Franz Lemmermeyer
Algebraic Geometry
Topics
- I. Algebraic curves parametrization, affine and projective plane,
coordinate rings, function fields, valuations,
singular points, rational curves, intersection multiplicity,
Bezout's theorem.
- II. Riemann-Roch.
Here's a
link
to last year's course.
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
- M. Reid, Undergraduate algebraic geometry
- P. Samuel, Projective Geometry
- H. Schenck, Computational Algebraic Geometry
The books by Gibson and Reid are very elementary and recommended
for giving you a first idea of what to expect.
Schedule
- 02.02.06 Unit circle. Chapter 1
- 06.02.06 Unique Factorization in K[X], Mason's Theorem
- 09.02.06 Unique Factorization in K[X,Y]
- 13.02.06 Hilbert's Basis Theorem Chapter 2
- 16.02.06 It looks as if from now on we will meet from 11:40 - 12:30.
Classes on Wednesday do not work for quite a few students, and so
far (Mo 23:30) no one has problems with the 11:40 class.
- 20.02.06 Nullstellensatz. I have updated the pdf files.
- 23.02.06 Nullstellensatz
- 27.02.06 Homework discussion
- 06.03.06 Coordinate rings. Polynomial Maps.
- 09.03.06 Review
- 10.03 or 11.03 or 13.03 (morning): Additional tutorial
- 13.03.06 Midterm 1, covering Chapters 1 and 2: parametrization,
Mason's Theorem, basic definitions, simple proofs concerning
maximal, prime, radical ideals. Understanding the correspondence
between radical (prime; maximal) ideals and affine algebraic sets
(varieties; points). Unique factorization, noetherian rings.
Understanding the statements of Hilbert's basis theorem and
the various forms of the nullstellensatz, as well as knowing
what they are good for. Zariski topology. The first three homeworks.
Here are lots of problems to get
some practice. I have added a few hints
here.
Midterm 1 solutions.
Average: 62/100.
- 16.03.06 Discussion of Midterm
- 20.03.06 Rational maps, dominant maps, categories.
Also, please feel free to check out
this
page, in particular the first chapter (the rest will be over
your head). Also, here are the notes by
Bart Snapp, giving an overview over
much that we have been doing (related to Smith et al, Introduction
to algebraic geometry -- it's in the library). We have now covered
Reid's Chapter II, so you may have a look there too.
- 23.03.06 NO CLASS. See April 29 and 30.
Also note that from now on, Thursday's classes are back at 10:40.
- 03.04.06 Dominant maps; projective planes.
Here is the final version of Chapter 3.
- 06.04.06 10:40 -- 11:40 Projective Spaces
- 11.04.06 Projective spaces, tangents, singularities.
Here's a draft of Chapter 4.
- 13.04.06 homework.
- 17.04.06 homework, review
- 20.04.06 NO CLASS. see April 29 and 30.
- 24.04.06 Local rings
- 27.04.06 review; here are some problems.
- 29.04.06 14:00 - 15:00 Cafe in
- 30.04.06 14:00 - 15:00 Cafe in
- 01.05.06 AZ32B, 13:30 - 15:40 Midterm 2: polynomial and rational maps,
domains of rational maps,
and affine and projective varieties (tangents, singularities,
parametrizations, . . .).
Midterm 2 solutions.
Average 55/100.
- 04.05.06 Discussion of midterm 2
- 08.05.06 Multiplicities
- 11.05.06 Multiplicities
- 15.05.06 review
- 17.07.06 final 12:15, SAZ18
- Here's Chapter 5 (minus some proofs)
- Here are some problems.
- Final problems
solutions; average 66/100
- 18.05, 10:30 - 12:00 I'll be in my office
- Next week: Oberwolfach. Mathematicians at work.
- 29.05, 13:30 make up