## Module Handbook

• Dynamischer Default-Fachbereich geändert auf MAT

# Module MAT-40-28-M-4

## Module Identification

Module Number Module Name CP (Effort)
MAT-40-28-M-4 Plane Algebraic Curves 4.5 CP (135 h)

## Basedata

CP, Effort 4.5 CP = 135 h 1 Sem. in SuSe [4] Bachelor (Specialization) [EN] English Gathmann, Andreas, Prof. Dr. (PROF | DEPT: MAT) Gathmann, Andreas, Prof. Dr. (PROF | DEPT: MAT) Schulze, Mathias, Prof. Dr. (PROF | DEPT: MAT) [MAT-AGCA] Algebra, Geometry and Computer Algebra [MAT-88.105-SG] M.Sc. Mathematics [NORM] Active

## Notice

Without a proof of successful participation in the exercise classes, only 3 credit points will be awarded for the module.

## Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
2V+1U MAT-40-28-K-4
Plane Algebraic Curves
P 42 h 93 h
U-Schein
- PL1 4.5 SuSe
• About [MAT-40-28-K-4]: Title: "Plane Algebraic Curves"; Presence-Time: 42 h; Self-Study: 93 h
• About [MAT-40-28-K-4]: The study achievement "[U-Schein] proof of successful participation in the exercise classes (ungraded)" must be obtained.

## Examination achievement PL1

• Form of examination: oral examination (20-30 Min.)
• Examination Frequency: each semester
• Examination number: 84112 ("Plane Algebraic Curves")

## 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

Upon successful completion of the module, the students have learnt basic concepts of algebraic geometry by studying a class of algebraic varieties which is accessible by selected and simple methods. They are able to name and to prove the essential statements of the lecture as well as to classify and to explain the connections.

By completing the given exercises, the students have developed a skilled, precise and independent handling of the terms, propositions and techniques taught in the lecture. In addition, they have learnt how to apply these techniques to new problems, analyze them and develop solution strategies independently or by team work.

## 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.

## Registration

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

## Requirements for attendance of the module (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 Module / Module Number [MAT-40-28-M-4]

Course of Study Section Choice/Obligation
[INF-88.79-SG] M.Sc. Computer Science [Core Modules (non specialised)] Formal Fundamentals [WP] Compulsory Elective
[MAT-88.105-SG] M.Sc. Mathematics [Core Modules (non specialised)] Pure Mathematics [WP] Compulsory Elective
[MAT-88.706-SG] M.Sc. Mathematics International [Core Modules (non specialised)] Pure Mathematics [WP] Compulsory Elective
Module-Pool Name
[MAT-GM-MPOOL-5] General Mathematics (Introductory Modules M.Sc.)