Math 6421, Algebraic Geometry

Fall 2009

• Instructor: Matt Baker
• Time and place: TuTh 9:30 - 11:00, Skiles 254
• E-mail: mbaker@math.gatech.edu
• Office: Skiles 212
• Office Hours: Tuesday 2-3, Wednesday 3-4

See also:

• Andreas Gathmann's course notes
• J.S. Milne's course notes

Final projects are due Tuesday, December 8.

• Homework assignment #5 (due Thursday, December 3)
• Solutions to Homework #4
• Homework assignment #4 (due Thursday, November 5)
• Solutions to Homework #3
• Homework assignment #3 (due Thursday, October 8)
• Homework assignment #2 (due Thursday, Sept. 24 )
• Solutions to Homework #2
• Homework assignment #1 (due Thursday, Sept. 3 )
• Solutions to Homework #1
• Handout on extension theorems for homomorphisms

Course outline:

This graduate-level course in algebraic geometry. Topics to be covered will include: Affine algebraic sets, affine varieties, the Zariski topology, Hilbert's basis theorem, Hilbert's Nullstellensatz, morphisms between algebraic varieties, regular maps and regular functions, function fields, affine algebras, projective and quasiprojective varieties, abstract varieties, sheaves and locally ringed spaces, introduction to scheme theory, products of varieties, Noether Normalization, dimension theory, Krull's Principal Ideal Theorem, tangent and cotangent spaces, differential forms, smoothness and regularity, regular local rings, separated and complete varieties, blowing up, resolution of singularities, discrete valuation rings, complete nonsingular curves, ramification theory for curves, divisors, intersection multiplicity and Bezout's theorem, the Riemann-Roch theorem and applications, introduction to elliptic curves.

Prerequisites:

Math 6121 and 6122, or permission of instructor.

Exams and final project:

There will be no exams. However, you will be required to write a 5-10 page final paper on a topic of interest to be chosen with the help of the instructor.

Homework:

Homework will be assigned on a regular basis. On the homework sets, collaboration is both allowed and encouraged. However, you must write up yourself and understand your own homework solutions.

Grading Policy:

Homework will count for 50% of your grade, and the final project will count for the other 50%.

Supplemental reading:

There are many good books on algebraic geometry and commutative algebra which would make good supplemental reading, including: "Basic Algebraic Geometry I" by Shafaravich, "Algebraic Geometry" by Harris, "Algebraic Geometry" by Hartshorne, "Algebraic Geometry I" by Ueno, "The Red Book of Varieties and Schemes" by Mumford, "Using Algebraic Geometry" by Cox, Little, and O'Shea, "Undergraduate Algebraic Geometry" by Reid, "An Introduction to Commutative Algebra" by Atiyah and Macdonald, "Commutative Algebra with a View Toward Algebraic Geometry" by Eisenbud, and "Steps in Commutative Algebra" by Sharp.