Part I is very elementary and should be a lot of fun. I will
discuss the arithmetic of the rational function field in more
detail than usually: resultants and discriminants, Fermat's Last
Theorem for polynomials, Mason's ABC theorem, quadratic extensions,
the Pell equation, and eventually valuations, divisors, and the
Theorem of Riemann-Roch.
Part II is an experiment and should also be a lot of fun even
though the subject may seem very abstract at first.
Prerequisites include the elementary
notions of algebra and topology.
- I. Algebraic curves parametrization, affine and projective plane,
group law, singular points, rational curves, Bezout's theorem.
- II. Coordinate ring, rational function field, valuations, divisors,
- III. Schemes: category theory, affine varieties, spectrum,
Zariski topology, regular functions, sheaves, local rings,
ringed spaces, affine schemes.
- You will need the
program for drawing some of the curves in homework 1. Here's how it
works: after the page has loaded, change the `algebraic surface' in the
`new' menu on the main window into `algebraic curve'. In the control
panel, pull down `inspector' and camera and then click on `Top(X-Y)'.
To get rid of the colors, pull down inspector and display, then
disable asurf in the window `visible geometry'.
You can also display the axes by clicking the appropriate box in
`Inspector' and `Display'. Afterwards, click `Project' in the
`Inspector' menu. Type in the equation of the curve, and don't forget
the ; at the end of your equation. Also, you might want to modify the
domain in the control panel in order to see more of the curve. You
will need a browser with java.
If you want to print curves, right click the main window and select
a new display. Then pull down file and save (as ps); copy the content
of the window into a file and call it curve.ps or something.
- Here's a windows executable of pari.
If you type in ?, you'll get a list of chapters; ?4 lists e.g. the
number theoretical functions, and ?gcd tells you what gcd does.
You can find a more detailed manual at the pari homepage in
Homework is always due one week after hand-out except when stated
Here are a few books covering what we did in the first part (and much more):
Also, check out the lecture notes by David Marker on
- E. Brieskorn, H, Knörrer, Plane Algebraic Curves
- C.G. Gibson, Elementary Geometry of Algebraic Curves
- C. Musili, Algebraic Geometry for Beginners
- M. Reid, Undergraduate algebraic geometry
- P. Samuel, Projective Geometry
Extra Credit Problems
Solutions to the following problems will be accepted until the
end of the semester.
- In Lecture 3 I proved, using the ABC theorem, that the
elliptic curve y2 = x3 + x cannot be
parametrized. Give a similar proof of the more general result
that y2 = x3 + ax2 + bx
cannot be parametrized if the polynomial on the right hand
side does not have multiple roots.
- Assume that you are given a nondegenerate conic defined over Q
with a rational point. Can it be rationally parametrized
using trigonometric functions? I suspect the answer is yes. Here
are a few examples: the hyperbola X2 - Y2 = 1
is parametrized by X = 1/cos t and Y = tan t; the parabola
Y = X2 is parametrized by X = cos t and Y = (cos 2t +1)/2;
and the circle X2 + Y2 = 2 is parametrized
by X = cos t + sin t, Y = cos t - sin t.
- Mo 09.02.04, Lecture 1:
- Th 12.02.04, Lecture 2:
group law on the unit circle.
- Mo 16.02.04, Lecture 3:
Mason's ABC Theorem, FLT for polynomials
Here's the homepage of the
ABC conjecture. Also, here's
N. Snyder's proof of Mason's ABC
- Th 19.02.04, Lecture 4:
affine and projective planes, points at infinity.
If you want to read more about elementary projective geometry,
try the very nice book Projective Geometry by Pierre Samuel.
- Mo 23.02.04, Lecture 5:
projective closure of lines, conics, and other algebraic curves.
Here's a short article on finite fields.
- Th 26.02.04 (13:40!) Lecture 6:
- Mo 01.03.04 no class
- Th 04.03.04 (13:40!) Lecture 7: multiplicity
- Mo 08.03.04 Lecture 8: Projective
transformations; here's a page with a Java applet illustrating
Pascal's Theorem; it remarks that Pascal's original proof is lost,
but gives a proof based on the theorem of
- Th 11.03.04 Lecture 9: Resultants
- Mo 15.03.04 Lecture 10:
Statement of Bezout's Theorem and consequences
- Th 18.03.04 (13:40) Lecture 11:
- Mo 22.03.04 Midterm 1 (Review; here are
I accidentally switched the files for review problems
and those for the exam with the result that the midterm
problems were almost all available online here as practice
problems. Luckily, no one noticed, and the average was
- Th 25.03.04 No class
- Mo 29.03.04 Lecture 12: the genus of curves.
- Th 01.04.04 Lecture 13:
- Mo 05.04.04 Lecture 14:
coordinate rings, polynomial maps
- Th 08.04.04 Lecture 15:
equivalence of categories of plane affine curves
and of their coordinate rings
- Mo 12.04.04 Lecture 16:
function fields, local rings
- Th 15.04.04 Lecture 17:
discrete valuation rings
- Mo 19.04.04 Lecture 18:
exact sequences, vector spaces, . . .
- Th 22.04.04 No class; moved to 29.04.04
- Mo 26.04.04 Lecture 19: multiplicities;
discussion of homework and review problems
- Th 29.04.04 (13:40) Midterm 2 (average 68/100);
solutions; here are the
- Mo 03.05.04 Lecture 20:
Intersection Multiplicities; Riemann Roch (without proofs)
- Th 06.05.04 spectrum of rings
- Mo 10.05.04 Lecture 21: Zariski topology;
- Th 13.05.04 (13:40) Lecture 22:
sheaves and schemes
- Mo 17.05.04 (I will be here
during this week).
- Th 20.05.04 9:00, SAZ-03 (C. Koca) Final