# Topics in modern geometry 3/4 (Autumn 2017)

MATH30001/MATHM0008

**Lecturers:** Matthew Palmer (Weeks 1–2 and 6) and Simon Peacock (Weeks 3–5).

**Lectures:** Monday 9am and Tuesday 2pm in SM4.

**Problems classes:** Thursday 4pm in SM4.

**Office hours:** by appointment by email (mp12500@bristol.ac.uk or simon.peacock@bristol.ac.uk).

## Outline of course

- Affine varieties
- Some commutative algebra
- Projective spaces
- Tangent spaces, smoothness and singularities
- Blowups and resolution of singularisation

## Lecture notes

Lecture 2: polynomial rings and affine varieties.

Lecture 3: irreducible varieties, dimension and morphisms.

Lecture 4: Noetherian rings and the ideal/variety correspondence.

Lectures 5–7: Ideal/variety correspondence, Hilbert’s Nullstellensatz and coordinate rings.

Lecture 8: Projective geometry.

Lectures 9–10: Tangent spaces and singularity.

Lecture 11: Desingularisation and blowing up at a point.

Lecture 12: Blowing up varieties at the origin.

## Homework

Homework 1. PhD students: submit solutions to 1cd, 2cd, 3cd, 5, 6 by Tuesday 10th October. (Solutions)

Homework 2. PhD students: submit solutions to exercises 3 and 9 by Tuesday 17th October. (Solutions)

## Notes from problems classes

Problems Class 1: describing varieties, and proving some basic properties

## Mock Exams

## Projects

List of potential projects: submit at least three ranked preferences by email by 5pm on Tuesday 31st October. Any emails received after this point will have projects allocated on a first-come-first-serve basis.

Project allocations have now been made. Both Matthew and Simon will be at the Problems Class on Thursday 2nd November to give out reading material and briefly discuss what is required of the project.

Projects should be submitted by 5pm on Friday 17th November, and you will get marks and feedback by Friday 1st December.

## Recommended reading

None of these books are required for the course, but could be helpful to supplement the lectures and notes if you get confused.

**Smith et al.,**This course grew out of a reading group centred around this book, so all of the material can be found in here. The book is easy to follow, and presents a variety of examples to illustrate the various topics.*An Invitation to Algebraic Geometry.*

**Reid,**Another good book on the subject.*Undergraduate algebraic geometry*.