Module Handbook

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Module MAT-40-12-M-7

Algebraic Geometry (M, 9.0 LP)

Module Identification

Module Number Module Name CP (Effort)
MAT-40-12-M-7 Algebraic Geometry 9.0 CP (270 h)

Basedata

CP, Effort 9.0 CP = 270 h
Position of the semester 1 Sem. in WiSe
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.
4V+2U MAT-40-12-K-7
Algebraic Geometry
P 84 h 186 h - - PL1 9.0 WiSe
  • About [MAT-40-12-K-7]: Title: "Algebraic Geometry"; Presence-Time: 84 h; Self-Study: 186 h

Examination achievement PL1

  • Form of examination: oral examination (20-30 Min.)
  • Examination Frequency: each semester
  • Examination number: 84110 ("Algebraic Geometry")

Evaluation of grades

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


Contents

Compulsory Topics:
  • affine and projective varieties (especially: dimension, morphisms, smooth and singular points, blow-ups of points, applications and examples),
  • sheaves and sheaf cohomology with applications (Riemann-Roch Theorem for curves, projective embedding of a curve).

In addition, a selection of the following topics is covered:

  • schemes,
  • differential forms,
  • further aspects of Algebraic Geometry.

Competencies / intended learning achievements

Upon successful completion of this module, the students have studied the basic terms, propositions and techniques of algebraic geometry. They have developed a profound understanding of the interplay between geometric and algebraic problems. They are able to name and to prove the essential propositions 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. They have learnt how to apply these techniques to new problems, analyze them and develop solution strategies independently or by team work. Moreover, they are able to present their solutions and to support their arguments using these in discussions.

Literature

  • D. Cox, J. Little, D. O'Shea: Ideals, Varieties, and Algorithms,
  • I. Dolgachev: Introduction to Algebraic Geometry,
  • J. Harris: Algebraic Geometry,
  • R. Hartshorne: Algebraic Geometry,
  • D. Mumford: The Red Book of Varieties and Schemes,
  • H.A. Nielsen, Algebraic Varieties,
  • M. Reid: Undergraduate Algebraic Geometry,
  • I. R. Shafarevich: Basic Algebraic Geometry 1: Varieties in Projective Space.

Requirements for attendance (informal)

Knowledge from the course [MAT-40-28-K-4] is useful but not necessarily required.

Modules:

Courses

Requirements for attendance (formal)

None

References to Module / Module Number [MAT-40-12-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.)