... Irreducibility, Singularity of a h ypersurface section In this section we are interested in the intersection of a variety with a generichypersurface. We will now begin by recalling the definition ... adjoining (v). The polynomial ring in y0, ,ynover k(v)will be denoted by k(v)[y ]. The general linear transformation establishes an isomorphismbetween two polynomials rings k(v )[x]andk(v)[y]whenineverypolynomialofk(v)[y ]the substitutionyi=n3j=0vijxj,i=0, ... polynomial in the vij, and primitive in them, so that f(y0, ,yd+1)isdefined to within a factor in k(u, v). By alinear projective transformation, we can choose f(y0, ,yd+1)sothatitisregularinyd+1.The...