On the equations defining tangent cones

This paper treats the local study of singularities by means of their tangent cones, more specifically the study of graded rings associated to an ideal of a local ring. We recall some basic facts: let ( R , ) be a local ring, I, J ideals of R , such that J ⊆ I ; then G R / J ( I / J ), the graded ring associated to I / J , is canonically isomorphic to the quotient of G R ( I ) modulo a homogeneous ideal, which is called J *, and which is generated by the so-called ‘initial forms’ of the elements of J . Let us consider the following example: Let k be a field, R = k [ X, Y, Z ]( x, y, Z ), I = ( X, Y, Z ) R, J the prime ideal generated by f l , f 2 where f 1 = Y 3 − Z 2 , f 2 = YZ − X 4 . Then and it is easily seen that J * properly contains the ideal generated by the initial forms f * 1 f * 2 of f 1 , f 2 ; namely f * 1 = − Z 2 , f * 2 = YZ and ( Yf 1 + Zf 2 )* = Y 4 ∉ (− Z 2 , YZ ).