Example of A Space Curve That Is Not A Complete Intersection
The twisted cubic is a set-theoretic complete intersection but is not an ideal-theoretic complete intersection, i.e. its homogeneous ideal cannot be generated by 2 elements.
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 and the twisted cubic has degree 3, plus a line of degree 1.
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