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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" />.
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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.==
 
==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,
+
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,
  
<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>
+
$$
 +
y  ^ {2}  = \
 +
4x  ^ {3} - g _ {2} x - g _ {3} ,
 +
$$
  
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
+
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
  
<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>
+
$$
 +
 
 +
\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
 
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" />.
+
\frac{1728g _ {2}  ^ {3} }{g _ {2}  ^ {3} - 27g _ {3}  ^ {2} }
  
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>
+
takes arbitrary values in  $  k $
 +
and depends only on the curve  $  X $;  
 +
it is called the absolute invariant of  $  X $.
  
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).
+
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
  
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 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027270/c02727036.png" /> is none other than the modular invariant.
+
$$
 +
( x _ {1} , x _ {2} )  \rightarrow \
 +
x _ {0} \circ ( x _ {1} \circ x _ {2} )
 +
$$
  
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]].)
+
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.==
 
==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 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 <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.
+
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.==
 
==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
+
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
  
<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>
+
$$
 +
H  ^ {1,2} ( V, \mathbf C )/H  ^ {3} ( V, \mathbf Z ),
 +
$$
  
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]]].
+
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 <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.
+
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 <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:
+
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:
  
<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>
+
$$
 +
\sum _ {i = 0 } ^ { {2m }  + 1 }
 +
a _ {i} x _ {i}  ^ {3}  = 0,\ \
 +
m \geq  2.
 +
$$
  
 
====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Revision as of 17:31, 5 June 2020


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

[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

Comments

References

[a1] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
How to Cite This Entry:
Cubic hypersurface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cubic_hypersurface&oldid=46563
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article