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. Riemann-Roch.
Software
- You will need the SingSurf program for drawing some
of the curves in your homework. Unfortunately, the official
site is not working anymore, so I put my version
here
(it should work on windows platforms).
Download it, unzip and install it.
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 will do (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
- H. Stichtenoth, Arithmetic of Function Fields
Also, check out the lecture notes by David Marker on
this
page.
Schedule
- Th 03.02.05 The Unit Circle
- Tu 08.02.05 The group law on the unit circle;
Finite Fields
and Unique Factorization Domains. Here's a
link
to a page with a more detailed explanation.
- Th 10.02.05 Mason's Theorem
- Tu 15.02.05 Projective Spaces
- Th 17.02.05 *Extra Class* Questions, Answers, Problems
- Tu 22.02.05 Study's Lemma; Tangents
- Th 24.02.05 No class
- Tu 01.03.05 Tangents in the Projective Plane
- Th 03.03.05 Multiplicity
- Tu 08.03.05 Projective Transformations
- Th 10.03.05 Conics and Cubics
- Tu 15.03.05 Resultants (corr. 02.04.05)
- Th 17.03.05 no class (CHANGE OF SCHEDULE!)
- Tu 22.03.05 Bezout's Theorem
- Th 24.03.05 review problems for the midterm
- Tu 29.03.05 The genus
- Th 31.03.05 midterm 1,
covering material up to March 15.
Solutions
- Tu 05.04.05 algebraic varieties.
See also Reid's book.
- Th 07.04.05 no class
- Tu 12.04.05 spring break
- Th 14.04.05 spring break
- Tu 19.04.05 Correspondence between ideals and varieties
- Th 21.04.05 Hilbert's Nullstellensatz
- Tu 26.04.05 Coordinate rings
- Th 28.04.05 polynomial maps
- Tu 03.05.05 abstract nonsense: functors and categories
- Th 05.05.05 No class
- Tu 10.05.05 local rings and smooth points;
CHANGE OF SCHEDULE! midterm 2 postponed.
- Th 12.05.05 midterm 2 (review sheet;
updated 03.05.05) solutions
Average: 54.
- Mo, 23.05.05 final
- For some of you (and me), the final was a disaster.
Here's my offer: there is a make-up exam Wed May 25
at 15:00 (we meet in my office) for two students who
missed the final. If you think you can do better, feel
free to join in. The make-up will cover about the same
material as the final but will be more difficult. If
you take the make-up, I will count only the make-up,
whether you do better than on the final or not.
Send me an email so I can prepare the exams.