Cubic hypersurface

A projective algebraic variety defined by a homogeneous equation of degree three with coefficients in some ground field .

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 over an algebraically closed field of characteristic or 3 can be reduced by birational transformations to Weierstrass form, which is, in terms of non-homogeneous coordinates on the -plane,

where , . Two cubic curves with coefficients and in Weierstrass form are isomorphic if and only if

The function

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

One can define a binary composition law on the set of points of a cubic curve: is the third point of intersection of with the straight line through and . If one fixes some point , the composition

turns into an Abelian group with neutral element . A cubic curve endowed with this structure is a one-dimensional Abelian variety (an elliptic curve).

If is the field of complex numbers, is a Riemann surface of genus 1, i.e. a one-dimensional complex torus — a quotient group , where is a two-dimensional period lattice. The field of rational functions of the curve is then isomorphic to the field of elliptic functions on with period lattice . The coefficients 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 is none other than the modular invariant.

A rich arithmetic theory has also been developed for cubic curves over algebraically non-closed fields (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 , every irreducible cubic surface (that does not degenerate into a cone) is a rational surface. The class of a hyperplane section of a surface is precisely the canonical class . Any smooth cubic surface can be obtained from the projective plane 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 is determined by the linear system of cubic curves passing through the 6 points. There are 27 straight lines on , each of which is exceptional (see Exceptional subvariety); they are the only exceptional curves on . 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 . Cubic surfaces belong to the class of del Pezzo surfaces — projective surfaces with an ample anti-canonical class.

Over an algebraically non-closed field , there are smooth cubic surfaces which are not birationally isomorphic to over (i.e. is not rational over ). Among these surfaces one finds surfaces possessing -points, and these are unirational over . Such cubic surfaces provide a counterexample for the Lüroth problem on surfaces over non-closed fields. There exist fields over which there are minimal cubic surfaces. Segre's minimality criterion [6]: . 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 one appeals to non-associative structures, such as quasi-groups and Moufang loops.

Cubic hypersurfaces of dimension 3.

All smooth cubic hypersurfaces of dimension 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 there exists a principal polarized five-dimensional Abelian variety — the intermediate Jacobian . If it is defined as as the complex torus

where is the corresponding Hodge component in the decomposition of the homology space . In order to prove that is non-rational, it was shown that 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 is uniquely determined by its Fano surface . For one has the Torelli theorems (which are also valid for 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 is rational. Rationality has been proved in this case for certain hypersurfaces of a special type; for example:

References

[1]	A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1964) MR0173749 Zbl 0135.12101
[2]	J.W.S. Cassels, "Diophantine equations with special reference to elliptic curves" J. London Math. Soc. , 41 (1966) pp. 193–291 MR0199150 Zbl 0138.27002
[3]	C.H. Clemens, P.A. Griffiths, "The intermediate Jacobian of the cubic threefold" Ann. of Math. , 95 (1972) pp. 281–356 MR0302652 Zbl 0245.14010
[4]	Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic" , North-Holland (1986) (Translated from Russian) MR0833513 Zbl 0582.14010
[5]	J.P. Murre, "Reduction of the proof of the non-rationality of a non-singular cubic threefold to a result of Mumford" Comp. Math. , 27 (1973) pp. 63–82 MR0352088 MR0352089 Zbl 0271.14020
[6]	B. Segre, "The non-singular cubic surfaces" , Clarendon Press (1942) MR0008171 Zbl 0061.36701 Zbl 68.0358.01
[7]	A.N. Tyurin, "Five lectures on three-dimensional varieties" Russian Math. Surveys , 27 (1972) pp. 1–53 Uspekhi Mat. Nauk , 27 : 5 (1972) pp. 3–50 Zbl 0263.14012
[8]	A.N. Tyurin, "The geometry of the Fano surface of a nonsingular cubic 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

References

[a1]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001