Special Topics in

ALGEBRAIC GEOMETRY 1

Math 689-601 Fall 1993

Instructor: Peter Stiller
Office: Milner Hall 215
Office Phone: 5-5727
E-mail: stiller@alggeo.tamu.edu
Office Hours: By appointment

Time: Tuesdays and Thursdays 11:10 to 12:25
Place: Fran 305
Text: "Algebraic Geometry: A first course" by Joe Harris, Springer Graduate Text in Mathematics (1992)
Supplementary Material: See bibliography below
Prerequisites: The equivalent of Math 416 or Math 654 (Rings and Modules) and background in Linear Algebra.

Overview: Algebraic Geometry is a subject with historical roots in analytic geometry. At its most naive level it is concerned with the geometry of the solutions of a system of polynomial equations. In its early days the subject developed around the classification problem, the search for invariants of transformations, intersection problems, and the study of families of points on a curve or curves on a surface (known as linear systems). It made use of techniques from geometry (projective geometry), number theory (Diophantine equations), and analysis (elliptic and abelian integrals). Today it is a powerful synthesis of those algebraic, geometric, and analytic techniques. Results can be universally applied to a range of problems from the discrete (such as the recent proof of Fermat's Last Theorem) to the continuous (global complex analysis). It subsumes most of commutative algebra and much of algebraic number theory, and overlaps with differential geometry, modern "analytic geometry" (complex manifolds), Lie groups, representation theory, theoretical physics, and to a lesser extent the theory of partial differential equations. In addition to being one of the central disciplines of pure mathematics, algebraic geometry has developed an applied side which is linked to problems in computational complexity and the theory of algorithms, symbolic computation, robotics, control theory, computational geometry, geometric modeling, image recognition, computer vision, and scientific visualization.

Syllabus:

8/30 Review of Commutative Algebra
9/2 Hilbert Basis Theorem, Noether Normalization, Nullstellensatz
9/7 Primary Decomposition and Discrete Valuation Rings
9/9 Affine and Projective Varieties
9/14 Affine and Projective Varieties (continued)
9/16 Regular Functions and Regular Maps
9/21 Veronese and Segre Maps, Fiber Products
9/23 Cones, Projections, and Products
9/28 Ideals of Varieties
9/30 Irreducible Decomposition
10/5 Dimension
10/7 Rational Functions
10/12 Rational Maps
10/14 Blow-ups
10/19 Examples
10/21 Rational Functions and Maps
10/26 Rational Maps and Blow-ups
10/28 Exam
11/2 Exam
11/4 Dimension
11/9 More on Dimension
11/11 Smoothness and Tangent Spaces
11/16 More on Smooth Varieties
11/18 Elliptic Curves
11/23 Elliptic Curves (cont.)
11/25 Thanksgiving
11/30 Curves
12/2 Curves (cont.)
12/3 RESEARCH PROJECTS DUE
12/7 Bezout's Theorem, Bertini's Theorem, Riemann-Roch Theorem

Basis for Grade: Mid-term Exam (20%), Final Exam (20%), Minor Research Project (15%), Major Research Project (30%), Homework Sets (15%). Both exams will be take-home exams. Homework problems will be assigned weekly.

Bibliography:

Hartshorne, "Algebraic Geometry", Springer GTM #52

Griffiths and Harris, "Principles of Algebraic Geometry" Wiley Interscience

Shafarevich, "Basic Algebraic Geometry", Springer

Kirwan, "Complex Algebraic Curves", Cambridge Univ. Press

Silverman, "The Arithmetic of Elliptic Curves" Springer GTM #106

Assignment #1 Read Sections 1,2, and 3 of the Notes. Also look at Section 0 which gives a preview of how commutative algebra relates to geometry - you may not understand everything in Section 0, but Sections 1,2, and 3 should be mostly familiar territory.

Try all the exercises especially the following:

Section 1: 1-6, 8,9, 13, 14
Section 2: 2,3,6,7,9,10,
Section 3: 3,6, 9,10

Assignment #2 Read Section 4 of the Handout and Lecture 1 in the Text. Do Exercises 1.3, 1.6, 1.12, 1.19, 1.21, 1.28 . These are due in class on Tuesday, September 7.

Assignment #3 (for 9/16) Read pages 17 - top of 23 from Lecture 2 in the Text. Section 1.1.4 (pages 6-8) of the xeroxed handout might be helpful if you find Harris tough going. Also read through the homework solutions I handed out.

Try the following exercises:

Try to calculate the ring of regular functions on P.
What are all possible regular maps from P to A ?

Do Exercises 2.2 and 2.3. These are due in class on Thursday, September 16.

Assignment #4 (for 9/21) Read pages 23 - 31 from Lecture 2 in the Text. Also read the xerox handout Section 2 and 3 pages 14-30.

Try the following exercises:
Text 2.5, 2.15, 2.20. 2.22, 2.24, 2.26
Handout Section 2 Exercises 6,14

Do the following exercises:
Text 2.8 and 2.9
Handout Section 2 Exercises 4,5
Handout Section 3 Exercises1,5.
These are due in class on Tuesday, September 21.

Assignment #5 (for 9/23) Read the xerox handout Section 4 pages 30-34. Also review primary decomposition from the commutative algebra handout from earlier in the course

Do the following exercises:
Text 2.20. 2.22, 2.24
Handout Section 3 Exercises 8,9.
Handout Section 4 Exercises 1,2.
These are due in class on Tuesday, September 28.

Lecture on Fermat's Last Theorem on Monday, September 27

Assignment #6 (for 10/5) Read the xerox handout Section 4 pages 34-41. Also review Lecture 2 in Harris and see if it is clearer to you.

Lecture on Fermat's Last Theorem on Monday, October 4, 4:00 pm

Assignment #7 (for 10/21) Read the xerox handout "Some Topological Considerations" and do the exercises therein to turn in on Thursday 10/21.
Reminder: Take Home Midterm will be handed out on Tues. 10/26.

Assignment #8 (for 10/26) We will begin the study of the concept of rational map. As a review of some of the algebra read through Lecture 5 of the Text and then read and study Lecture 7 pages 72 to 81 including Example 7.17 Blowing Up Points. This material is also covered in Shafarevich pages 24-30 and 38-40.

Do the following exercises (due Tuesday 10/26):
Handout Section 4 Exercises 7,9,10,11
Text Exercises 7.7, 7.12, 7.13, 7.14 (You might need to look at Examples 3.1 and 3.3 to do 7.14)

Assignment #9 (for 11/4) We will begin the study of the concept of dimension. Look at the beginning of Lecture 11 in the Text and then read and study pages 53-65 of the xeroxed handout. Also read the handout on the Projective Geometry of Quadrics and other topics. There will be no class on Thursday 10/28 or Tuesday 11/2. Your exam will be due in class on Thursday 11/4.

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