Algebraic Geometry

Welcome to the course webpage for Part III algebraic geometry, taught in the 2023 Michaelmas term at Cambridge. 

Instructor: Dhruv Ranganathan (E 1.01). Meeting time: MWF 10-11 AM (MR5)

Overview: The course will be an introduction to the theory of schemes, sheaves. These form the language and toolkit for modern research in algebraic geometry in its full range from geometry to arithmetic. The goal will be for us to understand the basic concepts in scheme theory and gain enough familiarity with a sufficient range of examples that more advanced topics in algebraic geometry move within reach. 

Resources and their use: The standard text is Algebraic Geometry by Hartshorne and The Rising Sea by Vakil, freely available here. It is important to have access to at least one: there will be occasions where some details will be outsourced to the book (though when this is done, the proof will be non-examinable).

I am fond of the Geometry of Schemes by Eisenbud and Harris and Algebraic Geometry for its plentiful examples. 

SAGES: If you would like to enrich your experience of learning algebraic geometry, I provide potential topics for a student-run seminar in this document. I will not be involved in the seminar.

Prerequisites: The main technical requirement will be an understanding of rings and modules; the parallel course in commutative algebra is a recommended co-requisite though it is possible to learn the relevant concepts on the fly. A number of other notions will be used in an elementary fashion, including basic point-set topology and manifolds. 

The theory of affine and projective varieties will arise as motivation, and remain the main source of examples but are not a technical pre-requisite. Students without experience with these ideas may wish to have Chapter I of Hartshorne's text or Reid's Undergraduate Algebraic Geometry on the side to get a sense for why people care about this stuff. 

Notes: My notes will be linked here. In a perfect world, these would be a complete reflection of what is said lecture. We will use this to deduce that the world is not perfect. 

Example Sheet: First Second Third Fourth

Lecture Schedule: 

Extra notes and materials: