MATHEMATICS W4045, Fall 2016

Algebraic Curves

Room: TBA

Time: 1:10-2:25, Tuesday and Thursday

Office Hours: Tuesday and Thursday, 10-11 and by appointment, room number 521

Teaching Assistant: TBA

This is an introduction to algebraic geometry, the study of the geometry of solutions to polynomial equations in several variables. The course will present theorems about algebraic curves of small degree and will develop the interplay between the geometry of affine and projective varieties and the algebra of polynomial rings.

Each of the topics listed below will occupy roughly two weeks of course time:

1. Plane conics

2. Cubic plane curves (elliptic curves) and the group law

3. Affine algebraic sets and the Nullstellensatz

4. Rings of functions on varieties

5. Projective varieties and birational equivalence

6. The 27 lines on a cubic surface

Prerequisites: Basic algebra through Galois theory. Familiarity with complex analysis is not required but algebraic geometry is much closer to complex analysis than to the analytic geometry of multivariable calculus.

Textbook: Miles Reid, Undergraduate Algebraic Geometry (Cambridge University Press, 1988 edition) The text is available online from the author.

We will also be using William Fulton's classic Algebraic Curves (also available online from the author)

Homework will be graded and should be handed in on time; the lowest grade will be dropped.

Final grades will be based on homework (20%), the midterm (30%), and the take-home final (50%).

Midterm: October 25

Final: FINAL EXAM (online December 8, due December 15)

Homework assignments

1st week (due September 15)

2nd week (due September 22)

3rd week (due September 29)

4th week (due October 6)

5th week (not to hand in)

6th week (due November 1)

(Midterm: no homework)

7th week (due November 10)

8th week (due November 17)

9th week (due December 1)

10th week (not to hand in)

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