Obstructions to choosing distinct points on cubic plane curves

TL;DR

This paper investigates the topological problem of continuously selecting multiple points on smooth cubic plane curves, proving it is impossible unless the number of points is a multiple of 9, thereby resolving a conjecture by Benson Farb.

Contribution

It establishes a topological obstruction to continuous point selection on cubic curves, confirming a conjecture and linking classical algebraic geometry with fiber bundle theory.

Findings

No continuous section exists unless n is a multiple of 9.

Resolves Benson Farb's conjecture on point selection.

Connects algebraic geometry with topological fiber bundle properties.

Abstract

Every smooth cubic plane curve has 9 inflection points, 27 sextatic points, and 72 ``points of type nine". Motivated by these classical algebro-geometric constructions, we study the following topological question: Is it possible to continuously choose $n$ distinct unordered points on each smooth cubic plane curve for a natural number $n$? This question is equivalent to asking if certain fiber bundle admits a continuous section or not. We prove that the answer is no when $n$ is not a multiple of 9. Our result resolves a conjecture of Benson Farb.