Mastermath course, Spring 2014


Robin de Jong, Leiden University

Ben Moonen, Radboud University Nijmegen.


Jinbi Jin (jinmengjun at and Stefan van der Lugt (stefanvdlugt+ag at

The lectures

The lectures are on Tuesday afternoons from 2pm till 4.45pm at campus "de Uithof" of the University of Utrecht, in room 079 of the BBL building. The third hour will be an exercise class, supervised by the assistents.


We shall follow course notes written by Ben Moonen. Robin de Jong has written some supplementary course notes. These notes are subject to revision, and comments are very welcome. As a second text we recommend Hartshorne's classic Algebraic Geometry, cited as HAG.

For commutative algebra we recommend Introduction to commutative algebra by Atiyah and MacDonald.


We expect that you have a good knowledge of basic Algebra (groups, rings, fields) and that you are willing to study further notions (e.g., modules) on your own, if necessary. Further we need some basic Topology, and it helps if you have already seen some basic notions from differential geometry, for instance in a first course on differentiable manifolds. We will use some terminology from category theory, but as we won't need more than some basic notions you can learn this as we go.


Every week (with some exceptions due to holidays) a few homework exercises are given, which are graded as "pass" or "fail". (Needless to say, if you don't hand in, this counts as "fail".) The exercises are to be handed in by the next Tuesday at 4pm, also if you're handing in by email. You can check here your grades thus far.

In contrast to what was announced earlier, there will be a written (not oral!) retake exam on June 24, 1:30 - 4:30 pm in the Went building of the UU, hall "Green". (Students with dyslexia can continue one more hour.) During the exam you are not allowed to use the lecture notes or other literature. A condition for admittance to this exam is that you have at most three "fails" for the weekly exercises and that you have done the larger homework assignments.

The main purpose of the exam is to test if

We are of course aware of the fact that in 3 hours you cannot do long and complicated calculations. This will be taken into account when we prepare the exam.

For the final grade, the result of the written (retake) exam and the average grade of the larger homework assignments both count for 50 %.

Overview of the lectures

Date Topics Exercises
February 4 (BM) Affine varieties (Chapter 1). 1.8 and 1.9. Due date: Feb 11.
February 11 (BM) Regular functions and morphisms. I've covered Chapter 2 until 2.21. 2.2 plus HAG, Section I.3, exercise 3. Due date: Feb 18.
February 18 (RdJ) Rest of Chapter 2 (read § 4 yourself). Chapter 3 until 3.9. 3.2 and 3.3. Due date: Feb 25.
February 25 (RdJ) Rest of Chapter 3. 3.10 and 3.4. Due date: March 4.
March 4 (BM) Sheaves. (Chapter 4 and/or HAG, Section II.1.) Exercises for this week. Due date: March 18.
March 11 No lecture
March 18 (RdJ) Sections 5.1, 5.2 and 5.4. The midterm assignment is now available. Due date: April 1.
March 25 (BM) Section 5.3 (birational geometry). Dimension theory. The tangent space.
April 1 (RdJ) Rest of Chapter 6. 6.4, 6.5 and HAG, Section I.4 Exercise 10. Due date: April 8. In Exercise 6.5 there is a typo: in the definition of S the second X0 should be Xn. Also, in part (i) it is of course assumed that n is at least 2. In case you don't have a copy of HAG at hand, here is the text of Exercise I.4.10.
April 8 (BM) Chapter 7 until 7.14. 7.3 and 7.5. Due date: April 15.
April 15 (RdJ) Sections 8.1 and 8.2. 8.1 and 8.3. Due date: April 29. In 8.1(i) the Vi are assumed to be affine. In 8.3(ii), replace 'f not in X' by 'f not in A'.
April 22 No lecture
April 29 (RdJ) Section 8.3. 8.2 and 8.4. Due date: May 6.
May 6 (RdJ) Robin de Jong has written some supplementary course notes. Done this week: Section 1 of the supplementary notes, and a beginning with sections 2 and 3. The second assignment is now available. Due date: May 20.
May 13 (BM) The canonical divisor class. Riemann-Roch.
May 20 (BM) Linear systems of divisors. Applications of Riemann-Roch.
June 10, 1:30 - 4:30 pm Written exam - see above
June 24 Retake exams in Utrecht.