Course: C3.4 Algebraic Geometry (2026-27)

Lecturer: Profile: Dominic Joyce

Course information

General prerequisites:

A3 Rings and Modules and B2.2 Commutative Algebra are essential. Noetherian rings, the Noether normalisation lemma, integrality, the Hilbert Nullstellensatz and dimension theory will play an important role in the course.
B3.3 Algebraic Curves is useful but not essential. Projective spaces and homogeneous coordinates will be defined in C3.4, but a working knowledge of them would be useful. There is some overlap of topics, as B3.3 studies the algebraic geometry of one-dimensional varieties.
Courses closely related to C3.4 include C2.2 Homological Algebra, C2.7 Category Theory, C3.7 Elliptic Curves, C2.6 Introduction to Schemes; and partly related to: C3.1 Algebraic Topology, C3.3 Differentiable Manifolds, C3.5 Lie Groups.

Course term: Michaelmas

Course lecture information: 16 lectures.

Course weight: 1

Course level: M

Assessment type: Written Examination

Course overview:

Algebraic geometry is the study of algebraic varieties: an algebraic variety is roughly speaking, a locus defined by polynomial equations. One of the advantages of algebraic geometry is that it is purely algebraically defined and applied to any field, including fields of finite characteristic. It is geometry based on algebra rather than calculus, but over the real or complex numbers it provides a rich source of examples and inspiration to other areas of geometry.

Learning outcomes:

Students will have a good understanding of affine, projective, and quasiprojective varieties over an algebraically closed field, and will be familiar with examples of these. They will understand morphisms between varieties, birational maps, and blow-ups. They will know about the Zariski topology, tangent spaces, singular points, smooth points, degree, and dimension, and be able to work with these in examples. They will be able to work with the dictionary between algebraic concepts (for example, ideals) and geometric concepts (for example, subvarieties).

Course synopsis:

Affine algebraic varieties, the Zariski topology, morphisms of affine varieties. Irreducible varieties.

Projective space. Projective varieties, affine cones over projective varieties. The Zariski topology on projective varieties. The projective closure of affine variety. Morphisms of projective varieties. Projective equivalence.

Veronese morphism: definition, examples. Veronese morphisms are isomorphisms onto their image; statement, and proof in simple cases. Subvarieties of Veronese varieties. Segre maps and products of varieties.

Coordinate rings. The geometric form of Hilbert's Nullstellensatz. Correspondence between affine varieties (and morphisms between them) and finitely generated reduced K-algebras (and morphisms between them). Graded rings and homogeneous ideals. Homogeneous coordinate rings.

Discrete invariants of projective varieties: degree, dimension, Hilbert function. Statement of theorem defining Hilbert polynomial.

Quasi-projective varieties, and morphisms between them. The Zariski topology has a basis of affine open subsets. Rings of regular functions on open subsets and points of quasi-projective varieties. The ring of regular functions on an affine variety is the coordinate ring. Localisation and relationship with rings of regular functions.

Tangent space and smooth points. The singular locus is a closed subvariety. Algebraic re-formulation of the tangent space. Differentiable maps between tangent spaces.

Function fields of irreducible quasi-projective varieties. Rational maps between irreducible varieties, and composition of rational maps. Birational equivalence. Correspondence between dominant rational maps and homomorphisms of function fields. Blow-ups: of affine space at a point, of subvarieties of affine space, and of general quasi-projective varieties along general subvarieties. Statement of Hironaka's Desingularisation Theorem. Every irreducible variety is birational to a hypersurface. Re-formulation of dimension. Smooth points are a dense open subset.

Section outline

Select section General

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General

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- Select activity Announcements
  
  Announcements Forum
- Select activity Ritter lecture notes
  
  Ritter lecture notes File
  
  These are lecture notes for C3.4 Algebraic Geometry written by Alexander Ritter in 2019. They are the primary reference for the course.
- Select activity Rossler lecture notes
  
  Rossler lecture notes File
  
  These are lecture notes for C3.4 Algebraic Geometry written by Damian Rossler in 2025. They are a very useful reference, and please do read them, but the approach I intend to take is a bit closer to Alexander Ritter's notes than to these.
- Select activity Baldwin-Berczi lecture notes
  
  Baldwin-Berczi lecture notes File
  
  These are lecture notes for C3.4a Algebraic Geometry written by Elizabeth Baldwin and Gergely Berczi in 2014. You may find them useful.
- Select activity Rossler B2.2 Commutative Algebra notes
  
  Rossler B2.2 Commutative Algebra notes File
  
  These are lecture notes for B2.2 Commutative Algebra (Oxford second year algebra course) written by Damian Rossler in 2025. They are not the notes for this course! But they are the most important background material for this course. If you don't know the material in them (for example, commutative rings, ideals, Noetherian rings, Hilbert's Nullstellensatz) it would be helpful for you to read them.
- Select activity Slides of lectures (preliminary version)
  
  Slides of lectures (preliminary version) File
  
  These are a preliminary version of the lecture slides I intend to use. They are a work in progress, and I will be posting updated versions as the term goes on.
- Select activity Sheet 0
  
  Sheet 0 File
  
  This is a short introductory problem sheet covering some of the themes of the course, using material that hopefully you may already know. Solutions are not provided.
- Select activity Sheet 1
  
  Sheet 1 File
  
  Here is Problem Sheet 1, for the first class. This sheet will not be marked.
- Select activity Sheet 1 with answers to Parts A and C
  
  Sheet 1 with answers to Parts A and C File
  
  Here is Sheet 1 together with answers for Parts A and C. Answers for Part B will be available after the class.
- Select activity Sheet 2
  
  Sheet 2 File
  
  Here is Problem Sheet 2. This sheet will be marked.
- Select activity Sheet 2 with answers for Parts A and C.
  
  Sheet 2 with answers for Parts A and C. File
  
  Here is Sheet 2 together with answers for Parts A and C. Answers for Part B will be available after the class.
- Select activity Sheet 3
  
  Sheet 3 File
  
  Here is Problem Sheet 3. This sheet will be marked.
- Select activity Sheet 3 with answers for Parts A and C
  
  Sheet 3 with answers for Parts A and C File
  
  Here is Sheet 3 together with answers for Parts A and C. Answers for Part B will be available after the class.
- Select activity Sheet 4
  
  Sheet 4 File
  
  Here is Problem Sheet 4. This sheet will not be marked.
- Select activity Sheet 4 with answers for Parts A and C
  
  Sheet 4 with answers for Parts A and C File
  
  Here is Sheet 4 together with answers for Parts A and C. Answers for Part B will be available after the class.
- Select activity Sheet 1
  
  Sheet 1 Assignment
  
  Don't hand in solutions to this sheet. Model solutions will be available here after the class.
- Select activity Sheet 2
  
  Sheet 2 Assignment
  
  Opened: Thursday, 22 October 2026, 12:00 AM
  
  Please upload your solution to Sheet 2 here. Model solutions will be available here after the class.
- Select activity Sheet 3
  
  Sheet 3 Assignment
  
  Opens: Thursday, 5 November 2026, 12:00 AM
  
  Please upload your solution to Sheet 3 here. Model solutions will be available here after the class.
- Select activity Sheet 4
  
  Sheet 4 Assignment
  
  Don't hand in solutions to this sheet. Model solutions will be available here after the class.
- Select activity Reading List
  
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  Discussion Forum
Select section Intercollegiate Classes

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Intercollegiate Classes
- Select activity Class Signup
  
  Class Signup Class registration
  
  Registration start: Monday, 5 October 2026, 12:00 PM
  
  Registration end: Friday, 6 November 2026, 12:00 PM
- Select activity Class Tutor's Comments
  
  Class Tutor's Comments Assignment
  
  Class tutors will use this activity to provide overall feedback to students at the end of the course.
  
  Not available unless: You have signed up to a class

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