Lifting tropical bitangents

Yoav Len; Hannah Markwig

doi:10.1016/j.jsc.2019.02.015

Lifting tropical bitangents

Yoav Len, Hannah Markwig

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study lifts of tropical bitangents to the tropicalization of a given complex algebraic curve together with their lifting multiplicities. Using this characterization, we show that generically all the seven bitangents of a smooth tropical plane quartic lift in sets of four to algebraic bitangents. We do this constructively, i.e. we give solutions for the initial terms of the coefficients of the bitangent lines. This is a step towards a tropical proof that a general smooth quartic admits 28 bitangent lines. The methods are also appropriate to count real bitangents, however the conditions to determine whether a tropical bitangent has real lifts are not purely combinatorial.

Original language	English
Pages (from-to)	122-152
Journal	Journal of Symbolic Computation
Volume	96
Early online date	1 Mar 2019
DOIs	https://doi.org/10.1016/j.jsc.2019.02.015
Publication status	Published - Jan 2020

Keywords

Tropical geometry
Bitangents of quartics

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10.1016/j.jsc.2019.02.015

Len-2019-Lifting-tropical-bitangents-JSC-AAM
Copyright © 2019 Elsevier Ltd. All rights reserved. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1016/j.jsc.2019.02.015
Accepted author manuscript, 595 KBLicence: CC BY-NC-ND

Cite this

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AB - We study lifts of tropical bitangents to the tropicalization of a given complex algebraic curve together with their lifting multiplicities. Using this characterization, we show that generically all the seven bitangents of a smooth tropical plane quartic lift in sets of four to algebraic bitangents. We do this constructively, i.e. we give solutions for the initial terms of the coefficients of the bitangent lines. This is a step towards a tropical proof that a general smooth quartic admits 28 bitangent lines. The methods are also appropriate to count real bitangents, however the conditions to determine whether a tropical bitangent has real lifts are not purely combinatorial.

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KW - Bitangents of quartics

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DO - 10.1016/j.jsc.2019.02.015

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VL - 96

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EP - 152

JO - Journal of Symbolic Computation

JF - Journal of Symbolic Computation

ER -

Lifting tropical bitangents

Abstract

Keywords

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