On the geometry of generalized quadrics

Abstract

Let {f₀,…,f_n;g₀,…,g_n} 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=Σⁿ_i=0f_ig_i 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_C²ⁿ⁺¹ for n≥1 are reduced.