Sommersemester 2018, JGU Mainz
A flat family of subschemes of the projective space


Dino Festi

Davide Cesare Veniani

Time and venue

The lectures are given on
Wednesdays, 10 - 12 in Room 04-432 and on
Thursdays, 10 - 12 in Room 04-432.


Sheaf cohomology — and, in particular coherent sheaf cohomology — is a fundamental tool in modern algebraic geometry. Not only does it allow to prove and even state basic theorems such as the Serre Duality Theorem and the Riemann—Roch Theorem, but it also provides crucial invariants for many classification problems.

Following the third chapter of Hartshorne’s book, we will first introduce sheaf cohomology from the point of view of derived functors. Then we will study the connection to Čech cohomology. As applications, after computing the cohomology of twisted line bundles on the projective space, we will prove the Serre Duality Theorem and we will study flat morphisms of schemes. If time permits, we will also review the Theorem on Formal Functions and the Semicontinuity Theorem, with a view towards their classical corollaries, namely Zariski’s Main Theorem, Stein Factorization and Grauert’s Theorem.


We shall follow Hartshorne's Algebraic Geometry [HAG], the third chapter.

Additional references are the books by Ueno, Algebraic geometry 2 & 3.


The exam will be an oral exam at the end of the course.


Date Topics Lecturer Homework
We. 18 April Section III.1 Dino
Th. 19 April Section III.2 Dino Exercises III.2.1a, 2, 7a, II.1.16
We. 25 April Section III.2 Dino
Th. 26 April Discussion on the exercises Dino
We. 2 May Section III.3 Dino
Th. 3 May Section III.3 Dino Exercises III.3.1, 2, 8.
We. 9 May Discussion on the exercises Dino
We. 16 May Section III.4 Dino Exercises III.4.1, 3, 7.
Th. 17 May Discussion on the exercises Dino
We. 23 May Section III.5 Dino
Fr. 25 May Section III.5 Dino
We. 30 May Section III.6 Davide Exercises
We. 6 June Section III.7 Davide Solutions
Th. 7 June Section III.7 Davide
We. 13 June Section III.8 Davide Exercises
Th. 14 June Cohen-Macaulay rings Davide Solutions
We. 20 June Section III.9 Davide
Th. 21 June Section III.9 Davide Exercises
We. 27 June Section III.12 Davide
Th. 28 June Section III.12 Davide
We. 4 July Section III.11 Davide
Th. 5 July Discussion on the exercises Davide Solutions