Obstructions to choosing distinct points on cubic plane curves

Weiyan Chen

doi:10.1016/j.aim.2018.09.040

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arXiv · 1806.10207

Obstructions to choosing distinct points on cubic plane curves

Weiyan Chen

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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.

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Weiyan Chen. 2018-06-26. Obstructions to choosing distinct points on cubic plane curves. https://doi.org/10.1016/j.aim.2018.09.040

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