Computing tropical bitangents to smooth quartic curves in polymake

Geiger, Alheydis; Panizzut, Marta

doi:10.48550/arxiv.2112.04447

Cited by 1 publication

(4 citation statements)

References 8 publications

(14 reference statements)

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“…Compared to [24], we require an additional genericity assumption on our tropicalized quartics: we subdivide the cone in the secondary fan corresponding to the unimodular triangulation according to the types of tropical bitangent classes that can occur and require our tropicalized quartic to correspond to a point in the interior of a cone in this subdivision. This subdivision of the secondary fan is computed in [11]. Put differently, we require the edge lengths of Trop(C) to be generic in the sense that no unexpected alignment of vertices happens, see Figure 6.…”

Section: Tropicalizations Of Plane Quarticsmentioning

confidence: 99%

“…Here we can make use of the Legendre symbols. This is particularly useful, because lifting over the real numbers can be checked computationally using the polymake-extension of Geiger and Panizzut [10,11,12]. Corollary 3.17.…”

Section: Using Theorem 33 We Can Relate the Lifts Over Different Fiel...mentioning

confidence: 99%

“…using the software system Polymake [10] or the libraby tropical.lib in the Computeralgebrasystem Singular [15,7]. • Using the Polymake-extension on tropical bitangents of quartics by Geiger-Panizzut [11], compute all its tropical bitangent classes. • For each liftable tropical bitangent, use Theorem 4.13 to compute its Qtype L∞ .…”

Section: 3mentioning

confidence: 99%

“…bitangent shapes which appear in the open cones of the subdivided secondary fan, where the subdivision is chosen such that the bitangent shapes are constant in each cell (see Remark 2.17). This subdivision has been computed in [11].…”

Section: Appendix a Classification Computations And Tablementioning

confidence: 99%

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

Markwig¹,

Payne²,

Shaw³

2022

Preprint

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We explore extensions of tropical methods to arithmetic enumerative problems such as 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 A 1 -enumeration of bitangents as defined by Larson and Vogt [21], 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 2H, twice the class of the hyperbolic plane, to the A 1 -enumeration, providing evidence for a conjecture of Larson and Vogt on the A 1 -enumeration of bitangents over the reals. 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.

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Section: Tropicalizations Of Plane Quarticsmentioning

confidence: 99%

Section: Using Theorem 33 We Can Relate the Lifts Over Different Fiel...mentioning

confidence: 99%