Prerequisite Course: Math 2200 or equivalent (diff. calc.) (first 4 chapters of EP): You must have the knowledge taught in 2200, ready for use, not just a passing grade from there.

Course Objectives: acquire a fundamental working knowledge of the ideas and uses of one variable integral calculus, (including definitions, computations, theorems and proofs), of the Riemann integral and its application to practical problems such as finding area, volume, arc length, force and work, and differential equations governing exponential growth.
We cover chapters 5,6,7,8 of E-P. You are responsible for reading these, even if not lectured on, and keeping up with the current lecture topic.

Tests (tentative): 1st 9/10, 2nd 10/6, 3rd 10/29, 4th 11/24.
Final Exam: Friday Dec 12, noon-3pm (in our room).
IMPORTANT: The final cannot be moved. No makeups of missed tests. If you have a valid medical reason that you cannot attend a test, or if you are on a varsity team and must be out of town, tell me IN ADVANCE of the test.
The number of tests, and test dates are tentative and may change, so it is unwise to plan to miss class, especially near a scheduled test date. Check NOW for possible exam conflicts in your courses at
http://www.reg.uga.edu/or.nsf/public/fallkeydates?OpenDocument

GRADING FORMULA: Your grade will not be lower than that given by the formula: 15% HW & quizzes + 60% Test Average + 25% FINAL EXAM. Letter grades (normally): 90-100=A; 80-89=B; 70-79=C; 60-69=D; 0-59=F.
Attendance is Required. 5 absences and you may be dropped (with a WF if failing or if after midpoint 10/14/03). Homework and test problem solutions should be written up clearly legibly in complete sentences, explaining your reasoning (not just calculations) for full credit. I will not grade what I cannot read, and I will not use a magnifying glass.

Academic Honesty: In all work for credit, do your own research, thinking, computations and writeup. You may "brainstorm" with others during problem assignments and you should. Notes, books, and calculators are normally not allowed on tests. Read the University policy at http://www.uga.edu/ovpi/academic_honesty/academic_honesty.htm
NOTE: This syllabus provides a general plan for the course; deviations may
be necessary.
EXPECTATIONS AND ADVICE:
1) LEARN THE BASIC INFORMATION.
This means studying the book and the lectures until you know and understand
all the definitions, theorems, formulas and procedures.
This involves both memorizing and understanding.
Thus you should be able to rattle off from memory the definition of limit,
derivative, continuous function, equation for a tangent line, max, min,
partition, upper and lower sum, Riemann sum, Riemann integral; statement of
product, quotient and chain rules for derivatives, trig identities,
derivatives of elementary functions, inverse function rule, properties of
logs, exponentials, intermediate value theorem, mean value theorem,
fundamental theorem of calculus, substitution rule, integration by parts,
etc... with perfect accuracy. Then you should be able to explain clearly
and correctly what each of these things means.

2) DEVELOP COMPUTATIONAL POWER.
Learn to solve problems and to make detailed and accurate calculations.
This can only be done by working large numbers of problems, not just the
few that are to be handed in. Spend as much time as you need to learn to
work as many problems in the book as possible. I will frequently choose
problems from the book, or similar ones, to put on tests. Study the worked
out examples, attempting them yourself first.

3) PRACTICE LOGICAL REASONING.
One of the main benefits of a mathematics course is learning to make
logical arguments. (This can help in arguing with a judge, the IRS, or
your boss.) It means knowing why the procedures actually work, and
understanding how to use the ideas of the course to solve problems which we
have not explicitly treated in the lectures. It also means learning to
make a clear statement and prove it. Practice by understanding my proofs
and the book's, and attempt some "prove" or "show" problems.

4) ASK QUESTIONS. Get any troublesome points explained well before the test
on that topic. (I am never available for help on the day of a test.) If
you cannot meet my office hours, email me with questions, (at both my
addresses to be sure to reach me). This works very well since I will
answer it from home 7 days a week.
I will test on your understanding of each topic, not just your ability to
repeat things exactly like ones shown on the board. You must be able to
state general principles correctly, apply them to old and new situations,
and write up your solutions in understandable, correct form, using words in
complete sentences. Keep up, and study for the final, since people who do
not do well earlier, or who do not restudy for the final, do not do well on
the final.

Math 8300, introduction to Algebraic Geometry
Fall 2003, Instructor: Smith, Boyd 410, MWF 10:10am
Office Boyd 448, 11:15am-12:15pm, & appt,
542-2595, roy@math.uga.edu
Texts: Basic Algebraic Geometry, Shafarevich, vol. I(primary text)
The Red Book of Varieties and Schemes, 2nd expanded edition, Mumford.
Undergraduate Algebraic Geometry, Miles Reid.
I will also hand out my notes that essentially cover the entire course.

Outline of Topics
We will describe the most basic tools and concepts of algebraic
geometry, in roughly the order of topics below. I want to introduce
enough language to state the important Riemann Roch problem, and to
understand the ingredients in the statement of its (partial) solution, the
Riemann Roch theorem. The proof of the Riemann Roch theorem is best done
with sheaf cohomology, the topic of another course. The importance of the
Riemann Roch theorem cannot be overstated. I will say one thing about it.
If we want to classify all the projective algebraic varieties in the world,
there is a standard approach:
1) classify all abstract algebraic varieties,
2) for each abstract variety, determine all its projective embeddings.
The Riemann Roch theorem is the primary tool in step 2), which can
be studied for a given variety independently of step 1).
My goal is to cover parts I-V of the outline, i.e. to prove Bezout's theorem.
Outline
I. Algebraic sets
decomposition into irreducible components.
the dimension of an irreducible algebraic set.
the differences between affine and projective sets.
Dimension of intersections
II. Algebraic maps
finite maps
Veronese and Segre maps
universal finite maps (normalization)
closedness of all maps on projective sets
birational equivalence
III. Nonsingularity
Zariski tangent spaces
Unique factorization in the local ring of a non singular point
Local equations for subvarieties
IV. Divisors
Weil divisors vs Cartier divisors D, linear equivalence
A principal divisor on a curve has degree zero
Bezout theorem for curves, applications to rationality of curves
The role of divisors in describing maps to projective space
The Riemann Roch problem (find the dimension of the space L(D))
V. Intersection numbers on non singular varieties
Definition of intersection numbers of divisors in general position
bilinearity of intersection product
Invariance under linear equivalence
Moving divisors up to linear equivalence
Definition of intersection numbers of divisors not in general position
Bezout's theorem
Applications to real division algebras
VI. Differentials
Rational and regular differential forms
The canonical divisor class K on a non singular variety
Statement of Riemann Roch for n.s. curves: chi(D) - chi(0) = deg(D),
Statement of Riemann Roch for n.s. surfaces:
2(chi(D) - chi(0)) = D\(D-K); and (over ^) 12chi(O) = K2 + chitop
VII Birational maps
Blowing up a point
Birational maps of surfaces

Prerequisites: Basic concepts of rings and ideals, some elementary field
theory, including the concept of a transcendence basis.

Grading will be based on various possible combinations of HW, tests, and
most important: class presentations of prepared material from my notes and
other books. Regular attendance is expected.