Algebraic Geometry

Welcome to the course webpage for Part III algebraic geometry, taught in the 2021 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, and the basics of sheaf cohomology. 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, such as moduli theory and etale cohomology 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 and Arithmetic Curves by Qing Liu. 

A basic guide to preparing for the exam for this course is available here.   

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 here. In a perfect world, these would be a fairly good reflection of what is said lecture. We will therefore use this to deduce that the world is not perfect. Notes for the first examples class are here. 

You can find the example sheets here. The solutions to sheet I are here. 

Lecture Schedule: 

• October 8: Introduction to the course; the Weil conjectures and the need for a more general theory than varieties. 

• October 11: The spectrum of a ring, its topology, distinguished opens and function theory. 

• October 13: Remarks on the function theory of Spec A; introduction to presheaves and sheaves. 

• October 15: Determining sheaves and morphisms on stalks; sheafification; kernels, cokernels, and images. James Bond discussions.

• October 18: More remarks on sheaves, pullback and pushforward. Towards affine schemes. 

• October 20: The structure sheaf of an affine scheme.

• October 22: Schemes in general and examples of schemes. 

• October 25: Navid will lecture on projective schemes and the Proj construction. 

• October 27: There will be no lecture on this day!

• October 29: Navid will continue discussion projective schemes. Sadly, Dhruv will return on November 1. 

• November 1: Morphisms of schemes and morphisms of affine schemes. 

• November 3: A few basic types of morphisms and fibre products. A toe dipped into the functor of points. 

• November 5: Further discussion of fiber products and examples.

• November 8: Separated morphisms and the valuative criterion. 

• November 10: More on separated and proper morphisms.

• November 12: Final words on proper morphisms; modules over the structure sheaf.

• November 15: Quasi-coherent and coherent sheaves. 

• November 17: More on quasi-coherent sheaves. Vector bundles and global constructions. 

• November 19: Weil and Cartier divisors. 

• November 22: More on divisor theory. Class groups and Picard groups. 

• November 24: Global sections, derived functors, and cohomology. 

• November 26: Cech cohomology for coherent sheaves.  

• November 28: Duality theory and the statement of the Serre duality theorem. 

• December 1: Representable functors, moduli problems, and the Hilbert schemes.