Section problems for configurations of points on the riemann sphere

Abstract

We prove a suite of results concerning the problem of adding m distinct new points to a configuration of n distinct points on the Riemann sphere, such that the new points depend continuously on the old. Altogether, these results provide a complete answer to the following question: given n ≠ 5, for which m can one continuously add m points to a configuration of n points? For n > 6, we find that m must be divisible by n(n 1)(n 2), and we provide a construction based on the idea of cabling of braids. For n = 3; 4, we give some exceptional constructions based on the theory of elliptic curves.