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High Quality Content by WIKIPEDIA articles! 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.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.