples of hyperplanes in $ \mathbf P ^ {N} $ that intersect at points of $ X $, and so $ F _ {X} $ uniquely determines $ X $. The form $ F _ {X} $ is called the Cayley form of $ X $.

The Cayley form is also frequently known as the Chow form, or the associated form, of the variety $ X $. The idea of defining $ X $ by the complex of linear subspaces of dimension $ N - n - 1 $ in $ \mathbf P ^ {N} $ that intersect $ X $ goes back to A. Cayley [5], who applied it to the case $ n = 1 $, $ N = 3 $. The coefficients of the Cayley form are known as the Chow coordinates of the variety $ X $.

The Cayley form $ F _ {X} $ is homogeneous in each of the $ n + 1 $ coordinate systems of the space

$$ {\widetilde{\mathbf P} } {} ^ {N} \times \dots \times {\widetilde{\mathbf P} } {} ^ {N} $$

( $ n + 1 $ factors) (the $ i $- th coordinate system is the coordinate system of the $ i $- th factor in this space). The form $ F _ {X} $ has the same degree of homogeneity with respect to each system; this common degree $ d $ is denoted by $ \mathop{\rm deg} X $ and is called the degree of the subvariety $ X $. It admits a geometrical interpretation: $ d $ is the maximum number of points of intersection of $ X $ with all possible $ ( N - n) $- dimensional linear spaces $ L $ in $ \mathbf P ^ {N} $ such that $ X \cap L $ is a finite set (i.e. $ d $ is the number of points in the intersection of $ X $ with a "general" $ ( N - n) $- dimensional linear subspace).

The set of all forms (up to multiplication by a non-zero constant) in $ n + 1 $ groups of variables consisting of $ N + 1 $ variables and having degree $ d $ in each group, forms a projective space $ \mathbf P ^ {\nu _ {N, n, d } } $ of dimension $ \nu _ {N, n, d } $. The Cayley form $ F _ {X} $ may be identified with a point of $ \mathbf P ^ {\nu _ {N, n, d } } $. The set $ C _ {N, n, d } $ of all points in $ \mathbf P ^ {\nu _ {N, n, d } } $ which are Cayley forms of $ n $- dimensional closed subvarieties of degree $ d $ in $ \mathbf P ^ {N} $ is a quasi-projective variety; it parametrizes the family of all such subvarieties in such a way that the family is algebraic relative to the parametrization. In general, $ C _ {N, n, d } $ need not be closed in $ \mathbf P ^ {\nu _ {N, n, d } } $.

The construction of the Cayley form carries over in a natural way to closed $ n $- dimensional cycles in $ \mathbf P ^ {N} $, i.e. to formal linear combinations $ m _ {1} X _ {1} + \dots + m _ {s} X _ {s} $ of closed $ n $- dimensional subvarieties $ X _ {1} \dots X _ {s} $ in $ \mathbf P ^ {N} $ with integer coefficients $ m _ {i} > 0 $. In fact, one puts

$$ \mathop{\rm deg} X = \ \sum _ {i = 1 } ^ { s } m _ {i} \mathop{\rm deg} X _ {i} $$

and $ F _ {X} = F _ {X _ {1} } ^ {m _ {1} } \dots F _ {X _ {s} } ^ {m _ {s} } $. The set $ {C _ {N, n, d } } bar $ of all Cayley forms of $ n $- dimensional cycles of degree $ d $ in $ \mathbf P ^ {N} $ is closed in $ \mathbf P ^ {\nu _ {N, n, d } } $.

The investigation of Cayley forms and the properties of the varieties $ C _ {N, n, d } $ and $ {C _ {N, n, d } } bar $ constitute an important aspect in the problem of classifying subvarieties and cycles in $ \mathbf P ^ {n} $. The first stage in this classification is to study the decomposition of $ {C _ {N, n, d } } bar $ into irreducible components. For example, if $ N = 3 $, $ n = 1 $, $ d = 2 $( curves of degree 2 in three-dimensional space), the variety $ {C _ {3, 1, 2 } } bar $ decomposes into two irreducible $ 8 $- dimensional components. The first corresponds to planar curves of the second order, the second to pairs of straight lines. The birational classification of the varieties $ C _ {N, n, d } $ is an important problem (in all known examples these varieties are rational).

References

[1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2] W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , 2 , Cambridge Univ. Press (1952) MR0048065 Zbl 0048.14502
[3] P. Samuel, "Méthodes d'algèbre abstraite en géométrie algébrique" , Springer (1955) MR0072531
[4] W.-L. Chow, B.L. van der Waerden, "Zur algebraische Geometrie IX" Math. Ann. , 113 (1937) pp. 692–704 Zbl 0016.04004
[5] A. Cayley, "On a new analytical representation of curves in space" , Collected mathematical papers , 4 , Cambridge Univ. Press (1891) pp. 446–455 (Quart. J. Pure Appl. Math. 3 (1860), 225–236)

Comments

The varieties $ C _ {N,n,d} $ are special cases of Hilbert schemes (cf. Hilbert scheme).

How to Cite This Entry:
Cayley form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cayley_form&oldid=11497
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article