On the geometry of generalized quadrics

Let {f0,…,fn;g0,…,gn} be a sequence of homogeneous polynomials in 2n+2 variables with no common zeros in P2n+1and suppose that the degrees of the polynomials are such that Q=Σni=0figi is a homogeneous polynomial. We shall refer to the hypersurface X defined by Q as a generalized quadric. In this note, we prove that generalized quadrics in PC2n+1 for n≥1 are reduced