Algebraic geometry reference

Question

I believe similar reference requests have been asked previously but I think mine is somewhat specific. I am interested in learning algebraic geometry.

My experience so far has been with complex projective varieties and schemes of finite type over $\mathbb{C}$, with divisors, sheaves, vector bundles, and cohomology on Riemann surfaces, and analytic sheaves. I am comfortable with commutative algebra at the level of Atiyah-MacDonald and Kleiman's texts. My interest leans towards complex geometry, but I would enjoy getting a good understanding of the algebraic/functorial picture too.

I tried reading Hartshorne but I found the style to be dry and not insightful. I do plan to work through it eventually, but would want a reference that covers similar amounts of theory and feels intuitive given my background.

My current picks are Cutkosky's book, "Introduction to algebraic geometry"; for schemes, I am reading Ulrich and Wedhorn's text, and for cohomology, my pick is Serre's FAC.

I'm wondering if these will cover roughly the same material as the first three chapters of Hartshorne; what I would be missing out on, would there be overlap bits that I can gloss over, and what else will be an intuitive reference for learning the subject?

Answer 1

I think that The Geometry of Schemes by Eisenbud and Harris is considered to be a book fitting your needs. I haven't read too much of the book myself, but I have it on good authority that it is a good supplement to Hartshorne and gives more insight into some of the basic constructions.

It should be noted that there is no section on cohomology, but the treatment of cohomology in Hartshorne is reasonably clear, I think. Also, since you already have experience with cohomology in the analytic case, that shouldn't be a major concern.

With regards to the first question, I'm not familiar with Ulrich and Wedhorn or Cutkosky, so can't offer much of an opinion.