Realizing tropical curves via mirror symmetry

Abstract: The tropicalization map associates to each curve in the algebraic n-torus a piecewise linear object (tropical curve) in real n-dimensional space. Given a tropical curve, a natural question is if it can arise as the tropicalization of some algebraic curve. If this is the case we say that the tropical curve is realizable. Determining good realizability criteria for tropical curves remains an important part of tropical geometry since Mikhalkin provided examples of non-realizable tropical curves. We explore the following strategy for realizing tropical curves: (1) Produce a Lagrangian submanifold of the cotangent bundle of the torus whose moment map projection approximates the tropical curve; (2) Use homological mirror symmetry to obtain a mirror algebraic sheaf; (3) Show that the tropicalization of the support of this sheaf is the original tropical curve. We will give full answers to (1) and (3), and explain why (2) is fairly subtle. As applications, we will obtain some new and known realizability statements for tropical curves.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html