View source for Cubic hypersurface

A projective algebraic variety defined by a homogeneous equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c0272701.png" /> of degree three with coefficients in some ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c0272702.png" />.

==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|Moduli theory]]). Every smooth cubic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c0272703.png" /> over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c0272704.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c0272705.png" /> or 3 can be reduced by birational transformations to Weierstrass form, which is, in terms of non-homogeneous coordinates on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c0272706.png" />-plane,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c0272707.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c0272708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c0272709.png" />. Two cubic curves with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727011.png" /> in Weierstrass form are isomorphic if and only if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727012.png" /></td> </tr></table>

The function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727013.png" /></td> </tr></table>

takes arbitrary values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727014.png" /> and depends only on the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727015.png" />; it is called the absolute invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727016.png" />.

One can define a binary composition law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727017.png" /> on the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727018.png" /> of a cubic curve: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727019.png" /> is the third point of intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727020.png" /> with the straight line through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727022.png" />. If one fixes some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727023.png" />, the composition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727024.png" /></td> </tr></table>

turns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727025.png" /> into an Abelian group with neutral element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727026.png" />. A cubic curve endowed with this structure is a one-dimensional Abelian variety (an elliptic curve).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727027.png" /> is the field of complex numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727028.png" /> is a Riemann surface of genus 1, i.e. a one-dimensional complex torus — a quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727030.png" /> is a two-dimensional period lattice. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727031.png" /> of rational functions of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727032.png" /> is then isomorphic to the field of elliptic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727033.png" /> with period lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727034.png" />. The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727035.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727036.png" /> is none other than the modular invariant.

A rich arithmetic theory has also been developed for cubic curves over algebraically non-closed fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727037.png" /> (see [[#References|[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|Elliptic curve]].)

==Cubic surfaces.==
Over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727038.png" />, every irreducible cubic surface (that does not degenerate into a cone) is a rational surface. The class of a hyperplane section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727039.png" /> of a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727040.png" /> is precisely the canonical class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727041.png" />. Any smooth cubic surface can be obtained from the projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727042.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727043.png" /> is determined by the linear system of cubic curves passing through the 6 points. There are 27 straight lines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727044.png" />, each of which is exceptional (see [[Exceptional subvariety|Exceptional subvariety]]); they are the only exceptional curves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727045.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727046.png" />. Cubic surfaces belong to the class of del Pezzo surfaces — projective surfaces with an ample anti-canonical class.

Over an algebraically non-closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727047.png" />, there are smooth cubic surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727048.png" /> which are not birationally isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727049.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727050.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727051.png" /> is not rational over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727052.png" />). Among these surfaces one finds surfaces possessing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727053.png" />-points, and these are unirational over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727054.png" />. Such cubic surfaces provide a counterexample for the [[Lüroth problem|Lüroth problem]] on surfaces over non-closed fields. There exist fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727055.png" /> over which there are minimal cubic surfaces. Segre's minimality criterion [[#References|[6]]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727056.png" />. 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 [[#References|[4]]]. In order to describe the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727057.png" /> one appeals to non-associative structures, such as quasi-groups and Moufang loops.

==Cubic hypersurfaces of dimension 3.==
All smooth cubic hypersurfaces of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727058.png" /> 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 [[#References|[3]]]. This also provides a negative solution to the Lüroth problem for three-dimensional varieties. For every smooth three-dimensional cubic hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727059.png" /> there exists a principal polarized five-dimensional Abelian variety — the intermediate Jacobian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727060.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727061.png" /> it is defined as as the complex torus

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727062.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727063.png" /> is the corresponding Hodge component in the decomposition of the homology space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727064.png" />. In order to prove that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727065.png" /> is non-rational, it was shown that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727066.png" /> 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 [[#References|[5]]].

A cubic hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727067.png" /> is uniquely determined by its [[Fano surface|Fano surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727068.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727069.png" /> one has the [[Torelli theorems|Torelli theorems]] (which are also valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727070.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727071.png" /> is rational. Rationality has been proved in this case for certain hypersurfaces of a special type; for example:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727072.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1964) {{MR|0173749}} {{ZBL|0135.12101}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.W.S. Cassels, "Diophantine equations with special reference to elliptic curves" ''J. London Math. Soc.'' , '''41''' (1966) pp. 193–291 {{MR|0199150}} {{ZBL|0138.27002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> C.H. Clemens, P.A. Griffiths, "The intermediate Jacobian of the cubic threefold" ''Ann. of Math.'' , '''95''' (1972) pp. 281–356 {{MR|0302652}} {{ZBL|0245.14010}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic" , North-Holland (1986) (Translated from Russian) {{MR|0833513}} {{ZBL|0582.14010}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> 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 {{MR|0352088}} {{MR|0352089}} {{ZBL|0271.14020}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> B. Segre, "The non-singular cubic surfaces" , Clarendon Press (1942) {{MR|0008171}} {{ZBL|0061.36701}} {{ZBL|68.0358.01}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> 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 {{MR|}} {{ZBL|0263.14012}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.N. Tyurin, "The geometry of the Fano surface of a nonsingular cubic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727073.png" /> 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 {{MR|}} {{ZBL|0215.08201}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR></table>