Module Handbook

  • Dynamischer Default-Fachbereich geändert auf MAT

Module MAT-41-21-M-7

Curves in Projective Space (M, 4.5 LP)

Module Identification

Module Number Module Name CP (Effort)
MAT-41-21-M-7 Curves in Projective Space 4.5 CP (135 h)

Basedata

CP, Effort 4.5 CP = 135 h
Position of the semester 1 Sem. irreg.
Level [7] Master (Advanced)
Language [EN] English
Module Manager
Lecturers
Area of study [MAT-AGCA] Algebra, Geometry and Computer Algebra
Reference course of study [MAT-88.105-SG] M.Sc. Mathematics
Livecycle-State [NORM] Active

Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
2V MAT-41-21-K-7
Curves in Projective Space
P 28 h 107 h - - PL1 4.5 irreg.
  • About [MAT-41-21-K-7]: Title: "Curves in Projective Space"; Presence-Time: 28 h; Self-Study: 107 h

Examination achievement PL1

  • Form of examination: oral examination (20-30 Min.)
  • Examination Frequency: irregular (by arrangement)
  • Examination number: 86187 ("Curves in Projective Space")

Evaluation of grades

The grade of the module examination is also the module grade.


Contents

  • singularities of plane curves, Puiseux series,
  • projective curves in n-dimensional space, Castelnuovo's inequality,
  • classification of curves and moduli spaces,
  • Jacobian variety, Abel’s Theorem.

Competencies / intended learning achievements

Upon successful completion of this module, the students have learnt the central concepts, results and methods of a selected branch of algebraic geometry. They have thereby developed a deeper understanding of algebraic curves and of their classification.They are able to name the essential propositions of the lecture as well as to classify and to explain the connections. They understand the proofs presented in the lecture and are able to comprehend and explain them. In particular, they can critically assess the conditions and assumptions that are necessary for the validity of the statements. Moreover they can apply the techniques of the proofs to other questions in algebraic geometry.

Literature

  • R. Miranda: Algebraic curves and Riemann surfaces,
  • F. Kirwan: Complex algebraic curves,
  • W. Fulton: Algebraic curves.

Requirements for attendance (informal)

Basic knowledge of projective geometry, e.g from the courses [MAT-40-28-K-4] or [MAT-40-12-K-7]; basic knowledge of Complex Analysis, e.g. from the course [MAT-12-24-K-3].

Modules:

Requirements for attendance (formal)

None

References to Module / Module Number [MAT-41-21-M-7]

Module-Pool Name
[MAT-41-MPOOL-7] Specialisation Algebraic Geometry and Computer Algebra (M.Sc.)
[MAT-RM-MPOOL-7] Pure Mathematics (Advanced Modules M.Sc.)