Section problems for configurations of points on the Riemann sphere

This paper contains 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, the results of the paper provide a complete answer to the following question: given $n \ne 5$, for which $m$ can one continuously add $m$ points to a configuration of $n$ points? For $n \ge 6$, 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.