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Bitangents to symmetric quartics

Candace Bethea, Thomas Brazelton

Abstract

Recall that a non-singular planar quartic is a canonically embedded non-hyperelliptic curve of genus three. We say such a curve is symmetric if it admits non-trivial automorphisms. The classification of (necessarily finite) groups appearing as automorphism groups of non-singular curves of genus three dates back to the last decade of the 19th century. As these groups act on the quartic via projective linear transformations, they induce symmetries on the 28 bitangents. Given such an automorphism group $G=\mathrm{Aut}(C)$, we prove the $G$-orbits of the bitangents are independent of the choice of $C$, and we compute them for all twelve types of smooth symmetric planar quartic curves. We further observe that techniques deriving from equivariant homotopy theory directly reveal patterns which are not obvious from a classical moduli perspective.

Bitangents to symmetric quartics

Abstract

Recall that a non-singular planar quartic is a canonically embedded non-hyperelliptic curve of genus three. We say such a curve is symmetric if it admits non-trivial automorphisms. The classification of (necessarily finite) groups appearing as automorphism groups of non-singular curves of genus three dates back to the last decade of the 19th century. As these groups act on the quartic via projective linear transformations, they induce symmetries on the 28 bitangents. Given such an automorphism group , we prove the -orbits of the bitangents are independent of the choice of , and we compute them for all twelve types of smooth symmetric planar quartic curves. We further observe that techniques deriving from equivariant homotopy theory directly reveal patterns which are not obvious from a classical moduli perspective.

Paper Structure

This paper contains 23 sections, 9 theorems, 55 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Data availability
  4. Acknowledgements
  5. Background
  6. Quartics, the basics
  7. Automorphisms of smooth quartics
  8. The Euler number for bitangents
  9. Symmetric quartics and their bitangents
  10. Type I: Klein quartics
  11. Type II: the Dyck quartic
  12. Type III: Order 48
  13. Type IV: The pencil of octahedral quartics
  14. Type V: Order 16
  15. Type VI: Order 9
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $G$ be any group which appears as the automorphism group of a genus three non-hyperelliptic curve. Then for any such curve $C$ with $G \cong \text{Aut}(C)$, the action of $G$ on the bitangents of $C$ is independent of the choice of $C$, and is given by the following $G$-set:See section:computati

Figures (13)

  • Figure 1: The Edge quartic, defined by the equation $25(x^4 + y^4 + z^4) -34(x^2y^2 + x^2z^2 + y^2z^2)$, and its 28 real bitangents graphed with their $S_4$ orbits.
  • Figure 2: An artistic rendering of the sevenfold symmetries on the Fano plane, drawn by Burkard Polster, found on John Baez' blog Baez.
  • Figure 3: The four real bitangents graphed on the Klein quartic. All lines lie in the same orbit.
  • Figure 4: The orbits of the real bitangents on the Type IV quartic with $a=-3$. Lines in green are in the $[S_4/C_2^o]$ orbit, and lines in pink are in the $[S_4/S_3]$ orbit.
  • Figure 5: The orbits of the real bitangents on a Type V quartic. The four in green form the the orbit whose isotropy is the central $C_4^Z$. The four lines in pink are part of an orbit with $C_2^{(1)}$ isotropy.
  • ...and 8 more figures

Theorems & Definitions (35)

  • Theorem 1.1
  • Example 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Theorem 2.2
  • Example 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 3.1
  • ...and 25 more