Hypersurface singularities, codimension two complete intersections and tangency sets (Q1104374)

The authors investigate the closed subvarieties of projective n-space over the field \({\mathbb{C}}\) which are not hypersurfaces but whose homogeneous ideal is generated by two polynomials P and Q. More precisely they consider the hypersurface singularity in affine \((n+1)\)-space defined by \(P+Q=0\) and give conditions (necessary and sufficient), in terms of the variety defined by P and Q, in order that the hypersurface has an isolated singularity at the origin. An application concerns the set of points where two complete intersections V and W in projective space have the same tangent space.