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:

  • October 7: Introduction to the course; the Weil conjectures.

  • October 10: Review of affine varieties; limitations of the theory.

  • October 12: The Zariski spectrum, the Zariski topology, basic examples.

  • October 14: Functions on the Zariski spectrum and towards sheaves.

  • October 17: Sheaves, morphisms between them, and basic constructions.

  • October 19: Sheafification, images, kernels, etc of sheaves. Examples of sheafification.

  • October 21: Moving sheaves between spaces by inverse image and pushforward. Localization and sheaves on a base.

  • October 24: Lecture cancelled

  • October 26: Tony lectured. He proved that the structure sheaf of an affine scheme is a sheaf. He defined schemes.

  • October 28: We saw various examples of interesting rings. Examples of non-affine schemes. Gluing sheaves.

  • October 31: Schemes by gluing. The bug-eyed line, plane, and a look at tthe projective line.

  • November 2: Graded rings and the Proj construction.

  • November 4: Completing the Proj construction and morphisms of locally ringed spaces.

  • November 7: Morphisms of schemes, basic types of morphisms, fibre products.

  • November 9: Existence of fibre products, examples.

  • November 11: Separatedness, basic properties, and examples.

  • November 14: Properness, examples, and valuative criteria. A word on more general properties of morphisms.

  • November 16: Algebras and modules over the structure sheaf I: Quasicoherent and examples.

  • November 18: Algebras and modules over the structure sheaf II: various important sheaves.

  • November 21: A few examples of quasicoherent sheaves. Weil and Cartier divisors and the class group via excision.

  • November 23: More on Weil divisors; invertible sheaves and Cartier divisors. Examples.

  • November 25: Finishing up divisors. Sheaf cohomology survival guide I: motivation.

  • November 28: Sheaf cohomology survival guide II: cohomology via resolutions and open covers.

  • November 30: Sheaf cohomology survival guide III: cohomology of projective space and more examples.

  • December 2: Representable functors and a bucket of examples of schemes.