Algebraic Geometry – Help Understanding Algebraic Geometry

Question

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 1

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

• Vakil - Foundations of Algebraic Geometry, Standford University.

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

• Mumford/Oda - Algebraic Geometry II: Schemes and Sheaf Cohomology, (draft).

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

• Beltrametti et al. - Lezione di Geometria Analitica e Proiettiva, Bollati Boringhieri 2003.
• Beltrametti et al. - Lectures on Curves, Surfaces and Projective Varieties: A Classical View of Algebraic Geometry, European Mathematical Society 2009.
• Mumford - Algebraic Geometry I: Complex Projective Varieties, Springer 1976.

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:

• Holme - A Royal Road to Algebraic Geometry, Springer 2012.

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):

• Badescu - Lezioni di Geometria Proiettiva (in Italian), Università di Genova.
• Badescu - Introduction to Algebraic Varieties, Università di Genova.
• Arrondo - Introduction to Projective Varieties, Universidad Complutense de Madrid.
• Manetti - Geometria Algebrica (in Italian), Università di Roma "La Sapienza".
• Marker - Topics in Algebra: Elementary Algebraic Geometry, University of Illinois at Chicago.
• Kerr - Lecture Notes Algebraic Geometry III/IV, University of Washington in St. Louis.
• Arapura - Notes on Basic Algebraic Geometry, Purdue University.
• Arapura - Crash Course on Complex Algebraic Varieties, Purdue University.
• Fulton - Algebraic Curves, An Introduction to Algebraic Geometry, University of Michigan.
• Skorobogatov - Algebraic Geometry, Imperial College London.
• Hacking - Introductory Lecture Notes in Algebraic Curves, University of Massachussetts.
• Hassett - Lecture Notes on Algebraic Geometry, Rice University.
• Gorodentsev - Algebraic geometry: a Start-up Course Independent University of Moscow.
• Bertram - Algebraic Geometry Notes, University of Utah.
• Dolgachev - Introduction to Algebraic Geometry, University of Michigan.
• Milne - Algebraic Geometry.
• Cutkosky - Introduction to Algebraic Geometry, University of Missouri.
• Debarre - Variétés Complexes (in French), École Normale Supérieure.
• Debarre - Introduction à la Géométrie Algébrique (in French), École Normale Supérieure.
• Gathmann - Notes for a Class in Algebraic Geometry, University of Kaiserslautern.
• Holme - Basic Modern Algebraic Geometry: Intro to Grothendieck's Theory of Schemes, Universitetet i Bergen.
• Ivorra Castillo - Geometría Algebraica (in Spanish), Universitat de València.
• Bruzzo - Introduction to Algebraic Topology and Algebraic Geometry, SISSA.
• Birkar - Algebraic Geometry Lecture Notes, University of Cambridge.
• Birkar - Topics in Algebraic Geometry, University of Cambridge.
• Voisin - Géométrie Algébrique et Géométrie Complexe (in French), Institute de Mathématiques de Jussieu.
• Voisin - Géométrie Algébrique et Spaces de Modules (in French), CNRS et IHÉS.
• Mumford/Oda - Algebraic Geometry II: Schemes and Sheaf Cohomology, (draft).
• Vakil - Foundations of Algebraic Geometry, Standford University.
• Gallier/Shatz - Complex Algebraic Geometry, University of Pennsylvania.
• Peters - An Introduction to Complex Algebraic Geometry, with Emphasis on the Theory of Surfaces, Institute Fourier Grenoble.
• Hacking - Introductory Lecture Notes in Algebraic Surfaces, University of Massachussetts.
• Reid - Chapters on Algebraic Surfaces, University of Warwick.
• Dolgachev - Classical Algebraic Geometry, A Modern View, University of Michigan.
• Debarre - Introduction to Mori Theory, Université Paris Diderot.
• Birkar - Lectures on Birational Geometry, University of Cambridge.