In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a cubic form is a
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables ...
of degree 3, and a cubic hypersurface is the
zero set
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or eq ...
of a cubic form. In the case of a cubic form in three variables, the zero set is a
cubic plane curve
In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation
:
applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an ...
.
In ,
Boris Delone and
Dmitry Faddeev showed that binary cubic forms with integer coefficients can be used to parametrize
orders
Order, ORDER or Orders may refer to:
* A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* H ...
in
cubic field In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of Degree of a number field, degree three.
Definition
If ''K'' is a field extension of the rational numbers Q of Degree of a field extensio ...
s. Their work was generalized in to include all cubic rings (a is a
ring
(The) Ring(s) may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
Arts, entertainment, and media Film and TV
* ''The Ring'' (franchise), a ...
that is isomorphic to Z
3 as a
Z-module),
[In fact, ]Pierre Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoor ...
pointed out that the correspondence works over an arbitrary scheme. giving a
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the zero of a function, roots without computing them. More precisely, it is a polynomial function of the coef ...
-preserving
bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
between
orbits
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificia ...
of a GL(2, Z)-
action on the space of integral binary cubic forms and cubic rings up to
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
.
The classification of real cubic forms
is linked to the classification of
umbilical point
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are e ...
s of surfaces. The
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es of such cubics form a three-dimensional
real projective space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in the real space It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.
Basic properti ...
and the subset of
parabolic forms define a surface – the
umbilic torus
The umbilic torus or umbilic bracelet is a single-edged 3-dimensional shape. The lone edge goes three times around the ring before returning to the starting point. The shape also has a single external face. A cross section (geometry), cross sectio ...
.
Examples
*
Cubic plane curve
In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation
:
applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an ...
*
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...
*
Fermat cubic
In geometry, the Fermat cubic, named after Pierre de Fermat, is a surface defined by
: x^3 + y^3 + z^3 = 1. \
Methods of algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly fr ...
*
Cubic 3-fold In algebraic geometry, a cubic threefold is a hypersurface of degree 3 in 4-dimensional projective space. Cubic threefolds are all unirational, but used intermediate Jacobians to show that non-singular cubic threefolds are not rational. The s ...
*
Koras–Russell cubic threefold In algebraic geometry, the Koras–Russell cubic threefolds are smooth affine complex threefolds diffeomorphic to \mathbf^3studied by . They have a hyperbolic action of a one-dimensional torus \mathbf^*with a unique fixed point, such that the q ...
*
Klein cubic threefold In algebraic geometry, the Klein cubic threefold is the non-singular cubic threefold in 4-dimensional projective space given by the equation
:V^2W+W^2X+X^2Y+Y^2Z+Z^2V =0 \,
studied by .
Its automorphism group is the group PSL2(11) of order 660 . ...
*
Segre cubic In algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 (or sometimes 5) dimensional projective space, studied by .
Definition
The Segre cubic is the set of points (''x''0:''x''1:''x''2:''x''3:''x''4:''x''5) of ''P''5 satisfying ...
Notes
References
*
*
*
*
*
*
Multilinear algebra
Algebraic geometry
Algebraic varieties
{{algebraic-geometry-stub