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Algebraic Geometry II

Prof. Dr. Anne Frühbis Krüger
M.Sc.-Math. Roberto Laface
Lectures

Tuesdays
14:00 - 16:00
(A410)

Thurdays
14:00 - 16:00
(G117)        

Exercise Sessions

Fridays
14:00 - 16:00
(G123)
Aim of the course

The aim of this class is to continue the study of schemes started in the course "Algebraic Geometry I", in order to get to the geometry underlying the abstract machinery introducted by Alexander Grothendieck.

Syllabus

After introducing the concept of projective scheme, which turns out to be the right setup for doing some true geometry, we will focus on applications of scheme theory to the geometry of varieties. Among the topics we will be dealing with, we have:
  • Bezout´s theorem
  • Divisors
  • Elliptic curves
  • Coherent and quasi-coherent sheaves
  • Vector bundles
  • Differentials
  • Cohomology of sheaves
  • Riemann-Roch theorem for curves
  • Riemann-Hurwitz formula
  • Intersection theory

Exercises & additional material

An exercise sheet will be uploaded here and on Stud.IP approximately one week before the class, whereas solutions to the problems will be made available shortly after the class. Also, some notes containing additional material will be made available, if necessary.

References

Algebraic Geometry involves ideas from both the worlds of Algebra and Geometry; thus, in order to understand it properly, one must have a good knowledge of both commutative algebra and elementary algebraic geometry.

As for commutative algebra, I strongly recommend Introduction to Commutative Algebra by Atiyah and MacDonald, a friendly reference if you don't want to get lost in algebraic questions; also Eisenbud's Commutative Algebra with a view towards algebraic geometry is not bad, although too chatty in my opinion. However, one must not forget the ultimate reference Commutative Algebra by Matsumura, as well as many others.

As for geometry, Hartshorne's Algebraic Geometry is the text most people forged themselves with; however there are many better textbooks, where the material is explained in a much clearer way. For instance, the series Algebraic Geometry 1-3 by Ueno is a very friendly reading, and gets the reader to important results quite quickly, without getting lost in algebraic nonsense. Indeed, there are many other sources I might quote, but if it's the ultimate reference you are seeking, then you should look up in Éléments de géométrie algébrique by Alexander Grothendieck (assisted by Jean Dieudonné); this one is impossible to read if you're not an expert, it's difficult to read if you are and the following rule applies: if a result is not written there, it must be false!!