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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|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 form]]s 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 [[#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  $  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|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|Lüroth problem]] on surfaces over non-closed fields. There exist fields  $  k $
over which there are minimal cubic surfaces. Segre's minimality criterion [[#References|[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 [[#References|[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 [[#References|[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 [[#References|[5]]].

A cubic hypersurface  $  V $
is uniquely determined by its [[Fano surface|Fano surface]]  $  \Phi ( V) $. 
For  $  \Phi ( V) $
one has the [[Torelli theorems|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====
<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>