Classical Algebraic Geometry
Lecture (3 hours per week)
Tutorial (1 hour per week)
Content
Algebraic geometry studies geometric properties of systems of
algebraic equations. Its technical foundations, including commutative algebra, are typically taught in the introductory class
(B5) and subsequent master courses at this university.
Classical algebraic geometry can be understood as that part of
algebraic geometry concerned with concrete classes of
individual algebraic varieties, like points, curves and
surfaces, determinantal varieties, Grassmannians, etc. Many of its concepts were already well developed in the
nineteenth century. While modern results are
extraordinarily useful in the study of classical questions, the focus
in this course will be on examples, geometric ideas, challenging
problems and
on showing the power of general results in applications, rather than on
developing the theory.
Prerequisites
Basic algebra is assumed. Prior exposure to algebraic geometry (B5) will be very
helpful, but is not strictly required. However, the difficulty level and
formal rigour of the presentation
may vary greatly.
Participants are expected to invest time and actively
explore the material together with the instructor.
Slides and problem sheets
Chapter 1: Projective Varieties -
(Slides, Printable A4)
Chapter 2: Review: Morphisms and
Rational Maps, Products and Projections -
(Slides, Printable A4)
Chapter 3: Grassmannians -
(Slides, Printable A4)
Chapter 4: Secant Varieties -
(Slides, Printable A4)
Chapter 5: Rational Functions and Maps -
(Slides, Printable A4)
Chapter 6: Smoothness and Tangent Spaces -
(Slides, Printable A4)
Chapter 7: Hilbert Polynomials and Degree -
(Lecture Notes)
Chapter 8: Curves -
(Lecture Notes)
Chapter 9: Cubic Surfaces -
(Lecture Notes)
1st problem sheet (for 23 April)
2nd problem sheet (for 30 April)
3rd problem sheet (for 5 May)
4th problem sheet (for 12 May)
5th problem sheet (for 19 May)
6th problem sheet (for 26 May)
7th problem sheet (for 2
June)
8th problem sheet (for 9
June)
9th problem sheet (for 16
June)
10th problem sheet (for 23
June)
11th problem sheet (for 30
June)
12th problem sheet (for 7
July)
13th problem sheet (for 14
July)
Twisted Cubic + Line (Macaulay2 code)
Secant variety to the Veronese surface (Macaulay2 code)
Literature
The primary source is
J. Harris: Algebraic Geometry: a first course. Springer GTM 133
(1992)
Further references:
I. Dolgachev: Classical Algebraic Geometry: a modern view. Cambridge
UP (2012)
P. Griffiths and J. Harris: Principles of Algebraic Geometry. Wiley
(1978)
R. Hartshorne: Algebraic Geometry. Springer GTM 52 (1977)
D. Mumford: Algebraic Geometry I: Complex Projective
Varieties. Springer Classics (1976)
Language
English
Time and Place
Tuesdays, 8:15am - 9:45am (Room: D 435)
Wednesdays, 10am - 11:30am (Room: D 435)
Degree programmes
Master (Wahlmodul/Spezialisierungsmodul)
Lehramt
ECTS Credits: 6
Examinations
Oral exams, to be scheduled individually
last edited 14 July 2015
