Taking hours on each exercise in Hartshorne chapter 1. What should I do? I understand this could be off-topic, but I did not know other place to ask for help. Sorry, but please bear with me.
I am at graduating year of undergrad, and ambitiously approached an introductory course on algebraic geometry with Hartshorne as a textbook, mainly studying chapter 1. It ended up being incredibly hard to me, even though it is just chapter 1.
As a result, I am currently having problem with solving Hartshorne's exercise problems. Each problem is taking like hours (short: 1 hour, long: 4~5 hours). Sometimes I think I might not be fit for mathematics at all.. I am curious if I am falling considerably behind.
Lastly, what should I learn/do to understand these subjects better, and attain good intuition? I would appreciate any advice!
 A: As someone who failed a beginning proofs class, a geometry class, and was close to failing a linear algebra class, I emphasize the feeling of falling behind. I'm really sorry you feel like the course is becoming out of your reach. I know the feeling of trying to play catch-up, especially since I imagine you probably are taking some other classes. I'm no psychologist, but I think the best way to get over that is by talking about it with your professor or talking about it with some friends/family just to get stuff off your chest. I'm sure they would be more than willing to help you out. Maybe talk to a counselor about it, too.
This is me being on the harsher side, but you say, "Sometimes I think I might not be fit for mathematics at all." That is not the attitude you should have. Anyone can do math regardless of how difficult it is as long as they keep trying over and over again. That being said, if you want to attain good intuition, you just have to keep struggling over and over again. It's probably not going to stick in your head for a few months, but I guarantee later down the line, probably months from now, things will click for you. That's the reality of how math and science are.
I'm sure you've heard that same sort of advice everywhere, but I still thought it was worth mentioning. I have never taken an algebraic geometry class, but here is some advice.
(1) If you find yourself getting stuck hours after attempting your proofs, you should ask for help. No one in this world was able to solve everything by themselves. They asked for help from other people to see if they could come up with ideas. That being said, you can: phone some friends for help; ask for help during office hours (as long as you show your professor the countless work you've written/typed); email your professor if you are doing this so-and-so proof correctly.
(2) You can also ask the questions you have on this site too. Whenever I ask a question on this site like this one, I always make sure to explicitly say the context, my really long attempt, and my question (in bold, too). That way, it shows you care about solving the problem and that you are part of the learning process. More experienced people would be drawn to help you out in the comments or even a full solution.
This answer wasn't a very structured one, but I hope the advice helps.
