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

Prof. Dr. Michael Rapoport
Dr. Eugen Hellmann
Kontakt: hellmann (ergänze @math.uni-bonn.de)

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.

## Prerequisites

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

## Program

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.

## References

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