Wintersemester 2017, JGU Mainz
A sketch of Spec Z[x] by David Mumford


Dino Festi

Davide Cesare Veniani

Time and venue

The lectures are given on
Mondays, 14 - 16 in Room 04-522 and on
Thursdays, 10 - 12 in Room 03-424.


In 1960, Alexander Grothendieck introduced the notion of Scheme in his treatise Éléments de géométrie algébrique. This notion developed a new formalism that led to the proofs of many long standing problems in algebraic geometry, such as, among others, the Weil conjectures and Fermat’s Last theorem, starting a new era for algebraic geometry. It also led to more rigorous proofs of classical results in algebraic geometry, like Bertini’s theorem.

The aim of this course is to make acquaintance with the definition of a divisor of a scheme. We will see how divisors are important to study maps of a given variety into a projective space, and also to study families of varieties. In doing so, we will follow Hartshorne's book Algebraic Geometry, the second half of Chapter II. If time permits, we will also introduce the notion of formal scheme.


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

The interested student may also read The red book of varieties and schemes, by Mumford [MVS].

For reference to commutative algebra, we suggest the book Introduction to Commutative Algebra, by Atiyah--MacDonald [AM], and/or Commutative Algebra with a view toward Algebraic Geometry by Eisenbud [EIS].


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


Date Topics Lecturer Homework
Th. October 19 Weil divisors Davide Exercises
Mo. October 23 Discussion on the exercises Davide
Th. October 26 Examples of Weil divisors and definition of Cartier divisors. Davide Exercises
Mo. October 30 Discussion on the exercises Davide
Mo. November 6 Cartier divisors and invertible sheaves Davide Exercises
Th. November 9 Discussion on the exercises Davide
Mo. November 13 Serre's twisting sheaves Davide Exercises
Th. November 16 Very ample divisors Davide
Mo. November 20 Ample divisors Davide
Th. November 23 Divisors on curves Davide
Mo. November 27 Local systems Davide Exercises
Th. November 30 Introduction to the study of curves Davide
Mo. December 4 Definition of Proj Dino Exercises
Th. December 7 Associated projective space bundle Dino
Mo. December 11 Blow up Dino Exercises
Th. December 14 Properties of the blow up Dino
Th. December 21 Discussion on the exercises Dino
Mo. January 8 Kähler differentials Dino
Th. January 11 Sheaves of differentials Dino Exercises
Mo. January 15 Nonsingular varieties Dino
Th. January 18 The geometric genus Dino Exercises
Mo. January 22 Kodaira dimension Dino
Th. January 25 Discussion on the exercises Dino
Mo. January 29 Discussion on the exercises Dino