# Studying Algebraic Geometry

I won’t start entirely from the beginning even though that wouldn’t be entirely useless to me. Here are the Wikipedia pages for groups, rings, ideals, and fields. In the introduction of page on fields you can see the chain of algebraic structures. These names sometimes come up, but since I personally don’t do a good job of keeping them straight I’ll try to mention definitions when they do. We also will skip over the definitions of normal subgroups, quotient groups, and homomorphisms. Algebraic geometry (at least in my limited understanding) has commutative algebra as a prerequisite so the article on commutative rings is also relevant, but I’ll actually write a bit about that myself. To clarify one possible issue, I will always assume that rings always have a one.

A very fair question is what applications algebraic geometry has or what it even is about. To this I have no good answer. The Wikipedia page lists some areas it is useful in, but I must be clear that I have zero understanding of how it is applied in those fields. Hartshorne’s book itself makes a very brief statement but then returns to the question after the first chapter. I will attempt to do the same. At the very least I have been informed that it finds some application in biology, and despite my abhorrence of the subject, it’s hard not to find that a little interesting. With our minimal assumptions of knowledge I will write up a little bit about basic algebra and the most basic of definitions in topology before attempting to dive into Hartshorne. If you’re interested in a real resource, in school I worked out of Algebra by Michael Artin and Abstract Algebra by Dummit and Foote, both of which I thought were perfectly reasonable.

Even when I get to algebraic geometry my commentary will of course be less insightful than Hartshorne or any other author (or most math students). I hope there will be some value in watching someone struggle to learn the topic rather than reading the perspective of someone who knows it inside and out. As always, any comments are more than appreciated.