**Algebraic
Curves**

Room: TBA

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

Instructor: Michael Harris

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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