Choosing points on cubic plane curves: rigidity and flexibility

TL;DR

This paper investigates the continuous selection of points on smooth cubic plane curves, confirming known algebraic structures suffice for certain cases and demonstrating limitations for others.

Contribution

It provides an affirmative answer for choosing 9 and 18 points, and a negative answer for infinitely many other values of n.

Findings

Confirmed algebraic structures suffice for n=9 and 18

Proved limitations for continuous point selection for infinitely many n

Addressed an open question posed by Farb

Abstract

Every smooth cubic plane curve has 9 flex points and 27 sextatic points. We study the following question asked by Farb: Is it true that the known algebraic structures give all the possible ways to continuously choose $n$ distinct points on every smooth cubic plane curve, for each given positive integer $n$? We give an affirmative answer to the question when $n=9$ and 18 (the smallest open cases), and a negative answer for infinitely many $n$'s.