Complete intersection
Encyclopedia
In mathematics
, an algebraic variety
V in projective space
is a complete intersection if it can be defined by the vanishing of the number of homogeneous polynomial
s indicated by its codimension
. That is, if the dimension of an algebraic variety
V is m and it lies in projective space Pn, there are homogeneous polynomials
in the homogeneous coordinates
Xj, with
such that on V we have
and for no other points of projective space do all the Fi all take the value 0. Geometrically each Fi separately define a hypersurface
Hi; the intersection of the Hi should be V, no more and no less.
In fact the dimension of the intersection will always be at least m, assuming as usual in algebraic geometry
that the scalars form an algebraically closed field
, such as the complex number
s. There will be hypersurfaces containing V, and any set of them will have intersection containing V. The question is then, can n − m be chosen to have no further intersection? This condition is in fact hard to satisfy, as soon as n ≥ 3 and n − m ≥ 2. When the codimension n − m = 1 then automatically V is a hypersurface and there is nothing to prove.
in P3. It is not a complete intersection: in fact its degree
is 3, so it would have to be the intersection of two surfaces of degrees 1 and 3, by the hypersurface Bézout theorem. In other words, it would have to be the intersection of a plane and a cubic surface
. But by direct calculation, any four distinct points on the curve are not coplanar, so this rules out the only case. The twisted cubic lies on many quadric
s, but the intersection of any two of these quadrics will always contain the curve plus an extra line, since the intersection of two quadrics has degree
and the twisted cubic has degree 3, plus a line of degree 1.
(properly though a multiset
) of the degrees of defining hypersurfaces. For example taking quadrics in P3 again, (2,2) is the multidegree of the complete intersection of two of them, which when they are in general position
is an elliptic curve
. The Hodge numbers of complex smooth complete intersections were worked out by Kunihiko Kodaira.
condition (like their tangent space
s being in general position at intersection points). The intersection may be scheme-theoretic
, in other words here the homogeneous ideal generated by the Fi(X0, ..., Xn) may be required to be the defining ideal of V, and not just have the correct radical
. In commutative algebra
, the complete intersection condition is translated into regular sequence
terms, allowing the definition of local complete intersection, or after some localization
an ideal has defining regular sequences.
proved Fermat's last theorem
by proving that a certain Hecke algebra
is a complete intersection.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V in projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
is a complete intersection if it can be defined by the vanishing of the number of homogeneous polynomial
Homogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...
s indicated by its codimension
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
. That is, if the dimension of an algebraic variety
Dimension of an algebraic variety
In mathematics, the dimension of an algebraic variety V in algebraic geometry is defined, informally speaking, as the number of independent rational functions that exist on V.For example, an algebraic curve has by definition dimension 1...
V is m and it lies in projective space Pn, there are homogeneous polynomials
- Fi(X0, ..., Xn)
in the homogeneous coordinates
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
Xj, with
- 1 ≤ i ≤ n − m,
such that on V we have
- Fi(X0, ..., Xn) = 0
and for no other points of projective space do all the Fi all take the value 0. Geometrically each Fi separately define a hypersurface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
Hi; the intersection of the Hi should be V, no more and no less.
In fact the dimension of the intersection will always be at least m, assuming as usual in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
that the scalars form an algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
, such as the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s. There will be hypersurfaces containing V, and any set of them will have intersection containing V. The question is then, can n − m be chosen to have no further intersection? This condition is in fact hard to satisfy, as soon as n ≥ 3 and n − m ≥ 2. When the codimension n − m = 1 then automatically V is a hypersurface and there is nothing to prove.
Example of a space curve that is not a complete intersection
The classical case is the twisted cubicTwisted cubic
In mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation...
in P3. It is not a complete intersection: in fact its degree
Degree of an algebraic variety
The degree of an algebraic variety in mathematics is defined, for a projective variety V, by an elementary use of intersection theory.For V embedded in a projective space Pn and defined over some algebraically closed field K, the degree d of V is the number of points of intersection of V, defined...
is 3, so it would have to be the intersection of two surfaces of degrees 1 and 3, by the hypersurface Bézout theorem. In other words, it would have to be the intersection of a plane and a cubic surface
Cubic surface
A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single polynomial which is homogeneous of degree 3...
. But by direct calculation, any four distinct points on the curve are not coplanar, so this rules out the only case. The twisted cubic lies on many quadric
Quadric
In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
s, but the intersection of any two of these quadrics will always contain the curve plus an extra line, since the intersection of two quadrics has degree
and the twisted cubic has degree 3, plus a line of degree 1.Multidegree
A complete intersection has a multidegree, written as the tupleTuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...
(properly though a multiset
Multiset
In mathematics, the notion of multiset is a generalization of the notion of set in which members are allowed to appear more than once...
) of the degrees of defining hypersurfaces. For example taking quadrics in P3 again, (2,2) is the multidegree of the complete intersection of two of them, which when they are in general position
General position
In algebraic geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible...
is an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
. The Hodge numbers of complex smooth complete intersections were worked out by Kunihiko Kodaira.
General position
For more refined questions, the nature of the intersection has to be addressed more closely. The hypersurfaces may be required to satisfy a transversalityTransversality
In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the idea of a generic intersection in differential topology...
condition (like their tangent space
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
s being in general position at intersection points). The intersection may be scheme-theoretic
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
, in other words here the homogeneous ideal generated by the Fi(X0, ..., Xn) may be required to be the defining ideal of V, and not just have the correct radical
Radical of an ideal
In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I. A radical ideal is an ideal that is its own radical...
. In commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, the complete intersection condition is translated into regular sequence
Regular sequence (algebra)
In commutative algebra, if R is a commutative ring and M an R-module, a nonzero element r in R is called M-regular if r is not a zerodivisor on M, and M/rM is nonzero...
terms, allowing the definition of local complete intersection, or after some localization
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
an ideal has defining regular sequences.
A Connection to number theory
Andrew WilesAndrew Wiles
Sir Andrew John Wiles KBE FRS is a British mathematician and a Royal Society Research Professor at Oxford University, specializing in number theory...
proved Fermat's last theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
by proving that a certain Hecke algebra
Hecke algebra
In mathematics, the Iwahori–Hecke algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a one-parameter deformation of the group algebra of a Coxeter group....
is a complete intersection.