Algebraic Geometry – Prerequisites to Begin Beauville’s Complex Algebraic Surfaces

Question

I'm going to read through the first three chapters of Arnaud Beauville's Complex Algebraic Surfaces. My background in algebraic geometry only consists of Gathmann's algebraic geometry notes (the part before schemes are introduced), but the point of view is quite different: Gathmann is more on the algebraic side, oriented towards schemes; instead Beauville has a more classical point of view, coming from complex geometry. For example, I don't know about complex manifolds, divisors, singularities, which seem to be prerequisites for Beauville's book. Is there any introduction to complex geometry / complex surfaces that you would recommend, in order to move on Beauville's book later? Thank you

Answer 1

I recently organized a reading course on algebraic surfaces using this book. To follow the book, you will probably need to understand some foundational concepts from algebraic/complex geometry:

1. line bundles, divisors, and their correspondence
2. cohomology of sheaves on algebraic varieties
3. knowledge of the theory of algebraic curves (e.g. Riemann-Roch and Riemann-Hurwitz)
4. at certain points it is also good to know a little bit about manifold topology; e.g. it is good to have some knowledge of singular cohomology and you should know about the exponential exact sequence on a complex manifold

As is sometimes the case in algebraic geometry, the treatment is more in the classical language of varieties and won't test your scheme theory knowledge very often. The best thing to have a handle on here is some basics on complex manifolds (in particular Riemann surfaces) and sheaf cohomology. Pretty much all of the necessary part of the theory related to complex manifolds and sheaf cohomology is in Huybrechts' book Complex Geometry: An Introduction. The Riemann surfaces are treated nicely in Miranda's book.