Section problems for configurations of points on the Riemann sphere

TL;DR

This paper investigates the conditions under which new points can be added continuously to point configurations on the Riemann sphere, providing a complete classification for all cases except when n=5.

Contribution

It offers a complete characterization of when and how points can be added continuously to configurations on the Riemann sphere, including novel constructions for specific cases.

Findings

For n ≥ 6, m must be divisible by n(n-1)(n-2).

Constructs based on cabling of braids for general n.

Exceptional constructions for n=3,4 using elliptic curves.

Abstract

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.