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: 

1. October 6: Introduction to the course; motivation from moduli theory.

2. October 9: A rapid review of the theory of varieties and its insufficiencies. 

3. October 12: The Zariski spectrum of a ring and its topology. Distinguished open sets. 

4. October 13: Quick recap of the spectrum. Basics of distinguished opens and localization. Presheaves and sheaves. 

5. October 16: Mark Gross will lecture while Dhruv is away. Basics of sheaves. 

6. October 18: Mark Gross will lecture while Dhruv is away.  Stalks, sheafification, image, kernel, etc.

7. October 20: Pushforward and inverse image sheaves. Sheaves on a base. Towards schemes. 

8. October 23: The structure sheaf of an affine scheme. The definition of a scheme. Examples. 

9. October 25: Open subschemes. A non-affine scheme. Gluing sheaves and schemes.

10. October 27: Basic examples: the line/plane with doubled origin, the projective line, and projective space via gluing.

11. October 30: Projective schemes again: graded rings and the Proj construction. 

12. November 1: Morphisms between schemes and basic properties. Morphisms between affine schemes are what you think they are.

13. November 3: Housekeeping: open and closed immersions. Fiber products of schemes. The diagonal morphism.

14. November 6: Separatedness and examples. 

15. November 8: More on separatedness, universal closedness and properness. 

16. November 10: Fundamental theorem of elimination theory. Valuation rings and valuative criteria. 

17. November 13: Consequences of valuative criteria. Modules and algebras over the structure sheaf. Basic operations.

18. November 15: Quasicoherent and coherent sheaves. Coherent sheaves on affine schemes. 

19. November 17: Coherent sheaves on projective schemes. Examples via homogeneous polynomials. 

20. November 20: Introduction to Weil divisors and principal divisors. 

21. November 22: Excision sequence and class group of Pn. Sheaf of rational functions.

22. November 24: Cartier divisors and how to think about them. Invertible sheaves, Cartier divisors, and Weil divisors. 

23. November 27: Overview of cohomology and how to calculate it. Euler characteristics. Cohomology of line bundles on projective space. 

24. November 29: Calculation of cohomology of line bundles. A few basic notions associated to sheaf cohomology. 

25. December 1: Cech cohomology is independent of the cover. Regularity of schemes and Serre duality.

Extra notes and materials:

• A summary of varieties. 

• Notes for my undergraduate course on varieties are here. 

• Visualising the Zariski spectrum of k[X,Y].