# Algebra 3 (Algebraic geometry)

**This is the website of the course taught in 2019.**

Welcome to Algebra 3! This year, algebra 3 will be algebraic geometry. We will study the geometry of subsets of the affine or projective space defined by the vanishing of polynomial equations, or in other words, (quasi)-projective varieties.

Some more practice problems, in addition to the previous practice problems.

## Homework

- Homework 10. Due 5pm, Friday, October 25 Submit
- Homework 9. Due 5pm, Friday, October 18
- Homework 8. Due 5pm, Friday, October 11
- Homework 7. Due 5pm, Friday, October 4
- Homework 6. Due 5pm, Friday, September 27
**Some students correctly observed that Problem 2 can be done very easily, even without using that Z is a projective variety. Yes! I miscalculated its difficulty.** - Homework 5. Due 5pm, Friday, August 30
**Here is a real picture of the conic fibration on the cubic.** - Homework 4. Due 5pm, Friday, August 23
**Here is an animated pencil of conics.** - Homework 3. Due 5pm, Friday, August 16
- Homework 2. Due 5pm, Friday, August 9
- Homework 1. Due 5pm, Friday, August 2.

I have tried to make sure that the homework solutions are correct, but some errors may have slipped in. If you find something that seems off, please let me know.

## Wattle links (ANU only)

## Outline

Here is a preliminary outline of the course. It is undergoing changes as the class progresses, so the later weeks may not be accurate. I will also upload my lecture notes and the workshop handouts here.

- Week 1: Lecture Notes 1 What is algebraic geometry?, Affine space, closed (algebraic) subsets of affine space. Ideals, Hilbert’s basis theorem, Zariski topology. Examples and non-examples. (Gathmann Chapter 0, Shafarevich Section 1.2.1)
- Week 2: Workshop 1, Lecture Notes 2 The ring of regular functions. The ideal associated to a subset of affine space. The nullstellensatz and consequences. (Shafarevich 1.2.2, Shafarevich A.9, Gathmann 1.2)
- Week 3: Workshop 2, Lecture Notes 3 Regular maps between affine algebraic sets, isomorphisms. Category of affine algebraic sets = Category of nilpotent-free, finitely generated algebras. Quasi-affine varieties. (Shafarevich 1.2.3, Danilov)
- Week 4: Workshop 3, Lecture Notes 4 Definition of abstract algebraic varieties. Projective and quasi-projective varieties. (Shafarevich 1.4.1, 1.4.2, Danilov)
- Week 5 Workshop 4, Lecture Notes 5 Regular functions and regular maps on quasi-projective varieties. Veronese and Segre embeddings. (Shafarevich 1.4.1, 1.4.2, 1.4.4, Danilov Harris). Math Stackexchange answer explaining how homogeneous polynomials in X, Y of degree d factor into d homogeneous linear factors. As a result, they have at most d zeros on P<sup>1</sup>.
- Week 6 Workshop 5, Lecture Notes 6 Continued from last week. Separatedness. Segre embedding. (Shafarevich 1.5.1)
- Week 7 Lecture notes 7 Closed image property, applications. (Shafarevich 1.5.2)
- Week 8: Workshop 7 Lecture notes 8 Irreducibility, irreducible components, rational maps. (Shafarevich 1.3.1, 1.3.2)
- Week 9: Workshop 8, Lectures notes 9 Rational maps continued, dimension. (Shafarevich 1.3.3, 1.5.3)
- Week 10: Workshop 9, Lecture notes 10 More on dimension. (Shafarevich 1.5.3, 1.5.4)
- Week 11: Workshop 10, Lecture notes 11 Dimension of fibers. Applications. Grassmannian. Harris, Bullock)
- Week 12: Lecture notes 12 Local ring at a point, tangent spaces, singularities.

## Prerequisites

Algebra 1 and algebra 2. Some knowledge of commutative algebra will help, but is not required.

## References

**Basic Algebraic Geometry, Part I**by I. Shafarevich.- The online notes by A. Gathmann.
**Algebraic varieties: Basic Notions**by V. Danilov.- Field theory notes by Alex Wright.
- Some practice questions for the midterm.

## Lectures and workshops

- Lecture on Wednesday, 12:00 to 13:00 in Hancock 2.27
- Lecture on Thursday, 9:00 to 10:00 in Hayden Allen G051
- Lecture on Friday, 12:00 to 13:00 in Hancock 2.27
- Workshop on Monday, 11:00 to 12:00 in Hanna Neumann 1.58 (starting week 2).

I will have office hours on Wednesday from 1 to 2, on Thusday from 10 to 11, and at other times by appointment.

## Assessment

There will be weekly homework assignments, a mid-semester exam, and a final exam. The exams will be worth 20% each (total 40%) and the assignments will be worth 6.66% each (total 60%). Submit your assignments through wattle by following the “submit” link as a single pdf file (handwritten and scanned or typed). Of the 10 assignments, I will drop the lowest score.

## Policies

### Collaboration

You are allowed, even encouraged, to work with others on assignments, but you must write up your solutions **on your own**. In other words, you **may not** copy someone else’s write-up and you **may not** write your solutions side by side someone else. On your submission, you must write the names of your collaborators. This is a matter of academic honesty; it will not affect your marks.

### Late assignments

I will grant extensions only for medical emergencies with a medical certificate. In accordance with the ANU policy, late assignments will incur a 5% penalty per working day. I will not accept any assignments later than a week. To mitigate the strict late policy, I will drop the lowest assignment score.

### Picture Credits

The images of the surfaces displayed above were created by Herwig Hauser using `surfer`.