The resultant R(f,g) of two polynomials f and g is an irreducible polynomial such that R(f,g) = 0 if and only if the equations f = 0 and g = 0 have one common root. When g = f′∕p, then D(f) = R(f,g) is called the discriminant of f and the discriminant hypersurface Dp = {f ∈ Cp,D(f) = 0} can be identified to the discriminant of a versal deformation of the simple hypersurface singularity Ap−1 : xp = 0. In particular, the fundamental group π = π1(Cp∖Dp) is the famous braid group and Cp∖Dp in fact a K(π,1) space. Here we prove the following. Theorem. π1(Cp+q∖Rp,q) = Z. As Cp∖Dp can be regarded as a linear section of Cp+q∖Rp,q, this theorem shows that by a nongeneric linear section the fundamental group may change drastically, in contrast with the case of generic section.