Complete intersection

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 polynomials 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

F_i(X₀, ..., X_n)

in the homogeneous coordinates X_j, with

1 ≤ i ≤ n − m,

such that on V we have

F_i(X₀, ..., X_n) = 0

and for no other points of projective space do all the F_i all take the value 0. Geometrically each F_i separately define a hypersurface H_i; the intersection of the H_i 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 numbers. 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 cubic 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 quadrics, 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 $2\times 2 = 4,$ and the twisted cubic has degree 3, plus a line of degree 1.

Multidegree

A complete intersection has a multidegree, written as the tuple (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.

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 transversality condition (like their tangent spaces being in general position at intersection points). The intersection may be scheme-theoretic, in other words here the homogeneous ideal generated by the F_i(X₀, ..., X_n) 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.

A Connection to number theory

Andrew Wiles proved Fermat's last theorem by proving that a certain Hecke algebra is a complete intersection.

References

• E. J. N. Looijenga (1984), Isolated Singular Points on Complete Intersections