TU
Kaiserslautern  Fachbereich
Mathematik AG Algebra, Geometrie und Computeralgebra Prof. Dr. W. Decker, Prof. Dr. M. Schulze, Dr. Janko Böhm 
Tel.: +49 (0)631/2052730 Zi. 435, 434, 430, Geb. 48 D67663 Kaiserslautern 
decker@mathematik.unikl.de mschulze@mathematik.unikl.de boehm@mathematik.unikl.de 
Seminar: Algorithms in toric and algebraic geometry
Summer Term 2013
The seminar will be Thursdays, 15:3017:00 in 48538. We will begin on 02.05.13.
We will have an organizational meeting on Friday 19.04.13 at 13:45  15:15 in 48538.
Please register online in the URM system.
The seminar will provide a systematic introduction to toric geometry and algorithms in toric and algebraic geometry. Toric varieties are special type of algebraic varieties which are particularly easy to handle. The reason is that they are described by convex polyhedral objects, from which lots of interesting information can be obtained in a direct manner (in contrast to an arbitrary algebraic variety where it is very hard to compute). As an example, the affine 2dimensional space ^{2} can be represented as a toric variety by the positive orthant with the standard integer lattice:
Specifically, we will take a look at the following topics:
Convex
polyhedral cones, affine toric varieties (Adrian Koch)
Projective
toric varieties, fans (Peter Chini)
Divisors
(Raul Epure)
Homogeneous
coordinate rings (Tien Mai Nguyen and Bin Nguyen)
Resolution
of singularities
Cohomology
of line bundles (Sasha Valentinova Bagryanova)
Betti numbers of
toric varieties (Corina Birghila)
Toric
intersection theory (Hiep Dang).
RiemannRoch
theorem (Florian Diebold).
Bezout's
theorem (Sebastian Muskalla).
Stanley's
theorem.
Algebraic
analogues of toric concepts.
Birational
geometry.
Each participant will give a talk in the seminar. There are topics which
are more theoretical or more practical in nature. The practical ones
come with some implementation work in Singular, which then will be
demonstrated in the talk.
The targeted audience are students who have attended the lecture Computer Algebra or the lecture Commutative Algebra in the winter term, or who are attending the lecture Algebraic Geometry in the current summer term.
However, anyone with a basic knowledge of commutative algebra is welcome. Knowledge in algebraic geometry is not required.
The seminar is a good starting point if you plan to write any sort of thesis in computer algebra or algebraic geometry.
W. Fulton 
Princeton University Press 
1993 
ISBN 9780691000497  
D. A. Cox, J. Little, H. Schenck 
AMS 
2010 
ISBN 9780821848197 

G.M. Greuel, G. Pfister  A SINGULAR Introduction to Commutative Algebra  Springer  2002  ISBN 9783540735410 
D. A. Cox, J. Little, D. O'Shea 
Springer  2007 
ISBN 9780387356501 

H. Schenck 
Cambridge University Press  2003 
ISBN 9780521536509 

W. Decker, Ch. Lossen 
Computing in Algebraic Geometry  A Quick Start using Singular  Springer  2005 
ISBN 9783540289920 
W. Decker, G. Pfister 
A First Course in Computational Algebraic Geometry  Cambridge University Press 
2012 
in print 
Module description for the Bachelor courses in Mathematics.
2 SWS seminar
3 credit
points
30h contact
hours for the seminar
60h
selfstudy hours
The seminar will be Thursdays, 15:30  17:00 in 48538.
We will begin on 02.05.13.
Please register online in the URM
system. Write an email to boehm@mathematik.unikl.de
if you have any issues with the date.
There will be organizational meeting in the first week of the semester.
Anyone is wellcome to listen to the seminar. To obtain credit points you will have to give a talk.
Singular:
The open source computer algebra system Singular is being developed in Kaiserslautern. It is one of the leading systems for calculations in polynomial rings. Algorithms implemented in Singular deal with Gröbner bases, free resolutions, polynomial factorization, primary decomposition, and many other problems in commutative algebra.
You can download Singular here, read the online manual is here, and obtain the source code here.
Polymake:
The open source computer algebra system Polymake is one of the leading systems for calculations in convex geometry. It has facilities for visualization and can be accessed directly from Singular.
You can download Polymake here, read the online manual is here.

 Prof.
Dr. Wolfram Decker Zi. 434, Geb. 48 D67663 Kaiserslautern Tel.: +49 (0)631/2055489 decker@mathematik.unikl.de 
 Prof.
Dr. Mathias Schulze Zi. 434, Geb. 48 D67663 Kaiserslautern Tel.: +49 (0)631/2055489 mschulze@mathematik.unikl.de 
 Dr. Janko
Böhm Zi. 430, Geb. 48 D67663 Kaiserslautern Tel.: +49 (0)631/2052730 boehm@mathematik.unikl.de 