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.