Graduate Seminar on Advanced Algebra (S4A3) - Algebraic Surfaces - Wintersemester 2015/16

Prof. Dr. Michael Rapoport
Dr. Eugen Hellmann
Kontakt: hellmann (ergänze

Algebraic surfaces are smooth projective varieties of dimension 2 over a fixed algebraically closed base field - classically the field of complex numbers. In this seminar we want to give an introduction to the theory of these surfaces which also will provide a large number of useful examples of algebraic varieties.

The aim of the classification of algebraic surfaces is to classify all surfaces up to birational equivalence. Two surfaces X and X' are called birationally equivalent if there exist open subsets U of X and U' of X' such that U and U' are isomorphic. Equivalently two algebraic surfaces X and X' are birationally equivalent if there exists a surface Y and morphisms f:Y → X and g:Y → X' that are compositions of blow-ups.

We will associate birational invariants to a surface X, e.g. the so called plurigenera or the Kodaira dimension

κ(X)=-1+trdeg Frac(Γ(X,ωX)).

It can be shown that the birational equivalence class of a surface of non-negative Kodaira dimension contains a unique minimal model. Surfaces of Kodaira dimension κ(X)<2 can be classified more explicitly.


We assume familiarity with the basic concepts of Algebraic Geometry, roughly in the amount of chapters II and III of Hartshorne's book.

Time and Place

Tuesday 16-18h, MZ Room 0.006


The detailed program can be found here.

Organizational meeting

Monday 13.07.2015 at 16h (c.t.), MZ Room 0.008.
If you can not come to the organizational meeting please e-mail in advance.


  • A.Beauville: Complex Algebraic Surfaces, LMS Student Texts, vol. 34, Cambridge University Press.
  • R.Hartshorne: Algebraic Geometry GTM, vol. 52, Springer

Last modified: 12. 07. 2015, Eugen Hellmann