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Being a not yet enrolled independently supervised graduate student in mathematics, with prospects of applying to American graduate schools hopefully in a 1-2 years' time, I have a background of having mastered Hartshorne - "Algebraic Geometry" with partial complements like Beauville - "Complex Algebraic Surfaces" and Reid - "Chapters on Algebraic Surfaces" to learn the Kodaira-Enriques classification, Kollár - "Lectures on Resolution of Singularities" to understand Hironaka's theorem, and bits of Mukai - "An Introduction to Invariants and Moduli" as introduction to GIT and moduli. Now, after studying minimal Cremona models for birational linear systems on rational surfaces from the article by Calabri and Ciliberto, I have become very interested in the general birational geometry of higher-dimensional algebraic varieties and their minimal model program. As this line of research has been very active and with important breakthroughs in recent years, I have tried to compile a large amount of fundamental articles, lecture notes, reviews and books on the subject, but I do not know which are better, which are outdated and in which order they should be studied. This is a selection of the material I think most important:

Books (added recommendations from the answers below):

  • Lazarsfeld - Positivity in Algebraic Geometry vol I & II, Springer 2004.
  • Debarre - Higher-Dimensional Algebraic Geometry, Springer 2001.
  • Miyaoka, Peternell - Geometry of Higher Dimensional Algebraic Varieties, Birkhauser 2004
  • Matsuki - Introduction to the Mori Program, Springer 2002.
  • Kollár; Mori - Birational Geometry of Algebraic Varieties, Springer 1998
  • Kollár - Rational Curves on Algebraic Varieties, Springer 2001.
  • Hacon; Kovács - Classification of Higher Dimensional Algebraic Varieties, Birkhauser 2010.
  • Kollár; Kovács - Singularities of the Minimal Model Program, CUP 2013 (to appear).
  • Corti (ed.) - Flips for 3-folds and 4-folds, OUP 2007.

Articles & Pre-Prints (added recommendations from the answers below):


I would like to get any advice on how to organize a deep study course on these topics for a 1-2 years period, as it were for an independent study program preparation for a future Ph.D. thesis on birational geometry. I would really appreciate if any of the professionals or advanced students in the field could provide hints for organizing such a guide, to make good use of the time I have till then, and get already a good general background preparation on these advanced topics.

  • For example, what would be a good approach to acquire the required background to work through the new book by Hacon and Kovács?

Thank you very much in advance for any hints on how to proceed.

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2 Answers 2

The standard answer to these type of questions on MO is "ask your (potential) PhD advisors", and I think here it applies even more so than usually. The most productive choice of topics to learn in detail depends on your choice of future research topics, and unless you want to do a PhD on your own, this will depend heavily on your advisor's skill set and interests. No PhD advisor would be equally qualified to supervise any topic in birational geometry.

Having said that, I think Lazarsfeld's books are a huge gap in your list.

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Thanks, I have added Lazarsfeld's book because I in fact have them and consider them great, but were not on my previous lists because not being exactly so specialized to the minimal model program. Besides, I have edited my last paragraph to reflect on my situation. –  Javier Álvarez Nov 21 '12 at 13:16

I would strongly agree with Arend that you should include Positivity in Algebraic Geometry I & II, by Lazarsfeld, (and that you should ask your potential advisor).

However, while you point out that some books have become outdated, it doesn't mean that they aren't worth going through. The book by Kollár and Mori I think is still the standard source for the foundations of the MMP (especially the earlier chapters). You could also see some exercises to supplement that book by Kollár http://front.math.ucdavis.edu/0809.2579

In particular, I would suggest that that is still the right place to start (possibly combined with Lazarsfeld's books). Debarre's book is also a good starting text.

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Thanks, but your link on the exercises points to a paper on computational complexity. –  Javier Álvarez Nov 21 '12 at 13:41
    
Fixed the link. –  Karl Schwede Nov 21 '12 at 15:13
    
Thanks a lot! That seems very useful! –  Javier Álvarez Nov 21 '12 at 15:18

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