By RADEMACHER

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Thus such points exist if and only if e1 = 3. CHAPTER 4 Singularities of multiplicity equal to degree divided by two Throughout this section B is the polynomial ring k [x, y], where k is a ﬁeld, and d = 2c is an even integer. 8 completely classiﬁes the parameterizations of rational plane curves C of degree d as a function of the conﬁguration of multiplicity c singularities which appear on, or inﬁnitely near, C. 8 is typical of classiﬁcation theorems in general in the sense that a classiﬁcation theorem is always the culmination of one project and is often the starting point of new projects.

In the present chapter we describe how one can read information about the inﬁnitely near singularities of C from the Hilbert-Burch matrix. 1 deals with inﬁnitely near singularities in the ﬁrst neighborhood of C. 5 is mainly concerned with inﬁnitely near singularities in the second neighborhood of C. 5 to classify singularities of multiplicity three (triple points) on rational plane curves of degree six and for that reason we call this result the Triple Lemma. 8. 1 with n = 3, and k an algebraically closed ﬁeld.

These results make sense in BalHd . (5) The action of G on Hd restricts to given actions of G on BalHd and also on BHd . 9. (6) The well known formula ξ −1 = (det ξ)−1 Adj ξ, where Adj ξ is the classical adjoint of ξ, expresses the inverse of the matrix as a rational function in the entries of ξ. Thus, the function Υ : G × Hd → Hd , which is deﬁned by Υ(g, ϕ) = gϕ, is a morphism of varieties. There are 7 conﬁgurations of multiplicity c singularities in CP, but 11 disjoint orbits in our decomposition of BalHd .