Please take this page in conjunction with the Part III Guide to Courses Geometry and Topology section

Prerequisite areas

## Algebraic Geometry

In theory, the Algebraic Geometry course usually starts from scratch, but you will find it impossible to keep up if you are not already familiar with basic algebra and point-set topology. It is also well worth gaining some exposure to simple concepts in classical algebraic geometry.

You will need this for the following Part III courses:

- Algebraic Geometry
- (Elliptic Curves)

Relevant undergraduate courses are:

**First level prerequisites**

- Elementary point-set topology: topological spaces, continuity, closure of a subset etc. See e.g. Ib Metric and Topological Spaces. You can check if you are at the required level by doing the following exercises: Met & Top 2015-16 Example Sheet 1.
- Commutative algebra, at roughly the level mentioned in Second level prerequisites of Basic Algebra: rings, ideals (including prime and maximal) and quotients, algebras over fields (in particular, some familiarity with polynomial rings over fields). See e.g. Ib Groups, Rings and Modules. You can check if you are at the required level by doing the following exercises: GRM 2015-16 Example Sheet 2.

**Second level prerequisites**

Basic affine algebraic geometry, in particular:

- affine space and algebraic sets;
- the Hilbert basis theorem and applications;
- the Zariski topology on affine space;
- irreducibility and affine varieties;
- the Nullstellensatz;
- morphisms of affine varieties;
- projective varieties.

You can use the following exercises to check your present level and to guide your study.

- Algebraic Geometry exercises provided by Jack Smith (thank you!).

**Useful books and resources**

- Notes from previous catch-up workshop on Algebraic Geometry, provided by Jack Smith (thank you!).
- Ravi Vakil's online notes Foundations of Algebraic Geometry
- Eisenbud
*Commutative Algebra with a view toward algebraic geometry*(covers all the algebra you might need, with a geometric flavour---it has pictures). - Atiyah, Macdonald
*Introduction to Commutative Algebra*. - Pelham Wilson's online notes for the `Preliminary Chapter 0' of his Part III Algebraic Geometry course from 2014 cover much of this catch-up material but are pretty brief. They do give further resources and book suggestions.
- Hartshorne `Algebraic Geometry' (classic textbook although it's quite dense; the workshop (notes above) mainly tried to match terminology and notation with Chapter 1 of this book).