# Cubic hypersurface

A projective algebraic variety defined by a homogeneous equation $F _ {3} ( x _ {0}, \dots, x _ {n} ) = 0$ of degree three with coefficients in some ground field $k$.

## Cubic curves.

An irreducible cubic curve is either smooth (in which case its canonical class is 0, its genus 1) or has a unique singular double point (in which case it is rational). Cubic curves are the curves of lowest degree for which there exist moduli (cf. Moduli theory). Every smooth cubic curve $X$ over an algebraically closed field $k$ of characteristic $\neq 2$ or 3 can be reduced by birational transformations to Weierstrass form, which is, in terms of non-homogeneous coordinates on the $( x, y)$-plane,

$$y ^ {2} = \ 4x ^ {3} - g _ {2} x - g _ {3} ,$$

where $g _ {2} , g _ {3} \in k$, $g _ {2} ^ {3} - 27g _ {3} ^ {2} \neq 0$. Two cubic curves with coefficients $( g _ {2} , g _ {3} )$ and $( g _ {2} ^ \prime , g _ {3} ^ \prime )$ in Weierstrass form are isomorphic if and only if

$$\frac{g _ {2} ^ {3} }{g _ {2} ^ {3} - 27g _ {3} ^ {2} } = \ \frac{g _ {2} ^ {\prime 2} }{g _ {2} ^ {\prime 3} - 27g _ {3} ^ {\prime 2} } .$$

The function

$$j = \ \frac{1728g _ {2} ^ {3} }{g _ {2} ^ {3} - 27g _ {3} ^ {2} }$$

takes arbitrary values in $k$ and depends only on the curve $X$; it is called the absolute invariant of $X$.

One can define a binary composition law $( x _ {1} , x _ {2} ) \rightarrow x _ {1} \circ x _ {2}$ on the set of points $X ( k)$ of a cubic curve: $x _ {1} \circ x _ {2}$ is the third point of intersection of $X$ with the straight line through $x _ {1}$ and $x _ {2}$. If one fixes some point $x _ {0} \in X ( k)$, the composition

$$( x _ {1} , x _ {2} ) \rightarrow \ x _ {0} \circ ( x _ {1} \circ x _ {2} )$$

turns $X ( k)$ into an Abelian group with neutral element $x _ {0}$. A cubic curve endowed with this structure is a one-dimensional Abelian variety (an elliptic curve).

If $k = \mathbf C$ is the field of complex numbers, $X ( \mathbf C )$ is a Riemann surface of genus 1, i.e. a one-dimensional complex torus — a quotient group $\mathbf C / \Gamma ( X)$, where $\Gamma ( X)$ is a two-dimensional period lattice. The field $k$ of rational functions of the curve $X$ is then isomorphic to the field of elliptic functions on $\mathbf C$ with period lattice $\Gamma ( X)$. The coefficients $g _ {2} , g _ {3}$ are interpreted as modular forms of weight 4 and 6, respectively, that are identical, up to a constant factor, with the forms defined by Eisenstein series of lowest weights. In that case the function $f$ is none other than the modular invariant.

A rich arithmetic theory has also been developed for cubic curves over algebraically non-closed fields $k$ (see [2]). Significant achievements in that respect are the Mordell–Weil theorem, the theory of complex multiplication and the homology theory of principal homogeneous spaces. The main unsolved problems (as of 1982) are: boundedness of the rank over an algebraic number field; the finiteness conjecture for the group of principal homogeneous locally trivial spaces; the conjecture of Birch and Swinnerton-Dyer on the zeta-function; Weil's uniformization conjecture, etc. (See also Elliptic curve.)

## Cubic surfaces.

Over an algebraically closed field $k$, every irreducible cubic surface (that does not degenerate into a cone) is a rational surface. The class of a hyperplane section $h$ of a surface $F$ is precisely the canonical class $(- K _ {F} )$. Any smooth cubic surface can be obtained from the projective plane $P ^ {2}$ by blowing-up (i.e. performing a monoidal transformation) of 6 points, no three of which are collinear, which do not lie on a single conic. The appropriate birational mapping $\phi : P ^ {2} \rightarrow F$ is determined by the linear system of cubic curves passing through the 6 points. There are 27 straight lines on $F$, each of which is exceptional (see Exceptional subvariety); they are the only exceptional curves on $F$. The configuration of these 27 lines is rich in symmetries: The automorphism group of the corresponding graph is isomorphic to the Weil group of type $E _ {6}$. Cubic surfaces belong to the class of del Pezzo surfaces — projective surfaces with an ample anti-canonical class.

Over an algebraically non-closed field $k$, there are smooth cubic surfaces $F$ which are not birationally isomorphic to $P ^ {2}$ over $k$ (i.e. $F$ is not rational over $k$). Among these surfaces one finds surfaces possessing $k$-points, and these are unirational over $k$. Such cubic surfaces provide a counterexample for the Lüroth problem on surfaces over non-closed fields. There exist fields $k$ over which there are minimal cubic surfaces. Segre's minimality criterion [6]: $\mathop{\rm Pic} ( F) \simeq \mathbf Z$. The group of birational automorphisms of a minimal surface has been determined (in terms of its generators and defining relations) and an arithmetic theory of cubic surfaces has been developed [4]. In order to describe the set of points $F ( k)$ one appeals to non-associative structures, such as quasi-groups and Moufang loops.

## Cubic hypersurfaces of dimension 3.

All smooth cubic hypersurfaces of dimension $\geq 2$ over an algebraically closed field are unirational. As far back as the eighties of the 19th century, the following question was posed: Is a smooth three-dimensional cubic hypersurface rational? A negative answer has been obtained [3]. This also provides a negative solution to the Lüroth problem for three-dimensional varieties. For every smooth three-dimensional cubic hypersurface $V$ there exists a principal polarized five-dimensional Abelian variety — the intermediate Jacobian $J _ {3} ( V)$. If $k = \mathbf C$ it is defined as as the complex torus

$$H ^ {1,2} ( V, \mathbf C )/H ^ {3} ( V, \mathbf Z ),$$

where $H ^ {1,2} ( V, \mathbf C )$ is the corresponding Hodge component in the decomposition of the homology space $H ^ {3} ( V, \mathbf C )$. In order to prove that $V$ is non-rational, it was shown that $J _ {3} ( V)$ is not the Jacobian of any curve of genus 5. The fact that a cubic hypersurface over a field of finite characteristic is non-rational was established in [5].

A cubic hypersurface $V$ is uniquely determined by its Fano surface $\Phi ( V)$. For $\Phi ( V)$ one has the Torelli theorems (which are also valid for $V$ itself). The following problem is unsolved: Given a three-dimensional cubic hypersurface, describe its group of birational automorphisms.

It is not known (1987) whether every smooth cubic hypersurface of dimension $\geq 4$ is rational. Rationality has been proved in this case for certain hypersurfaces of a special type; for example:

$$\sum _ {i = 0 } ^ { {2m } + 1 } a _ {i} x _ {i} ^ {3} = 0,\ \ m \geq 2.$$

#### References

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