Kommmutative Algebra und Algebraische Geometrie

Kommutative Algebra und Algebraische Geometrie (315.301)
Franz Winkler
WS 97/98

Time: Tue 16.30 - 18.00 Place: T 212
first lecture: October 7

Short description:

An introduction to commutative algebra and algebraic geometry is presented. Algebraic geometry deals with geometric properties of solution sets of polynomial equations, i.e. with algebraic curves, surfaces, etc. Commutative algebra is the theory of polynomial ideals. These two areas are closely related.

In particular we will deal with algebraic curves in affine and projective space. Interesting problems arise from the determination and analysis of singularities, the decomposition into irreducible components, different representations of curves, or the determination of intersections. The theorem of Bezout on intersection multiplicities will be discussed. Another topic will be the dimension of algebraic sets.

Participants are expected to be acquainted with basic algebraic concepts.

Relevant literatur:

E. Brieskorn, H. Knörrer: Ebene algebraische Kurven, Birkhäuser Boston Inc., 1981

D. Cox, J. Little, D. O'Shea: Ideals, Varieties, and Algorithms, Springer-Verlag, 1992

W. Fulton: Algebraic Curves, The Benjamin/Cummings Publ. Comp., Menlo Park, California, 1974

W. Gröbner: Algebraische Geometrie I, II, B.I. Hochschultaschenbücher, 1968 bzw. 1970

E. Kunz: Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser, Boston-Basel-Stuttgart, 1985

R.J. Walker: Algebraic Curves, Springer, New York-Heidelberg-Berlin, 1978

O. Zariski, P. Samuel: Commutative Algebra I, II, Springer, New York-Berlin-Heidelberg-Tokyo, 1958 and 1960