Franz Lemmermeyer

# Algebraic Geometry

### Topics

• 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, intersection multiplicity.
• III. Schemes: category theory, affine varieties, spectrum, Zariski topology, regular functions, sheaves, local rings, ringed spaces, affine schemes.
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.

### Software

• You will need the sing surf 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 Bordeaux.

### Homework

Homework is always due one week after hand-out except when stated otherwise.

### Books

Here are a few books covering what we did in the first part (and much more):
• 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
Also, check out the lecture notes by David Marker on this page.

### Extra Credit Problems

Solutions to the following problems will be accepted until the end of the semester.
1. 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.
2. 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.

### Schedule

• Mo 09.02.04, Lecture 1: unit circle.
• 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 theorem.
• 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: singular points
• 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 Menelaus.
• 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: Linear systems
• Mo 22.03.04 Midterm 1 (Review; here are some problems). exam; solutions. 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 54/100.
• Th 25.03.04 No class
• Mo 29.03.04 Lecture 12: the genus of curves.
• Th 01.04.04 Lecture 13: Parametrizing Surfaces
• 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); problems; solutions; here are the review problems
• 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 review, problems, solutions.