Divisors on curves and graphs

Description

Divisors are fundamental objects in algebraic geometry. A divisor on an algebraic curve is a finite linear combination of points with integer coefficients, and can be used to encode information about the regular functions on the curve. If we look at them sufficiently deeply, divisors tell us a compelling story that helps us to classify curves and to uncover many of their features.

In analogy with the case of curves, one can also define divisors on finite graphs as finite linear combinations of vertices. These apparently simple objects play a fundamental role in graph theory, and connections with game theory (via "chip-firing" on finite graphs) and theoretical physics (via the “abelian sandpile” model) were discovered independently from the case of curves.

At some point researchers realized that this correspondence between curves and graphs is much more than just an analogy, and one can actually “send” divisors from a curves to a graph, effectively building a bridge from the algebro-geometric to the combinatorial world.

The study of this topic leads naturally to many possible paths: after an initial exploration of the different avatars of divisors and their mutual interaction, students can decide to look more deeply at the algebro geometric aspects (Riemann-Roch theory, Brill-Noether theory, ...), the aspects related to commutative algebra (resolutions of binomial and monomial ideals) or the more combinatorial features (G-parking functions, spanning trees, matrix-tree theorem, ...).

Essential modules

It is essential that students have taken Algebra II and Geometry III.
Helpful modules are Galois Theory III and Codes & Cryptography III, but they are not essential for the success of the project.

Resources

“Divisors and sandpiles” by Scott Corry and David Perkinson [link: https://people.reed.edu/~davidp/divisors_and_sandpiles/]
“What is… a sandpile?” by Lionel Levine and James Propp [link: https://pi.math.cornell.edu/~levine/what-is-a-sandpile.pdf]
“Algebraic geometry and arithmetic curves” by Qing Liu, especially Chapter 7. [Library link]