Math 672: Algebraic Geometry

WHEN: MWF 3 – 3:50 pm WHERE: ENG E 104 // in the oval with good weather WHAT: The time has finally come when I have to teach my own subject! Yikes!

Togliatti Surface (quintic with 31 ordinary double points)

Algebraic geometry: the study of the geometry of solution sets of systems of polynomial equations. Sounds simple enough doesn’t it? It turns out that while the first steps and ideas in algebraic geometry are indeed simple and clear, in order to make further progress the theory took a turn in the direction of high abstraction. On the one hand, this expanded the reach of algebraic geometry to a much broader scope interacting with many more areas of mathematics (there is hardly any area of mathematics that has been left untouched by algebraic geometers, or that has not been affected by the developments of algebraic geometry); on the other hand, it made algebraic geometry a difficult to approach subject. The goal of this first semester is to go back to the origins of the subject in order to capture the initial geometric ideas and motivations – but keeping an eye to how these ideas give birth to the modern formulations and technological traits of the theory. The following book provides a nice skeleton for achieving this goal, and you are all encouraged to grab a copy of it!

HOMEWORK (really? Really!)

There are several good exercises in the book and we will be mostly drawing from that pool!

MONDAY Aug 27: Exercises 1.1.1 and 1.2.2.

FRIDAY Sep 7: Exercises 2.1.4, 2.1.5, 2.2.2, 2.3.3, 2.5.1 and 2.5.2.

MONDAY Sep 17: Exercises 3.1.1, 3.2.1, 3.3.1, 3.3.2, 3.5.1, 3.5.2

MONDAY Sep 24: Exercises 4.1.1, 4.2.1, 4.2.2, 4.2.3, 4.3.2.

MONDAY Oct 8: Exercises 5.6.1, 5.6.2, 5.6.3.

FRIDAY Oct 16: Exercises 6.1.2, 6.1.4, 6.4.1.

VARIOUS REFERENCES FOR THIS CLASS:

1. Joe Harris. Algebraic geometry, a first course.

2. David Eisenbud. Commutative algebra with a view towards algebraic geometry.

3. Miles Reid. Undergraduate Algebraic Geometry.

4. Igor Shafarevich. Basic Algebraic Geometry.