Complete intersection

In mathematics, an algebraic variety V in projective space is a complete intersection if the ideal of V is generated by exactly codim V elements. That is, if V has dimension m and lies in projective space Pⁿ, there should exist n − m homogeneous polynomials:^[1]

${\displaystyle F_{i}(X_{0},\cdots ,X_{n}),1\leq i\leq n-m,}$

in the homogeneous coordinates X_j, which generate all other homogeneous polynomials that vanish on V.

Geometrically, each F_i defines a hypersurface; the intersection of these hypersurfaces should be V. The intersection of n − m hypersurfaces will always have dimension at least m, assuming that the field of scalars is an algebraically closed field such as the complex numbers. The question is essentially, can we get the dimension down to m, with no extra points in the intersection? This condition is fairly hard to check as soon as the codimension n − m ≥ 2. When n − m = 1 then V is automatically a hypersurface and there is nothing to prove.