Algebraic geometry 2.
New: the students who want to pass the second exam of Algebraic Geometry 2 should register by writing me an e-mail until the 5th of August.
This lecture (4 hrs) + exercises (2 hrs) will be taught in english in the room B 251 Tuesday, Thursday 10:00-12:00 for the lectures and in the room B 046 (Tuesday 16:00-18:00) for the exercises. The first lecture will be on tuesday 18th April. No exercise class that week.
This lecture will give a sequel to the lecture algebraic geometry 1 given in the winter semester.
This Lecture is a sequel to the lecture Algebraic Geometry I of the winter semester 2022/2023. This course will focus on the notion of (quasi-)coherent sheaves and there associated cohomology. We will start with some recollections on projective/proper morphisms, introduced in the winter semester. Then the notion of quasi-coherent sheaf will be introduced and we will give and study the basic examples: quasi-coherent sheaves of ideals and closed subschemes, Picard group of invertible sheaves and divisors. Then we will study general properties of the category of quasi-coherent sheaves on a scheme: the fact it is abelian, its various functoriality. The cohomology of sheaves will then be introduced, and we will end the lectures by giving possible examples and applications: Be'zout theorem, Riemann-Roch theorem (at least for curves), higher direct images of coherent sheaves through a projective morphism, etc...
It is strongly recommended to have taken the course "algebraic geometry 1", although not obligatory. However a good knowledge of the notions of schemes, morphisms of schemes, and the basic examples is required.
References:
A. Grothendieck (et J. Dieudonné), Éléments de Géométrie algébrique, Publications Mathématiques de l' IHES.
U. Görtz and T. Wedhorn, Algebraic Geometry: Part I: Schemes. With Examples and Exercises. Springer.
R. Hartshorne, Algebraic Geometry, Springer
Exercise sheets .
Previous semesters .