Math 767 - Spring 2006 - Algebraic Surfaces

Plan: The course is naturally divided in two parts. In the first one I want to introduce the barely minimal amount of machinery needed to present the Minimal Model Program point of view on the classification of surfaces. This will give us a preliminary, coarse partition of algebraic surfaces into disjoint classes. In the second part of the course we will analyze in greater detail the surfaces provided to us by the Minimal Model Program, and go back to the classical theory.

Text: The course does not have a specific text, though I will mostly follow Debarre's Higher-Dimensional Algebraic Geometry for the first part of the course and Arnaud Beauville's Complex algebraic surfaces for the second part. There is also plenty of relevant material online:

Miles Reid's notes.
Rick Miranda also has a nice set of notes on algebraic surfaces available online.
Ravi Vakil has some class notes for a course he taught on algebraic surfaces (let me, or directly him, know if you have any comments on his notes).

A few comments on these references.
Miles Reid's notes are very informative and fun to read. He also discusses in the final chapters the approach to the classfication with which I want to start.
Rick Miranda's notes do not go into too much detail, but they are easy to read and are very useful to get a good flavour of Enriques classification of surfaces.
Ravi Vakil's class notes have a good amount of details and are fairly complete. They also have a pleasant way of not dwelving too much on the gory details.

Here is the computation of the cohomology of Pⁿ to which I referred to today in class (Mon. Feb 6).

Here is the computation of the splitting type of the pull-back of the tangent bundle of the Fermat quintic threefold X to a line on X (Tue. Feb 21).

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Damiano Testa
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Cornell University
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