Help understanding Algebraic Geometry

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!

Answer

Recently, the best freely available textbook on category-laden algebraic geometry seems to be:

The following reference is a great companion to the hard core of Vakil and/or Hartshorne:

For deep classical projective algebraic geometry, I cannot but eagerly recommend:

For a mixture of both, with a first half introduction to projective algebraic geometry and a second half heavily focused categorical introduction to schemes, this new book is a gem, and may be exactly what you are looking for, serving as a perfect introduction before/along with Hartshorne first chapters:

You can get part of the scheme theory of that book for free at Holme’s website. Definitely, Holme’s book will be more than enough (maybe along with Gathmann’s notes, see links below) to fill in geometric motivations for Hartshorne; jointly with Vakil’s course complementing the categorical side, you will have enough and almost self-conteined material to digest for a long time.

Besides the recommendations given already, I would suggest you check out the other useful posts I referred to in this other answer. The lecture notes by Kerr provide a lot of geometric motvation and intuitive pictures on projective algebraic curves, and Gathmann’s thorough course gives a highly insightful and motivated broad introduction to the more abstract approach, being an excellent detailed “overview” before approaching Hartshorne (as the author himself points out in his bibliography).

To clarify concepts on projective geometry, projective varieties and to supplement Hartshorne’s reading, either from a complex geometry or purely algebraic point of view, the following long list of freely available online courses may provide you with the extra bits you need on specific topics (warning! most of them are more elementary than Hartshorne but some of them go beyond it or supplement it on other topics, they are included for completeness of good references to have if you decide to go beyond Hartshorne):

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