# Cayley form

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)

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