It is a classical topic to study structures of certain special points on complex smooth cubic plane curves, for example, the 9 flex points and the 27 sextactic points. We consider the following topological 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? This work is joint with Ishan Banerjee.
