Algebraic Geometry


This is a one-semester course in Algebraic Geometry.


Math 2501 (Algebra II), or equivalent


The Rising Sea: Foundations of Algebraic Geometry, by Ravi Vakil, available for free download at http://math.stanford.edu/~vakil/216blog/index.html.  Please use the November 28, 2015 draft, so that we all refer to the same version.

Course website



Thomas Hales, Thackeray 416, lastname at pitt.edu
Office hours: Monday 10-11am, Friday 2-3pm.


Your grade will be based entirely on assigned readings, homework, and class participation. Students are expected to participate regularly in class discussions and to present material in class related to readings and homework.


This course is an introduction to the theory of schemes and Grothendieck-style algebraic geometry.

This course will follow our text The Rising Sea very closely. The book is a mammoth 799 pages long. (Students wanting a strong foundation in algebraic geometry should plan to spend a full year studying this book.) We will only cover the first portion of the book. The first five chapters are

  1. Some category theory
  2. Sheaves
  3. Toward affine schemes
  4. The structure sheaf, and the definition of schemes in general
  5. Some properties of schemes

Throughout the course, we will connect the general theory with concrete intuition developed from simple examples.

Understanding algebraic geometry is often thought to be hard because it consists of large complicated pieces of machinery. In fact, the opposite is true; to switch metaphors, rather than being narrow and deep, algebraic geometry is shallow but extremely broad. It is built out of a large number of very small parts, in keeping with Grothendieck’s vision of mathematics. – from the Rising Sea, preface, page 12.

Much of the book is structured as a long series of exercises. Many of these exercises are important for the proper understanding of the subject.

The class meetings will be discussions of the text and its exercises.


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