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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[8]	A.N. Tyurin, "The geometry of the Fano surface of a nonsingular cubic $F \subseteq \mathbb{P}^4$ and Torelli theorems for Fano surfaces and cubics" Math. USSR-Izv. , 5 : 3 (1971) pp. 517–546 Izv. Akad. Nauk SSSR Ser. Mat. , 35 (1971) pp. 498–529 Zbl 0215.08201
[9]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[a1]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001