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Tropical methods for building real space sextics with totally real tritangent planes

Maria Angelica Cueto, Yoav Len, Hannah Markwig, Yue Ren

TL;DR

The paper develops a comprehensive tropical framework to construct the classical 120 tritangent planes to a smooth space sextic, by lifting tropical tritangents of a tropical (3,3) curve on the Segre quadric. It identifies 15 tropical tritangent classes, each containing eight classical lifts, and proves that lift fields are quadratic over the base field; in the real setting, liftings come in eight-real-blocks tied to avoidance loci. The authors provide two explicit totally real realizations (64 and 120) via sign distributions and tropical refinements, and supply detailed local-lifting analyses (via multivariate Hensel’s lemma) for all tangency types, including higher multiplicities and 4-valent cases. They also develop constructive methods to pick tropical-modification parameters and to assemble full tritangent tuples, and study arithmetic aspects (quadratic extensions, comparisons of residue fields) as well as real-lift phenomena using avoidance loci. Altogether, the work bridges tropical, real-algebraic, and arithmetic geometry to realize maximal real counts and offers practical recipes for generating explicit totally real tritangents.

Abstract

This paper proposes the use of combinatorial techniques from tropical geometry to build the 120 tritangent planes to a given smooth algebraic space sextic. Although the tropical count is infinite, tropical tritangents come in 15 equivalence classes, each containing the tropicalization of exactly eight classical tritangents. Under mild genericity conditions on the tropical side, we show that liftings of tropical tritangents are defined over quadratic extensions of the ground field over which the input sextic curve is defined. When the input curve is real, we prove that every complex liftable member of a given tropical tritangent class either completely lifts to the reals or none of its liftings are defined over the reals. As our main application we use these methods to build examples of real space sextics with 64 and 120 totally real tritangents, respectively. The paper concludes with a discussion of our results in the arithmetic setting.

Tropical methods for building real space sextics with totally real tritangent planes

TL;DR

The paper develops a comprehensive tropical framework to construct the classical 120 tritangent planes to a smooth space sextic, by lifting tropical tritangents of a tropical (3,3) curve on the Segre quadric. It identifies 15 tropical tritangent classes, each containing eight classical lifts, and proves that lift fields are quadratic over the base field; in the real setting, liftings come in eight-real-blocks tied to avoidance loci. The authors provide two explicit totally real realizations (64 and 120) via sign distributions and tropical refinements, and supply detailed local-lifting analyses (via multivariate Hensel’s lemma) for all tangency types, including higher multiplicities and 4-valent cases. They also develop constructive methods to pick tropical-modification parameters and to assemble full tritangent tuples, and study arithmetic aspects (quadratic extensions, comparisons of residue fields) as well as real-lift phenomena using avoidance loci. Altogether, the work bridges tropical, real-algebraic, and arithmetic geometry to realize maximal real counts and offers practical recipes for generating explicit totally real tritangents.

Abstract

This paper proposes the use of combinatorial techniques from tropical geometry to build the 120 tritangent planes to a given smooth algebraic space sextic. Although the tropical count is infinite, tropical tritangents come in 15 equivalence classes, each containing the tropicalization of exactly eight classical tritangents. Under mild genericity conditions on the tropical side, we show that liftings of tropical tritangents are defined over quadratic extensions of the ground field over which the input sextic curve is defined. When the input curve is real, we prove that every complex liftable member of a given tropical tritangent class either completely lifts to the reals or none of its liftings are defined over the reals. As our main application we use these methods to build examples of real space sextics with 64 and 120 totally real tritangents, respectively. The paper concludes with a discussion of our results in the arithmetic setting.
Paper Structure (16 sections, 78 theorems, 205 equations, 21 figures, 12 tables)

This paper contains 16 sections, 78 theorems, 205 equations, 21 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Preliminaries: tropical tritangent curves and lifting methods
  3. Combinatorics of local tropical tangencies
  4. Local lifting of multiplicity two tangencies: the trivalent case
  5. Tropical hyperflexes
  6. Type (3) tangencies for trivalent tritangents
  7. Local lifting multiplicities for tropical 4-valent tritangents
  8. Tangencies of multiplicity six
  9. How to pick the parameters $m_1, n_1$ and $d_1$
  10. Finding the tuple $(\ell,p,p',p")$
  11. Proof of Theorems \ref{['thm:main2']} and \ref{['thm:main3']}
  12. Avoidance loci and real lifts of tropical tritangents
  13. Constructing totally real tritangents
  14. Arithmetic considerations
  15. Type (3) tangencies for tropical lines in $\mathbb{T}\mathbb{P}^2$
  16. ...and 1 more sections

Key Result

theorem 1

Up to $D_{4}$-symmetry, there are 38 possible local tangencies between a $(1,1)$-tropical curve $\Lambda$ and a smooth $(3,3)$-tropical curve $\Gamma$ in $\mathbb{T}\mathbb{P}^1\times \mathbb{T}\mathbb{P}^1$. They are depicted in fig:classificationLocalTangencies.

Figures (21)

  • Figure 1: Tropical $(1,1)$-curves with the corresponding Newton subdivisions of $\ell$.
  • Figure 2: Representatives of all possible local tangency types between a $(1,1)$-tropical curve $\Lambda$ (in dashed lines) and a smooth $(3,3)$-tropical curve $\Gamma$, under the action of $D_{4}$. The last row corresponds to the cases when $\Gamma$ is $4$-valent. For types (1), (2), (4) (5) and (6), we use the labels 'a' and 'b' to distinguish between tangencies of multiplicity two and four, respectively. The black dots show the location of the tropical tangency point. The numbers adjacent to each dot indicate possible tangency multiplicities. Unfilled dots in (3f) and (8) denote potential tangencies, and their total multiplicities are four and six, respectively.
  • Figure 3: Representatives of potential local tangency types between $\Lambda$ and $\Gamma$ that cannot occur for bidegree reasons.
  • Figure 4: From left to right and top to bottom: skeleton of $\Gamma$ with a type (8) tangency and its relevant edge lengths, and four tropical regular functions on it leading to the three possibilities for the distribution of the tangency points under the generic assumption that $L_1<L_2, L_4, L_5$ and that $\min\{L_3, L_5-L_1, L_4-L_1\}$ is attained exactly once. Planar coordinates for the 15 points $p_1, \ldots, p_{15}$ can be found in \ref{['eq:p1Top6']} and \ref{['eq:p7Top15']}.
  • Figure 5: From top to bottom: Relevant cells in the Newton subdivision of $f$ and notation for the corresponding coefficients for all type (2) tangencies of multiplicity two that occur for $(3,3)$-curves in $\mathbb{P}^1\times \mathbb{P}^1$, up to $D_{4}$-symmetry. We group them in pairs if they are connected by an edge dual to the one carrying the tangency. The first row corresponds to horizontal tangencies, whereas the second row marks tangencies occurring along an edge of $\Lambda$ with slope one. The marked lattice point on each case determines the polytope.
  • ...and 16 more figures

Theorems & Definitions (173)

  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • definition 5
  • definition 6
  • definition 8
  • definition 9
  • theorem 10
  • definition 11
  • ...and 163 more