Infinitely Near Points on Algebraic Surfaces

Abstract

1. Infinitely near points on plane algebraic curves were introduced by Max Noether [1] in the process of reduction of singularities. Noether's classical theorem is: a sequence of singular infinitely near points on a plane algebraic curve is necessarily finite. In the case of space curves and surfaces the first definitions and results were established by C. Segre [2] using quadratic transformations of the ambient space. Noether's result, though it remains valid for space curves, is no longer true for surfaces because of the appearance of singular curves. As a general rule, and even in the simplest case of plane algebraic curves, the analysis of infinitely near points is intricate and all proofs have to be carried out by handling a great number of details.2 In the case of algebraic surfaces B. Levi [3] proved the following theorem: if P, P1, P,, * is an infinite sequence of infinitely near points on an algebraic surface and all of the same multiplicity v > 1, then for any given p there exists q > p such that the point Pq lies on the transfornb of a v-fold curve passing through the point Pq-1 immediately preceding Pq. It is the purpose of this note to give a purely algebraic proof of Levi's theorem for arbitrary ground fields of characteristic zero. Our proof makes use of related results proved in the fundamental paper of Zariski [7] and it goes further by showing the existence of an index p such that for any q > p the point Pq lies on a v-fold curve.