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As the title suggest, I am trying to read the book mentioned, but I find that it uses a lot of material that I don't know yet. For example, it uses toric geometry and polytopes, topics that I've never seen in regular courses at my university.

So, I want to know from the experience of someone who has used the book, what are the prerequisites of algebraic geometry to understand the text. Is it necessary to know the language of schemes? Are schemes used at all?

My background is very modest, being a course in classic algebraic geometry, at the level of Fulton's book "Algebraic curves" and almost the second chapter of Hartshorne's "Algebraic Geometry".

I found the book too diffuse, as to the range of topics used is refered. If someone can answer my questions and give some suitable references (in terms of what is needed in the book) for the necessary background, I'll be very grateful. The thing is, I barely can see what is being done with the algebraic geometry in the text, and I'm looking for an "scheme theoretic" point of view of the topic.

I was reading it a year ago. It's important that you have a little experience with Hodge decomposition, Gauss-Manin connection and Kahler geometry in general (there are two volumes of Claire Voisin which provide the necessary background. Here's a review). Also you need to understand moduli space of algebraic curves (read for examples Morris and Harrison's "Moduli of curves". Virtual fundamental class is a stack-theoretic construction and thus has some relationship to schemes but I don't think it's important for a first read- so schemes are not important). Fulton has a book on toric varieties, you can also look at Batyrev's original work. Also you need to understand group actions on varieties and orbifold construction- I'm not really sure about reference for that. As for physical side (string theory or QFT) it is not really important but you can look through Clay monograph's physics chapters.