Choosing points on cubic plane curves: Rigidity and flexibility

Abstract

Every smooth cubic plane curve has 9 flex points and 27 sextatic points. We study the following question of 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.