Algebraic Geometry

Welcome to the course webpage for Part III algebraic geometry, taught in the 2022 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: The standard texts are Algebraic Geometry by Hartshorne and The Rising Sea by Vakil, freely available here. 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. The document will be periodically updated. 

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 are linked here. In a perfect world, these would be a complete reflection of what is said lecture. We will therefore use this to deduce that the world is not perfect. Notes for my undergraduate course on varieties are here

You will eventually find the example sheets linked on the official DPMMS page here. Example sheet I is here. Example sheet II is here. Example sheet III is here. Example sheet IV is here [but is subject to minor edits].

October Engine Check.

Basic Examples of Morphisms.

Lecture Schedule: