Topics

- Affine and projective varieties 

- Morphisms and rational functions 

- Smoothness 

- Dimension 

- Bezout theorem 

- Hilbert theorem on degree and dimension 

- Newton polytopes and a touch of toric varieties 

- A touch of category theory and schemes - Sheaves, vector bundles and divisors 

- Riemann-Roch theorem 

What's new!

o Welcome! First class is on January 5th. 

o I will mainly follow Karen Smith’s An invitation to algebraic geometry. Although I will cover material f rom other places, specially in the second half of the course. 

o I will assume some familiarity with algebra such as rings, ideals, modules etc. I will briefly recall notions such as localizations and integral closure but previous knowledge of these would help. You can find these standard material in many texts such as  

o Jan. 28: I gave a quick introduction to Grobner bases (of ideals in a polynomial ring).  A good reference is Dummit and Foote Section 9.6. A standard and detailed reference is Sturmfels’ Grobner bases and convex polytopes.

o So far we have covered Chap. 1 and 2 from Smith. We also gave complete proof of Hilbert Nullstellensatz (from Milne’s online book
o Feb. 1: HW1 posted.
o Feb. 3: I fixed a typo in Problem 7 in HW1. One of the equations should be x^2 - y^2 - z^2 + 1 =0.
o March 13: HW2 posted.
o April 10: HW3 posted. It is due on April 20. There are 5 problems. Let me know if you need more hints or help.
o Final exam is posted. It take-home and is due by Tuesday noon. You can give it to me in person (in my office) or email your solutions to me.

Some sources

o “An invitation to algebraic geometry” by Karen Smith et. al. 

o “Algebraic geometry” by Hartshorne 

Hartshorne is by now a classic text for algebraic geometry but keep in mind that it is not a first course in algebraic geometry. It implicitly presumes familiarity with the origins of algebraic geometry. It was one of the first texts to present algebraic geometry in Grothendieck’s (more abstract and powerful) language of “schemes”. 

o “Algebraic geometry” by J. S. Milne. This is a quite friendly introduction to algebraic geometry. As a book it is not yet complete. 

Milne’s “Algebraic geometry” 

o “Basic algebraic geometry” by I. R. Shafarevich 

o Ravi Vakil’s notes. He has two sets of notes, one introductory: Ravi Vakil’s course ntoes (“Introduction to algebraic geometry”) 

and the other one more advanced: 

Ravi Vakil’s notes (“Foundations of algebraic geometry”) 

o “Algebraic geometry: an introduction” by Daniel Perrin. Nice text book. The whole text book is online in Pitt library.

Homeworks

Homework 1 and some practice problems
Homework 2 and some practice problems
Homework 3

Take-home final exam

Course Information

o Text: An invitation to algebraic geometry by Karen Smith et. al. (although I will use other sources as well specially during the second half of the course). 

o Time: MWF 9:00AM - 9:50AM 

o Location: Thackeray 525 

o There will 3 homeworks and a take-home final exam. 

o Office hours: 10AM-11AM Mondays.

Instructor's Information

Email: kavehk AT pitt.edu 

Office: Thackeray #424 

Office hours: Mondays after the class or by appointment (I am usually in you can drop by). 

Office phone: 412-624-8331