This is the website of the seminar in algebraic geometry which is organised by Andrea Petracci in summer semester 2021 at Freie Universität Berlin.

Registration. The students who are interested in this module should register in Campus Management, should register in Whiteboard, and should write an e-mail to me.

Contact information. If you want to contact me, write me at andrea.petracci [at] fu-berlin [dot] de . Please use your FU e-mail address, otherwise your message will likely go into my spam folder.

Topics. The topic will be complex algebraic surfaces. Algebraic surfaces are complex projective varieties of dimension 2. Their geometry is particularly rich and quite different from the geometry of algebraic curves (Riemann surfaces). In particular there exist maps between algebraic surfaces which are isomorphisms on big open subsets, but are not isomorphisms; these maps are called birational maps. The ultimate achievement of the Italian school of algebraic geometry, which flourished in the first decades of the 20th century, was Enriques' birational classification of algebraic surfaces. This is what we aim to study in this seminar.
The birational classification of algebraic curves is just a classification up to isomorphism, because each birational map between smooth algebraic curves is an isomorphism. The birational classification of algebraic surfaces was done by Enriques around 1914; later this was slightly improved and generalised by Kodaira, Mumford and Bombieri, but still only for surfaces. The birational classification of algebraic varieties of dimension at least 3 started with the work of Mori in 1979 and is still a very active research topic nowadays; this research area is called the minimal model program (MMP). Birkar won the Fields medal in 2018 for obtaining extremely important results in birational geometry. For an introduction to birational geometry you can watch this video by Paolo Cascini and this poster by Andrea Fanelli and Diletta Martinelli.

Summary. This document contains the summary of the talks, with appropriate references, and the planned schedule. You can find the notes/slides of the talks below:
04_12_Petracci.pdf
04_19_Petracci.pdf
04_26_Petracci.pdf
05_03_Rasmussen.pdf
05_10_Garzon.pdf
05_17_Rotfogel.pdf
05_31_Petracci.pdf
06_07_Montes.pdf
06_14_Blagojevic.pdf
06_21_Brommer-Wierig.pdf
06_28_Li_Li.pdf
07_05_Li_Yumeng.pdf
07_12_Petracci.pdf

References. We will mainly follow the book Complex Algebraic Surfaces by A. Beauville (2nd edition, London Mathematical Society Student Texts, 34, Cambridge University Press, 1996). Also the following references might be useful.
- L. Bădescu, Algebraic surfaces, Universitext, Springer-Verlag, New York, 2001
- W. Barth, K. Hulek, C. Peters, A. Van de Ven, Compact complex surfaces, 2nd edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 2004
- Chapter V in R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York-Heidelberg, 1977
- M. Reid, Chapters on algebraic surfaces, arXiv:9602006

Prerequisites. Participants should know the basic notions of quasi-projective varieties as in the following books:
- Chapter I in R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York-Heidelberg, 1977
- M. Reid, Undergraduate algebraic geometry, London Mathematical Society Student Texts, 12, Cambridge University Press, Cambridge, 1988
- K. Smith, L. Kahanpää, P. Kekäläinen, W. Traves, An invitation to algebraic geometry, Universitext, Springer-Verlag, New York, 2000
The knowledge of schemes and of coherent sheaves is recommended, but not strictly necessary. Other useful notions might be rudiments of algebraic topology and of complex geometry.

Times and venues. This seminar will be held online on Mondays 2pm-4pm.

Rules. I give the first 3 talks about basic topics (sheaf cohomology, divisors, line bundles). Afterwards each participant have to present certain topics in a 1h30' talk, according to the schedule contained in this document.
Each speaker has to show me their slides on the Thursday or on the Friday before their talk. Please write me in order to fix an appointment. This is the occasion in which I can correct something or give some specific suggestions.
In addition to their talk, each participant is required to attend at least 9 talks.

General suggestions for the preparation of your talk. You have to study the topics which are written in this document. Always think of possible examples and non-examples (I will always ask for them!). You can also use other references and other books, but please be consistent with the usual notation of this seminar.
Your talk should be a coherent presentation. So if you use different references, please create something which totally makes sense; in particular, we do not care if you took theorem A from the book A' and theorem B from the book B' (this is useless information); you do not need to specify the references. You should feel free to rearrange the material and organise it in a different (but substantially equivalent) way, if you think that this improves the understanding of the topics.
If you use slides, please do not make them too full and you should be careful about the speed: slides talks might be too fast. It could be a good idea if you do a rehearsal of your talk, so that you understand if you need to slow down or to cut some topics.
Most importantly, the intended listeners for your talk are your fellow students, not me. So try to be easy and explain things clearly. It is also important that you conscientiously decide what to say (without writing) and what to write.
If you have questions and doubts, you could ask questions to your fellow students or to me - you can always write me an e-mail. In any case, I want to see what you have prepared some days before your talk (let's say on Thursday or on Friday). So please send me your slides/notes in a pdf file via email and we can fix an appointment to talk about them.