Bi-annual Algebraic and Tropical Meetings of
Brown and YaLE

Fall 2020 @ Brown (virtually)
November 9, 2020

The BATMOBYLE is a vehicle for bringing together the algebraic and tropical geometry groups of Brown and neighboring institutions for a biannual day of talks.

Hannah Markwig

Hannah Markwig

Counting bitangents of plane quartics - tropical, real and arithmetic

A smooth plane quartic defined over the complex numbers has precisely 28 bitangents. This result goes back to Plücker. In the tropical world, the situation is different. One can define equivalence classes of tropical bitangents of which there are seven, and each has 4 lifts over the complex numbers. Over the reals, we can have 4, 8, 16 or 28 bitangents. The avoidance locus of a real quartic is the set in the dual plane consisting of all lines which do not meet the quartic. Every connected component of the avoidance locus has precisely 4 bitangents in its closure. For any field k of characteristic not equal to 2 and with a non-Archimedean valuation which allows us to tropicalize, we show that a tropical bitangent class of a quartic either has 0 or 4 lifts over k. This way of grouping into sets of 4 which exists tropically and over the reals is intimately connected: roughly, tropical bitangent classes can be viewed as tropicalizations of closures of connected components of the avoidance locus. Arithmetic counts offer a bridge connecting real and complex counts, and we investigate how tropical geometry can be used to study this bridge.

This talk is based on joint work with Maria Angelica Cueto, and on joint work in progress with Sam Payne and Kristin Shaw.

Printable BATMOBYLE poster PDF or HI-DEF PNG

To access the talks, use this Zoom link.
The password is the first name of the speakers.
All times listed in EST.

09:30 - 10:45 Hannah Markwig (Tübingen)
Counting bitangents of plane quartics - tropical, real, and arithmetic
10:45 - 11:15 Break
11:15 - 12:15 Hannah Larson (Stanford)
Brill-Noether theory over the Hurwitz space
12:15 - 12:45 Break

Dan Abramovich (Brown), Kenny Ascher (MIT), Asher Auel (Dartmouth), Dori Bejleri (Harvard), Madeline Brandt (Brown), Melody Chan (Brown), Yuchen Liu (Yale), Kalina Mincheva (Tulane), Nathan Pflueger (Amherst), and Jeremy Usatine (Brown).

Previous BATMOBYLE models:
Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017, Fall 2017, Spring 2018, Fall 2018, Spring 2019, and Fall 2019.