Deformations of bitangent classes of tropical quartic curves

Abstract: Over an algebraically closed field a smooth quartic curve has 28 bitangent lines. Plücker proved that over the real numbers we have either 4, 8, 16 or 28 real bitangents to a real quartic curve. A tropical smooth quartic curve has exactly 7 bitangent classes which each lift either 0 or 4 times over the real numbers. The shapes of these bitangent classes have been classified by Markwig and Cueto in 2020, who also determined their real lifting conditions. However, for a fixed unimodular triangulation different choices of coefficients imply different edge lengths of the quartic and these can change the shape of the 7 bitangent classes and might therefore influence their real lifting conditions. In order to prove Plückers Theorem about the number of real bitangents tropically, we have to study these deformations of the bitangent shapes. In a joint work with Marta Panizzut we develope a polymake extension, which computes the tropical bitangents. For this we determine two refinements of the secondary fan: one for which the bitangent shapes in each cone stay constant and one for which the lifting conditions in each cone stay constant. This is still work in progress, but there will be a small software demonstration.

algebraic geometrycombinatorics

Audience: researchers in the topic

Tropical Geometry in Frankfurt/Zoom TGiF/Z

Series comments: Description: An afternoon seminar series on tropical geometry, known as the TGiZ ("Tropical Geometry in Zoom") or the TGiF ("Tropical Geometry in Frankfurt") seminar.

Please send an email to one of the organizers at the latest one day before a session if you wish to receive the Zoom link and password.

Videos of some past talks are available on YouTube here, while some slides can be found here.