A variety with a certain singular point

Let F be an irreducible variety of d dimensions in ^-dimensional affine space, defined over an algebraically closed ground field K of characteristic zero. A point of F is simple if there exists at the point a uniquely determined tangent [d] -space to V. I t is known that the ideal of non-units in the quotient ring of a point of F always has a basis of d or more elements, and that a point of V is simple if and only if the ideal of non-units in the quotient ring of the point has a basis of d elements. Let P be a point of V such that its quotient ring is integrally closed and the ideal of non-units in the quotient ring has a basis of d +1 elements. We show that we can find a point P' on a primal V, birationally equivalent to F, such that P and P' have the same quotient ring. Let (£1;...,£n) be a generic point of F, / = K[E,X,...,fn] the integral domain of V, and S = K(^lt ...,£n) the function field of F. We may choose the coordinate system (as is shown, for example, in a forthcoming book (l)) so that (i) ix, ...,£d are algebraically independent over K and £d+1, ...,£n are integrally dependent on K\gx,...,fd], (ii) £d+1 is a primitive element of K{E,X, ...,£d) over S, (iii) Pis given by p = I.(gv ...,£n) and / p . p = /p.(£1; ...,£d+i), (iv) the [n — d] space xx = 0,..., xd = 0 meets F in a finite set of points and xd+1 = 0 contains only P of this set. From the coordinate system we can show that any element w of p may be put in