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.