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Bitangents to plane quartics via tropical geometry: rationality, $\mathbb{A}^1$-enumeration, and real signed count

Hannah Markwig, Sam Payne, Kris Shaw

Abstract

We explore extensions of tropical methods to arithmetic enumerative problems such as $\mathbb{A}^1$-enumeration with values in the Grothendieck-Witt ring, and rationality over Henselian valued fields, using bitangents to plane quartics as a test case. We consider quartic curves over valued fields whose tropicalizations are smooth and satisfy a mild genericity condition. We then express obstructions to rationality of bitangents and their points of tangency in terms of twisting of edges of the tropicalization; the latter depends only on the tropicalization and the initial coefficients of the defining equation modulo squares. We also show that the GW-multiplicity of a tropical bitangent, i.e., the multiplicity with which its lifts contribute to the $\mathbb{A}^1$-enumeration of bitangents as defined by Larson and Vogt, can be computed from the tropicalization of the quartic together with the initial coefficients of the defining equation. As an application, we show that the four lifts of most tropical bitangent classes contribute $2\mathbb{H}$, twice the class of the hyperbolic plane, to the $\mathbb{A}^1$-enumeration. These results rely on a degeneration theorem relating the Grothendieck-Witt ring of a Henselian valued field to the Grothendieck-Witt ring of its residue field, in residue characteristic not equal to two.

Bitangents to plane quartics via tropical geometry: rationality, $\mathbb{A}^1$-enumeration, and real signed count

Abstract

We explore extensions of tropical methods to arithmetic enumerative problems such as -enumeration with values in the Grothendieck-Witt ring, and rationality over Henselian valued fields, using bitangents to plane quartics as a test case. We consider quartic curves over valued fields whose tropicalizations are smooth and satisfy a mild genericity condition. We then express obstructions to rationality of bitangents and their points of tangency in terms of twisting of edges of the tropicalization; the latter depends only on the tropicalization and the initial coefficients of the defining equation modulo squares. We also show that the GW-multiplicity of a tropical bitangent, i.e., the multiplicity with which its lifts contribute to the -enumeration of bitangents as defined by Larson and Vogt, can be computed from the tropicalization of the quartic together with the initial coefficients of the defining equation. As an application, we show that the four lifts of most tropical bitangent classes contribute , twice the class of the hyperbolic plane, to the -enumeration. These results rely on a degeneration theorem relating the Grothendieck-Witt ring of a Henselian valued field to the Grothendieck-Witt ring of its residue field, in residue characteristic not equal to two.
Paper Structure (20 sections, 37 theorems, 65 equations, 36 figures)

This paper contains 20 sections, 37 theorems, 65 equations, 36 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Field conventions
  4. Tropicalizations of plane quartics
  5. Bitangents to tropicalized quartics
  6. Lifting tropical bitangents and tropical points of tangencies
  7. Local lifting equations and their solutions
  8. Lifting conditions and twisted edges
  9. Comparing lifting for different fields
  10. The Grothendieck-Witt ring and $\mathbb{A}^1$-enumerative invariants
  11. The Grothendieck-Witt ring
  12. Grothendieck-Witt rings over valued fields
  13. The Qtype of a bitangent to a quartic
  14. The $\mathop{\mathrm{GW}}\nolimits$-multiplicity of a tropical bitangent class
  15. Comparing the $\mathrm{Qtype}_{L_{\infty}}$ for different fields
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $C = V(Q)$ be a quartic curve defined over $K$ with $\mathop{\mathrm{Trop}}\nolimits(C)$ a generic tropicalized quartic and suppose $\Lambda$ is a liftable tropical bitangent to $C$. Then whether or not $\Lambda$ lifts over $K$ is determined by $\mathop{\mathrm{Trop}}\nolimits(C)$ and the equiva

Figures (36)

  • Figure 1: On the left, the set $D$ for the polynomial $Q$ in Example \ref{['example-newtonsubd']}, on the right is the Newton subdivision $\Delta_Q$.
  • Figure 2: The tropicalization of the curve from Example \ref{['example-newtonsubd']} and the Newton subdivision $\Delta_Q$.
  • Figure 3: A local picture of a tropicalized quartic $\mathop{\mathrm{Trop}}\nolimits(C)$ together with the tropicalization of the two tangency points of a bitangent $\ell$ to $C$. Here the line $\ell$ is a lift of a tropical line $\Lambda$ whose vertex coincides with the center vertex of $\mathop{\mathrm{Trop}}\nolimits(C)$. The intersection of $\Lambda$ and $\mathop{\mathrm{Trop}}\nolimits(C)$ is a single connected component with tropical intersection multiplicity $4$. The tropical line does not appear in the figure.
  • Figure 4: Taken from CM20: Orbit representatives of all 41 shapes of bitangent classes to $\Gamma$, grouped by the dimension of its maximal cells. The numbers above each vertex indicate lifting multiplicities over the complex numbers, whereas the red ones above edges indicate slopes. The black cells of each bitangent class miss $\Gamma$, whereas the red ones lie on it. The unfilled dots are vertices of $\Gamma$.
  • Figure 5: A tropicalized quartic, with its dual Newton subdivision, and a tropical bitangent line. We can move the vertex of the tropical bitangent upwards or downwards maintaining the bitangency.
  • ...and 31 more figures

Theorems & Definitions (96)

  • Theorem 1.1: Theorem \ref{['thm:twistlift']}
  • Theorem 1.2: Theorem \ref{['thm-allornone']}
  • Theorem 1.3: Corollary \ref{['cor-totallyreal']}
  • Theorem 1.4: Theorem \ref{['thm-2H']}
  • Theorem 1.5: Theorem \ref{['thm-GWKGWk']}
  • Remark 1.6
  • Theorem 1.7: Theorems \ref{['thm-2H']} and \ref{['thm-2Hdetails']}
  • Theorem 1.8: Theorem \ref{['thm-realQtypes']}
  • Example 2.1
  • Definition 2.2
  • ...and 86 more