Tropical K_4 curves

Algebra Seminar
Wednesday, September 24, 2014 - 15:05
1 hour (actually 50 minutes)
Skiles 006
Harvard University
This is joint work with Pakwut Jiradilok.  Let X be a smooth, proper curve of genus 3 over a complete and algebraically closed nonarchimedean field.  We say X is a K_4-curve if the nonarchimedean skeleton G of X is a metric K_4, i.e. a complete graph on 4 vertices.We prove that X is a K_4-curve if and only if X has an embedding in p^2 whose tropicalization has a strong deformation retract to a metric K_4. We then use such an embedding to show that the 28 odd theta characteristics of X are sent to the seven odd theta characteristics of g in seven groups of four.  We give an example of the 28 bitangents of a honeycomb plane quartic, computed over the field C{{t}}, which shows that in general the 4 bitangents in a given group need not have the same tropicalizations.