## Module Handbook

• Dynamischer Default-Fachbereich geändert auf MAT

# Course MAT-40-28-K-4

## Course Type

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

## Basedata

SWS 2V+1U 4.5 CP = 135 h 1 Sem. in SuSe [4] Bachelor (Specialization) [EN] English Gathmann, Andreas, Prof. Dr. (PROF | DEPT: MAT) Schulze, Mathias, Prof. Dr. (PROF | DEPT: MAT) [MAT-AGCA] Algebra, Geometry and Computer Algebra [NORM] Active

## 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.

## Contents

Compulsory Topics:
• 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.

## Literature

• 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.

## Materials

Further literature will be announced in the lecture; Exercise material is provided.

## Registration

Registration for the exercise classes via the online administration system URM (https://urm.mathematik.uni-kl.de)

## Requirements for attendance (informal)

More advanced knowledge from the courses [MAT-12-22-K-3] und [MAT-12-26-K-3] is useful but not necessarily required.

None

## References to Course [MAT-40-28-K-4]

Module Name Context
[MAT-40-28-M-4] Plane Algebraic Curves P: Obligatory 2V+1U, 4.5 LP
Course-Pool Name
[MAT-40-2V-KPOOL-4] Elective Courses Algebra, Geometry and Computeralgebra (2V, B.Sc.)