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I have a copy of the book "Introduction to Toric varieties" by William Fulton, and over the next few months I'd like to make some progress on it.

As a first goal, I'd like to be able to read just the first 3 chapters, which I list some of the sections so people without the book get an idea of the contents:

  1. Definitions and Examples: Convex polyhedral cones, Affine toric varieties, Fans and torics varieties
  2. Singularities and compactness: Local properties of toric varieties, Compactness and properness, Resolution of singularities
  3. Orbits, topology and line bundles: Orbits, Fundamental groups and Euler Characteristics, Cohomology of line bundles

In the next two months I should have covered most of Atiyah-Macdonalds Commutative algebra book. In parallel I was planning to read Fulton's "Algebraic Curves" book (which can be found online here). I already know some cohomology. My question is, will reading "Algebraic Curves" give me enough background in Algebraic geometry to read the 3 chapters on toric varieties? If so, are there any sections of "Algebraic Curves" I could skip? If not, how much more do I need?

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Of the first three chapters, I think the 3rd chapter is a "step up" in difficulty and amount of prerequisites. With a solid background in the very basics (construction of affine and projective algebraic varieties, Zariski topology, definition of singularity, the gluing construction, morphisms...) you should be able to make it through chapter 2. There will be a few places where things don't make sense (for example there is a page and a half where Fulton explains why toric varieties are Cohen-Macaulay), but these can be skipped at first reading without losing the big picture.

Some facility with line bundles and divisors will get you through most of chapter 3, except for the last section on cohomology. Fulton assumes some familiarity with these ideas, and then shows how to work with them in the toric case.

Sometimes, if you run into a gap in your knowledge, you might still be able to follow the toric, or convex, construction. I.e., you can still try to understand what happens on the level of cones, polytopes and fans before working out the algebro-geometric side. After all, the book is a sort of dictionary between the geometry of convex sets and the geometry of algebraic varieties, and one must work to understand the constructions of both categories to get the full translation. Working out the toric side will provide you with examples to understand the general algebraic theory as well.

With regards to Fulton's curves book, there probably are some sections you could skip if your only goal is to read the first three chapters of "Toric Varieties". However, all the material in that book is worthwhile, so I would encourage you to read as much of it, or whatever other introductory book you choose (I like Shafarevich BAG) as you can. For example, part of chapter 2 of "Toric Varieties" is about blow-ups, which are covered in chapter 7 of "Curves". So if you just can't wait to start learning about toric varieties, maybe you can read only the first half of the curves book, then read chapter 1 of the toric book, then go back to curves, etc.

Good luck!

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