Seminar in Real and Complex Geometry

Thursday, April 20, 2023, 17:15-18:45, online via zoom

Hannah Markwig
(Universität Tübingen)

Counting bitangents of quartic curves - arithmetic, real, tropical


We showcase tropical geometry as a tool for geometric counting problems. A nice feature of tropical geometry is that many techniques can be applied simultaneously over various ground fields, e.g. for complex or real counting problems. Our prime example will be the count of bitangent liens to a smooth plane quartic. Already Plücker knew that a smooth complex plane quartic curve has exactly 28 bitangents. Bitangents of quartic curves are related to a variety of mathematical problems. They appear in one of Arnold's trinities, together with lines in a cubic surface and 120 tritangent planes of a sextic space curve. In this talk, we review known results about counts of bitangents under variation of the ground field. Special focus will be on counting in the tropical world, and its relations to real and arithmetic counts. We end with new results concerning the arithmetic multiplicity of tropical bitangent classes, based on joint work with Sam Payne and Kris Shaw.