I while ago I started reading Hartshorne's Algebraic Geometry and it almost immediately felt like I hit a brick wall. I have some experience with category theory and abstract algebra but not with algebraic or projective geometry.
I'm wondering if any of you out there know of any articles, blog posts or whatever offering a light, intuitive and geometric introduction the subject. I really wanna get back to Hartshorne's book cause I am very curious about the categorical description.
I have provided the first few problems I ran into to give you an idea of where I come from. Of course if you can answer any of the questions that would be welcome.
First of all I'm having trouble grasping the very basic notion of a continuous function with respect to the Zariski topology. I don't which they are or know how to conceptualize them. I get how the rational polynomials work but I don't know if they are a subclass of the continuous functions or if they exhaust them. Any help in this regard is welcome.
Further I couldn't really get the projective part. I guess part of my problem comes from the fact that this is a set theoretic quotient of an algebra, which is then interpreted as an algebraic object. At least that's what I read, might be wrong. I seem to get lost during this transition and I don't know how to relate, are there any universal properties involved, whats the big picture?
Thanks in Advance
Edit1: Also, where is the hyperbolic geometry in all this?
Edit2: I want to express my gratitude towards all the people who have takes their time to give me recommendations and sympathy. Thank you!