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.

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