Every exercise in the first three chapters of Hartshorne

Originally posted: May 14th, 2025

I have recently begun work on a new, wildly over-ambitious project: doing every exercise in the first three chapters of Hartshorne before my PhD begins in the Fall. I was inspired by this website. I don't even think this individual accomplished their goal, and I may not either (in fact, I'd say it is quite likely that I won't). However, as the adage goes, "shoot for the stars, land on the moon" or something like that.

My main goal with this project is this: I want to be as comfortable with the foundational aspects of algebraic geometry as I am with the foundational aspects of differential geometry and topology.

Unlike the linked website, I decided that I would additionally work through the Chapter 1 problems, rather than only Chapters 2 and 3. I took a class on algebraic curves during my final semester of undergrad, where my professor, Stephen Kudla, stressed the importance of understanding the "classical theory of varieties" in order to most effectively internalize the modern scheme-theoretic toolkit, which is exactly what Chapter 1 covers.

Here is a link to my solutions. I'll keep a tally of my progress here:

Chapter 1 problems completed: 16/90
Chapter 2 problems completed: 1/134
Chapter 3 problems completed: 0/88

If I have any extra thoughts of notes about particular problems, I will add them below. Suggestions and/or corrections are much much appreciated, I'm sure there will be typos!

May 14th, 2025: On the original publication date of this blog post, I have finished 9/90 (in other words, 10%) of the problems in Chapter 1. So far so good. If I manage to keep up the pace of solving 3 or 4 problems every day, then I'll easily finish this project by the Fall. Easier said than done!