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Choosing points on cubic plane curves: rigidity and flexibility

Ishan Banerjee, Weiyan Chen

Abstract

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.

Choosing points on cubic plane curves: rigidity and flexibility

Abstract

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 distinct points on every smooth cubic plane curve, for each given positive integer ? We give an affirmative answer to the question when and 18 (the smallest open cases), and a negative answer for infinitely many 's.

Paper Structure

This paper contains 22 sections, 39 theorems, 105 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Multisections and virtual sections
  3. Multisections and fundamental groups
  4. Fundamental group of $\mathcal{X}$: Dolgachev-Libgober's results
  5. Subgroups and quotients of $\pi_1(\mathcal{X})$ and $\pi_1(E)$
  6. Fundamental group of multisection covers
  7. Multisections and cohomology
  8. Proofs of Theorem \ref{['9']} and \ref{['18']}
  9. A general theorem
  10. Proof of Theorem \ref{['9']}: Multisections of degree 9 are unique
  11. Proof of Theorem \ref{['18']}: Multisections of degree 18 do not exist
  12. Algebraic constructions of multisections
  13. Algebraic constructions
  14. Multisections from torsion constructions
  15. Proofs of Theorem \ref{['connected new multi']} and \ref{['topo constr']}
  16. ...and 7 more sections

Key Result

Theorem 1.2

The universal cubic plane curve $\pi$ admits a multisection of degree $n$ when and $I$ is a set of positive integers, for example, when $n= 9, 27, 36, 72, 81, 99, 108...$

Figures (4)

  • Figure 1: The figure illustrates the images of $S$ and $H$ on a single fiber of the bundle $E^{(n)}\to \mathcal{X}$ over a point $F$ when $n=2$. In this case, the fiber is $C_F\times C_F$. Since $C_F$ is 2-dimensional, $S$ can always be homotoped to an $H$ whose image has empty intersection with the diagonal $\Delta_{12}\subset C_F\times C_F$.
  • Figure 2: An illustration of the 4-step construction of the vector $v(F,x)$ out of the homotopy $s_t(F,x)$ on a fixed cubic curve $C_F$.
  • Figure 3: The figure illustrates the construction of $\tau$ when $n=2$ and $k=3$. The first picture shows a single fiber of the torus bundle $\pi:E\to\mathcal{X}$ marked with a multisection of degree $n=2$. The second picture shows a nonvanishing vector field $v$ of $\xi$. The last picture shows the new multisection $\tau$ of degree 6. Here we choose $n=2$ to simplify the pictures. In reality, $n$ must be a multiple of 9 by Theorem \ref{['old thm']}.
  • Figure 4: The figure shows the construction of $\mu$ when $n=2$. The multisection of degree $2$ in the first picture is deformed to a multisection of degree $4$ in the last picture. Here again we choose $n=2$ to simplify the pictures. In reality, $n$ must be a multiple of 9 by Theorem \ref{['old thm']}.

Theorems & Definitions (84)

  • Theorem 1.2: Maclaurin, Cayley, Gattazzo; see Section \ref{['alg cons']} for exact attributions
  • Theorem 1.3: Chen, Theorem 2 in WC
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10: Strategy and difficulties
  • Remark 1.11: Algebraic multisections
  • ...and 74 more