I am looking for recommendations on an AG text to work through this summer, possibly with the help of a mentor. I would want this book to have some introduction to categories, and then develop the modern methods (some development of sheaves and schemes), hopefully up to some realization of Riemann-Roch. Any recommendations?
2026-03-28 12:48:29.1774702109
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Looking for an introductory Algebraic Geometry book
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These notes by Andreas Gathmann are precisely what you're asking for. He starts very gently, schemes being introduced in Chapter 5, but he ends with sheaf cohomology (including Riemann-Roch) and some intersection theory.
The book Foundations of Algebraic Geometry by Ravi Vakil is likely to become the standard as a comprehensive text in algebraic geometry. It may be more thorough than what you have in mind (and it certainly is not something that can be fully digested by mere mortals in a single summer of self-study), but it covers all the topics you want, is very well-written, and is freely available for download.