Professor - University of Utah

MATH 4800 - REU - SPRING 2015

Introduction to Tropical Geometry

Meeting: TH 09:10-10:30 - JTB 110

Office Hours: By appointment

Instructor: Tommaso de Fernex

Contact: Office JWB 322, email

Course Webpage:

Prerequisites: Enrollment in this class is by permission of the instructor only. While there are no fixed prerequisites, given the advanced topic of the course it is recommended to have a some familiarity with the basic notions in Algebra and Topology (rings, fields, polynomials, power series, open and closed sets, continuity, compact surfaces like sphere and torus, complex numbers, etc). No prior knowledge of tropical or algebraic geometry is needed: we will start from scratch.

Tuition Benefit: The RTG grant in Algebraic Geometry and Topology provides tuition benefit for up to 15 undergraduates who are US citizens, nationals, and permanent residents. Departmental support is also available for other students.

Course Content: The purpose of this course is to give an introduction to algebraic curves and tropical curves.

The origins of algebraic geometry lie in the study of zero sets of systems of polynomials. These objects are algebraic varieties, and they include familiar examples such as plane curves and surfaces in three-dimensional space.

In tropical algebra, the sum of two numbers is their minimum and the product of two number is their sum. This algebraic structure is known as the tropical semiring or as the min-plus algebra. It makes perfect sense to define polynomials and rational functions over the tropical semiring. The functions they define are piecewise-linear. Also, algebraic varieties can be defined in the tropical setting. They are now subsets of R^n that are composed of convex polyhedra. Thus, tropical algebraic geometry is a piecewise-linear version of algebraic geometry

Syllabus: Before starting with tropical geometry, we will spend some time introducing the basic concepts in algebraic geometry in the context of plane curves. We will then quickly move to tropical geometry. Rather than giving a rigorous, abstract treatment of the theory which would go beyond the purpose of this course, we will discuss many examples (mostly focusing on the case of plane curves) and explore some of the connnection between algebra, geometry, topology, and other fields.

A rought list of topics expected to be covered in the course is given below. Topics are subject to change according to the level of the class.

  • Algebraic Geometry: Real plane conics. From the real numbers to the complex numbers, and from the affine plane to the projective plane. Conics revisited, cubics, and their group law. Topology of curves and Riemann surfaces, genus of a smooth curve. Plane curves. Resultant, Study's lemma (special case of Hilbert's Nullstellensatz), and the Bezout theorem.
  • Tropical Geometry: Tropical ring, examples and applications. First examples of tropical curves. Amoebas and their tentacles. Formalize approach: fields and valuations, Puisuex power series. Tropical curves and hypersurfaces. Fundamental theorem of tropical geometry. Tropical conics and higher degree plane curves. Genus of a tropical curve. Intersection and tropical Bezout theorem.

References: Algebraic geometry:

Tropical geometry:

Coursework and Grading: There will be homework assignments throughout the semester, and as the course progresses I will encourage students to find and work on a project to a topic of their choice related to the course. Some references and possible topics will be suggested in class. The last classes at the end of the semester will be devoted to presentations given by the students on their projects. Grade will be assigned based on class participation, homework, and final presentation.

Communications: I will use the automatic system for email communications provided by the Campus Information System. Refer to General Calendar Dates for important dates.

ADA Statement: The Americans with Disabilities Act requires that reasonable accommodations be provided for students with physical, sensory, cognitive, systemic, learning, and psychiatric disabilities. Please contact me at the beginning of the term to discuss any such accommodation for the course.


April 21: Guest Lecture by Drew Ellingson

Title: Tropical Grab Bag

Abstract: We will explore two interesting topics in Tropical Geometry. First, we look to tropical theta characteristics. A classical theorem states that a plane quartic curve has exactly 28 odd theta characteristics, corresponding to exactly 28 bitangents. Recent work by Yoav Len and Matt Baker has discovered that tropical plane quartics have 7 effective theta characteristics, and conjectured that the classical 28 specialize to the tropical 7 in groups of 4.
Second, we turn to tropicalization of the trigonal locus in M_g. This tropicalization (almost always) defines a top dimensional cell in the moduli of tropical planar curves. To study this, we introduce Hirzebruch Surfaces, Petri's Theorem, and lots of really cool combinatorics!

Final Presentations:

April 23 Aaron Huston Elliptic integrals
Ethan Lake The group structure on smooth tropical cubics
April 28 Scott Neville Combinatorial tropical determinants
Taylor Pope Tropical Riemann-Roch
Hunter Simper Classification of tropical projective conics
May 5 Connor James Elliptic curves and cryptography
Brady Bowen Moduli space of elliptic curves
Oliver Richardson Minkowski sum reducibility of tropical curves