On the geometry of generalized quadrics

Let {f0,…,fn;g0,…,gn}{f0,…,fn;g0,…,gn} be a sequence of homogeneous polynomials in 2n+22n+2 variables with no common zeros in P2n+1P2n+1 and suppose that the degrees of the polynomials are such that Q=∑i=0nfigi is a homogeneous polynomial. We shall refer to the hypersurface XX defined by QQ as a generalized quadric  . In this note, we prove that generalized quadrics in PC2n+1 for n≥1n≥1 are reduced.