Module Handbook

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Course MAT-40-28-K-4

Plane Algebraic Curves (2V+1U, 4.5 LP)

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
CP, Effort 4.5 CP = 135 h
Position of the semester 1 Sem. in SuSe
Level [4] Bachelor (Specialization)
Language [EN] English
Lecturers
Area of study [MAT-AGCA] Algebra, Geometry and Computer Algebra
Additional informations
Livecycle-State [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.

Modules:

Courses

Requirements for attendance (formal)

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.)