Plane Algebraic Curves (2V+1U, 4.5 LP)
|SWS||Type||Course Form||CP (Effort)||Presence-Time / Self-Study|
|-||K||Lecture with exercise classes (V/U)|
|2||V||Lecture||3.0 CP||28 h||62 h|
|1||U||Exercise class (in small groups)||1.5 CP||14 h||31 h|
|(2V+1U)||4.5 CP||42 h||93 h|
Possible Study achievement
- Verification of study performance: proof of successful participation in the exercise classes (ungraded)
- Examination number (Study achievement): 84009 ("Exercise Class Plane Algebraic Curves")
- Details of the examination (type, duration, criteria) will be announced at the beginning of the course.
- Affine and projective spaces, in particular, the projective line and the projective plane,
- Plane algebraic curves over the complex numbers,
- Smooth and singular points,
- Bézout's Theorem for projective plane curves,
- Topological genus of a curve and the genus formula,
- Rational maps between plane curves and the Riemann-Hurwitz formula.
In addition, a selection of the following topics is covered:
- Polar and Hessian curves,
- Dual curves and Plücker’s formulas,
- Linear systems and divisors on smooth curves,
- Real projective curves,
- Puiseux-parameterizations of plane curve singularities,
- Invariants of plane curve singularities,
- Elliptic curves,
- Further aspects of plane algebraic curves.
Competencies / intended learning achievements
The students have studied and understand basic concepts of algebraic geometry in a selected and easily accessible class of algebraic varieties.
- G. Fischer: Ebene algebraische Kurven,
- E. Brieskorn, H. Knörrer: Plane Algebraic Curves,
- E. Kunz: Introduction to Plane Algebraic Curves,
- F. Kirwan: Complex Algebraic Curves,
- R. Miranda: Algebraic Curves and Riemann Surfaces.
Further literature will be announced in the lecture; Exercise material is provided.
Registration for the exercise classes via the online administration system URM (https://urm.mathematik.uni-kl.de)