On the Geometry of Generalized Quadrics

Let {f0, · · · , fn; g0, · · · , gn} be a sequence of homogeneous polynomials in 2n + 2 variables with no common zeroes in P and suppose that the degrees of the polynomials are such that Q = ∑n i=0 figi 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 P C for n ≥ 1 are reduced.