Simplicial Divisors

Course Overview

Welcome to the algebraic geometry section of Math 194, Spring 2021! This quarter we will study combinatorial objects called simplicial complexes which, when satisfying certain conditions, correspond with important objects studied in algebraic geometry. We will begin by discussing the fundamentals of algebraic geometry and then will study simplicial complexes and attempt to find ones that satisfy our desirable properties. Due to the combinatorial nature of this problem, we can also approach it from a computational perspective if you are interested.

Prerequisites

I don't necessarily assume any prerequisite knowledge going into this course and I am enthusiastically willing to take as much time as we need to fill in gaps but some knowledge of rings and vector spaces would be useful. For the sake of time, I may direct you to read about certain fundamental concepts on your own but I will help you through them as much as you need.

Course Materials

I don't believe cost should be a barrier to education so I have specifically chosen resources that are available for free online:

• My course notes: I will update these as often as I can. The idea is that I will focus on material relevant to our problem and I will explain those concepts in greater detail than one would normally find in a traditional algebraic geometry textbook.

• Andreas Gathmann's notes: these are notes intend for a graduate level of study so don't worry if you don't understand everything! They are, however, very well written and could be a useful resource to supplement what we discuss in class.

• Doran, Giansiracusa, and Jensen's paper which first introduced this idea.

• González, Gunther, and Zhang's paper building on this idea with very approachable exposition and some great computational work.