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

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.

  1. 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)
  2. 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)
  3. 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)
  4. 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)
  5. 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>.
  6. Week 6 Workshop 5, Lecture Notes 6 Continued from last week. Separatedness. Segre embedding. (Shafarevich 1.5.1)
  7. Week 7 Lecture notes 7 Closed image property, applications. (Shafarevich 1.5.2)
  8. Week 8: Workshop 7 Lecture notes 8 Irreducibility, irreducible components, rational maps. (Shafarevich 1.3.1, 1.3.2)
  9. Week 9: Workshop 8, Lectures notes 9 Rational maps continued, dimension. (Shafarevich 1.3.3, 1.5.3)
  10. Week 10: Workshop 9, Lecture notes 10 More on dimension. (Shafarevich 1.5.3, 1.5.4)
  11. Week 11: Workshop 10, Lecture notes 11 Dimension of fibers. Applications. Grassmannian. Harris, Bullock)
  12. 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

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

Lectures and workshops

  1. Lecture on Wednesday, 12:00 to 13:00 in Hancock 2.27
  2. Lecture on Thursday, 9:00 to 10:00 in Hayden Allen G051
  3. Lecture on Friday, 12:00 to 13:00 in Hancock 2.27
  4. 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`.

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