Sommersemester 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 Thursdays, 12 - 14 in Room 04-516,


Wednesday 24th May, instead of Thursday 25th May, and
Wednesday 14th June, instead of Thursday 15th June.
In these days the lectures are held in Room 04-426, at 10 - 12.


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.

The aim of this reading course is to make acquaintance with the definition of a scheme, investigate their first properties and study morphisms between schemes.

In doing so, we will follow Hartshorne's book Algebraic Geometry.


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 is a take away exam in two parts.

On the fourth and the eleventh class, a list of exercises will be given to the students.

The students will have one week to solve the exercises on the list. The due date to hand the exercises in is the beginning of the next lecture.

During the fifth and the twelfth class, the solutions of the exercises will be shown.

The graded homework can be picked up in Davide's office (Room 04-427)


Date Topics Lecturer Homework
April 20 [Hag 2.1] Dino (Optional) Read definitions of affine and projective varieties, and regular functions on varieties [HAG, pp. 3, 10, 15]. Do Exercises [HAG, II.1.9] and [HAG, II.1.17].
April 27 [Hag 2.1] Dino (Optional) Exercises [HAG, II.1.9],[HAG, II.1.17], and [HAG, II.1.22].
May 4 [Hag 2.2] Dino (Optional) Exercises [HAG, II.2.1,2,7,10]. Show that the line with two origins is not an affine scheme. Read the proofs of Lemma II.2.1, Proposition II.2.2, and Lemma II.2.4.
May 11 [Hag 2.2] Dino First assignment
May 18 Correction of the first assignment. Dino Solutions
May 24 (!!Wednesday, 10-12, Room 04-426!!) [Hag 2.3] Davide (Optional) Guidelines 1
June 1 [Hag 2.3] Davide (Optional) Guidelines 2
June 8 [Hag 2.4] Davide (Optional) Guidelines 3
June 14 (!!Wednesday, 10-12, Room 04-426!!) [Hag 2.4] Davide (Optional) Guidelines 4
June 22 [Hag 2.5] Davide (Optional) Guidelines 5
June 29 [Hag 2.5] Davide Second assignment
July 6 NO CLASS!
July 13 Correction of the second assignment. Davide Solutions