# Math 863: Introduction to Tropical Geometry

**Math 863 Section 001, Spring 2020**

**Lectures** Tuesdays and Thursdays 1:00 - 2:15 pm, Van Vleck B131.

**Instructor **Daniel Corey, Van Vleck 321, dcorey[at]math.wisc.edu

**Office Hours **Mondays and Tuesdays, 11:00 am - noon, Van Vleck 321 (or by appointment).

**Course Overview **This class is an introduction to tropical geometry. We will focus on the case of tropical curves, and include some background on algebraic curves. Topics include: divisors and Riemann-Roch on (metric) graphs, classification of special divisors on a generic chain of loops, Jacobians of (metric) graphs, specialization of divisors from curves to graphs, analytification and skeleta of curves, tropical proofs of the Brill-Noether and Gieseker-Petri theorems.

**Prerequisites** Logistically: Graduate/professional standing or member of the Pre-Masters Mathematics (Visiting International) Program. Mathematically: familiarity with basic notions of graph theory and algebraic geometry at the level of schemes. *No special knowledge of algebraic curves will be assumed. *

**Schedule **Here is a schedule of topics to be covered. This will likely change given the pace of the course.

**Jan 21, 23:**Introduction to the class, divisors on finite graphs.**Jan 28, 30:**Rank, v-reduced divisors, Dhar's burning algorithm.**Feb 4, 6:**Riemann-Roch, Rank-determining sets.**Feb 11, 13:**More rank-determining sets, divisors on a generic chain of loops.**Feb 18, 20:**More divisors on a generic chain of loops.**Feb 25, 27:**Divisors on metric graphs.**March 3, 5:**Riemann-Roch and rank determining sets for metric graphs, Jacobians of metric graphs.**March 10, 12:**Tropical Abel-Jacobi theorem and Abel-Jacobi map.**March 17, 19: Spring Recess.****March 24, 26:**Divisors on algebraic curves, and Riemann-Roch. Worksheet.**March 31, April 2:**Baker's specialization lemma. Worksheet.**April 7, 9:**Tropical proof of the Brill-Noether theorem. Worksheet.**April 14, 16:**Berkovich analytification of A^1. Worksheet.**April 21, 23:**Analytification and skeleta of curves. Worksheet.**April 28, 30:**Reserved for term paper.

**References **There will be no required textbook. We will largely follow the notes of a course led by David Jensen and Sam Payne at Yale University in the Spring of 2016:

http://web.ma.utexas.edu/users/sampayne/Math665.html

A nice overview of the subject matter is contained in the following expository paper:

M. Baker, D. Jensen. *Degeneration of Linear Series From the Tropical Point of View and Applications* (arXiv, chapter).

The source material is, for the most part, contained in the following papers.

- M. Baker.
*An introduction to Berkovich analytic spaces and non-Archimedean potential theory on curves.*(link). - M. Baker.
*Specialization of linear systems from curves to graphs.*(arXiv, journal). - M. Baker, X. Faber.
*Metric properties of the tropical Abel–Jacobi map.*(arXiv, journal). - M. Baker, S. Norine.
*Riemann–Roch and Abel–Jacobi theory on a finite graph.*(arXiv, journal). - M. Baker, S. Payne, J. Rabinoff.
*Nonarchimedean geometry, tropicalization, and metrics on curves.*(arXiv, journal). - F. Cools, J. Draisma, S. Payne, and E. Robeva.
*A tropical proof of the Brill-Noether Theorem.*(arXiv, journal) - J. Hladký, D. Král, S. Norine.
*Rank of divisors on tropical curves.*(arXiv, journal). - D. Jensen, S. Payne.
*Tropical independence I: Shapes of divisors and a proof of the Gieseker-Petri theorem.*(arXiv, journal). - Y. Luo.
*Rank-determining sets of metric graphs.*(arXiv, journal).

While familiarity with algebraic geometry is assumed, all relevant concepts will be reviewed in the course. Many of the facts from AG that will be cited may be found in the following texts.

- E. Arbarello, M. Cornalba, P. Griffiths, J. Harris,
*Geometry of Algebraic Curves.* - Q. Liu
*Algebraic Geometry and Arithmetic Curves.*

Here are some interesting chip-firing resources:

- https://markusfeng.com/projects/graph/ This is a useful tool (by Markus Feng) that allows the user to draw graphs, assign chips, and play the chip-firing game.
- https://thedollargame.io/ This is a game where the graph and chips are given, and you must clear all debt via chip-firing.

**Homework**

There will be a few problem sets for this course (~every 3 weeks). A number of the exercises are borrowed from Yoav Len and Dhruv Ranganathan from the MSRI Graduate Summer School: Chip-firing and tropical curves.

Homework 1: Due Feb 18.

Homework 2: Due March 24.

**Final Project (Due date: April 30)**

All students will choose a topic related to tropical geometry (loosely interpreted) and submit a 1-2 page summary. For instance, this could be an interesting example/computation worked out in detail, a theorem with the ideas behind its significance and proof, or a broad overview of a different branch of tropical geometry. Each student must choose a topic and notify me by email no later than **March 31, 2020**. Here is a list of suggested topics, but if you have something else in mind please feel free to write about that.

**Modification due to COVID-19**

Due to the COVID-19 pandemic, there is no face-to-face instruction starting March 23. Here is how remote instruction will run. Each Thursday, I will assign readings/supplementary material along with comprehension questions. For each reading, I ask students to send me questions about the material due by the following Thursday via email. I will create responses and send them on Tuesdays (I'll keep the asker's anonymous). Office hours will all be by appointment, just send me an email and we can do a google meets. Students may pose questions to each other via Piazza. However, the best way to contact me is by email.