Algebraic Geometry

Welcome to the Algebraic Geometry class (winter semester 25/26)!

Lectures

The first lecture is on Tuesday 14 October. We have lectures every Tuesday and Friday:

Tutorials

Each week there will be a tutorial with Constantin Podelski:

Course description

Algebraic geometry is a very active research area at the interface between algebra, geometry, number theory and complex analysis. Its basic objects are algebraic varieties, the zero loci of systems of polynomial equations in affine or projective space. Depending on the context these can be studied over the complex or real numbers, but also over the rational numbers, over the integers or over finite fields, which allows for a wide range of applications for instance in cryptography, algebraic statistics and robotics. From a mathematical viewpoint, the interplay between different aspects is the source of many deep relations between algebra, geometry and number theory at the frontier of modern research.

The course will give a gentle first introduction to the basic objects and tools of algebraic geometry: Affine and projective varieties, sheaves, smoothness, dimension, tangent spaces, ... and look at some classical examples such as algebraic curves, cubic surfaces etc. We will assume only very basic knowledge of algebra, all other prerequisites will be developed as needed.

Literature

There are many very good textbooks of various types: 

a) Introductions using the classical language of algebraic varieties:

• Klaus Hulek, Elementare Algebraische Geometrie. Springer Verlag (2012)
• Andreas Gathmann, Algebraic Geometry. Course Notes available here: https://agag-gathmann.math.rptu.de/de/alggeom.php
• Daniel Perrin, Algebraic Geometry: An Introduction. Springer Universitext (2008)
• James S. Milne, Algebraic Geometry. Course Notes available here: https://www.jmilne.org/math/CourseNotes/ag.html

b) Textbooks that pass from algebraic varieties to the modern language of schemes:

• Igor and Sophie Kritz, Introduction to Algebraic Geometry. Birkhäuser Verlag (2021)
• Robin Hartshorne, Algebraic Geometry. Springer Verlag (1977)
• David Mumford, The Red Book of Varieties and Schemes (1999)

c) Textbooks on scheme theory:

• D. Eisenbud and J. Harris, The geometry of schemes. Springer Verlag (2000)
• S. Bosch, Algebraic Geometry and commutative algebra. Springer Verlag (2022)
• Qing Liu, Algebraic Geometry and Arithmetic Curves. Oxford University Press (2002)
• U. Görtz and T. Wedhorn, Algebraic Geometry I: Schemes. Springer Verlag (2020)
• U. Görtz and T. Wedhorn, Algebraic Geometry II: Cohomology of Schemes. Springer Verlag (2023)
• The stacks project (encyclopaedic online reference): https://stacks.math.columbia.edu/

d) Background in commutative algebra:

• M. Atiyah and I. MacDonald, Commutative Algebra. Addison-Wesley (1969)
• D. Eisenbud, Commutative Algebra (with a view towards algebraic geometry). Springer GTM 150 (1995)
• S. Bosch, Algebraic Geometry and commutative algebra (see c)) also includes a self-contained part on commutative algebra.

Course material

Problem sheets, notes and other material will be uploaded on OPAL during the semester.