Choosing points on cubic plane curves: rigidity and flexibility

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.