Cayley form
A form in variables, where
, with
a closed algebraic subvariety of an
-dimensional projective space
, which is determined by
uniquely up to a constant factor and itself uniquely determines
. The exact definition is as follows. Let
be the
-dimensional projective space of all hyperplanes in
, and let
be the subset of the variety
![]() |
( factors
) consisting of all
-tuples
, where
is a point in the intersection of the hyperplanes
and
![]() |
( factors
each time) is the natural projection. Then
is an irreducible subvariety of codimension 1 in
![]() |
( factors) and so
is the zero variety for some form
on
![]() |
( factors).
It may always be assumed that has no multiple factors, and this condition defines
uniquely up to multiplication by a constant, given
. Conversely,
uniquely determines the set of all possible
-tuples of hyperplanes in
that intersect at points of
, and so
uniquely determines
. The form
is called the Cayley form of
.
The Cayley form is also frequently known as the Chow form, or the associated form, of the variety . The idea of defining
by the complex of linear subspaces of dimension
in
that intersect
goes back to A. Cayley [5], who applied it to the case
,
. The coefficients of the Cayley form are known as the Chow coordinates of the variety
.
The Cayley form is homogeneous in each of the
coordinate systems of the space
![]() |
( factors) (the
-th coordinate system is the coordinate system of the
-th factor in this space). The form
has the same degree of homogeneity with respect to each system; this common degree
is denoted by
and is called the degree of the subvariety
. It admits a geometrical interpretation:
is the maximum number of points of intersection of
with all possible
-dimensional linear spaces
in
such that
is a finite set (i.e.
is the number of points in the intersection of
with a "general"
-dimensional linear subspace).
The set of all forms (up to multiplication by a non-zero constant) in groups of variables consisting of
variables and having degree
in each group, forms a projective space
of dimension
. The Cayley form
may be identified with a point of
. The set
of all points in
which are Cayley forms of
-dimensional closed subvarieties of degree
in
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,
need not be closed in
.
The construction of the Cayley form carries over in a natural way to closed -dimensional cycles in
, i.e. to formal linear combinations
of closed
-dimensional subvarieties
in
with integer coefficients
. In fact, one puts
![]() |
and . The set
of all Cayley forms of
-dimensional cycles of degree
in
is closed in
.
The investigation of Cayley forms and the properties of the varieties and
constitute an important aspect in the problem of classifying subvarieties and cycles in
. The first stage in this classification is to study the decomposition of
into irreducible components. For example, if
,
,
(curves of degree 2 in three-dimensional space), the variety
decomposes into two irreducible
-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
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) |
[2] | W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , 2 , Cambridge Univ. Press (1952) |
[3] | P. Samuel, "Méthodes d'algèbre abstraite en géométrie algébrique" , Springer (1955) |
[4] | W.-L. Chow, B.L. van der Waerden, "Zur algebraische Geometrie IX" Math. Ann. , 113 (1937) pp. 692–704 |
[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 are special cases of Hilbert schemes (cf. Hilbert scheme).
Cayley form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cayley_form&oldid=11497