# Algebraic Geometry, Spring 2007

On Mastermath one can see a general description of this course, as well as a schedule (time and location). There will be no course on April 3 and May 1.

This page gives a progress report on what has been done (course and problem session), and gives a planning of what is to follow. The course will be taught by Bas Edixhoven and assisted by Johan Bosman and Peter Bruin.

Categories and functors. The language of category theory has turned out to be very important in algebraic geometry. Note that this is a very basic and very incomplete introduction of 3 pages into the terminology of categories. Other important notions are: contravariant functors, fibred products, representable functors, Yoneda's lemma.

Bas Edixhoven's handwritten notes for the course (36 scanned pages in the form of PNM files), available as a bzip2'ed tar file or a zip archive.

Each week we will present a set of exercises, mainly from Hartshorne's book. The students should select three exercises from the set to hand in. We will determine a number C in [0,10] that should reflect the quality of this submission. The grade awarded to the submission will then be computed as follows:
• grade = C if the submission is done within one week;
• grade = min(C,6) if the submission takes more than a week, but less than two weeks;
• grade = min(C,0) if the submission takes more than two weeks.
• In the last case we will use an efficient algorithm to determine the grade: we output 0 without first computing C. In the first two cases the number C is determined in the following way. You can get at most 9 points for the problems and possibly 1 extra point. For an easy problem you can get at most 2 points and for a difficult problem you can get at most 3 points. The extra point is awarded if all of the following conditions are met: your name appears on the paper and the submission is well-readable (mathematically as well as typographically).

### Final set of exercises

The final set of exercises can be found here. Solutions to all problems have to be handed in, to either Johan Bosman or Peter Bruin. After we read your written solutions, we will discuss them orally with you. You may choose a date yourself when to do this, but we prefer you do this the first week of July. You may also choose to let this assignment count for either 30% or 50% of your final grade, whichever percertage fits best for you.

### Weekly progress

We will start working in Chapter II of Hartshorne's Algebraic Geometry.

1. February 6.
We will start in [HAG, section II.1]: sheaves.

Exercises: 1.1 (3 pts), 1.2 (3 pts), 1.3 (3 pts), 1.4 (2 pts), 1.5 (2 pts) (all from chapter II).

2. February 13.

Exercises: I.1.7 (3pts), II.1.8 (2pts), II.1.16abc (3pts), additional exercise 1 (3pts). (Note that the first exercise is from chapter I).

3. February 20.
We will give examples of locally ringed spaces and see the connection with algebraic varieties.
4. February 27.
We will continue the formulation of algebraic varieties in terms of locally ringed spaces and give the definition of an abstract variety. The anti-equivalence of the category of affine varieties over an algebraically closed field k to the category of reduced k-algebras of finite type will be treated.
5. March 6.
We will define projective n-space Pn over an algebraically closed field k as a quotient of An+1 \ {0} in the category of varieties over k, and we will describe how Pn can be covered by copies of An.
6. March 13.
We will finish our treatment of projective spaces and start with section II.2: schemes.
7. March 20.
We will show that Spec is a functor from the category of rings to the category of locally ringed spaces, and that giving a ring homomorphism from a ring A to a ring B is ‘the same’ as giving a morphism of locally ringed spaces from Spec(B) to Spec(A) (i.e. Spec is fully faithful). We will also give some examples of schemes.

Exercises: 2.6 (2 pts), 2.7 (3 pts), 2.8 (3 pts), 2.9 (3 pts), 2.13 (3 pts) (all from Chapter II).

8. March 27.
We will make projective schemes by gluing affine schemes in the proper way. We will also treat the equivalence between reduced k-schemes of finite type and varieties over k, where k is an algebraically closed field.
9. April 3.
There will be no course this week. This means that the exercises from March 27 can be handed in the 10th of April without loss of points.

10. April 10.
We will treat various properties of schemes defined in Hartshorne, Section II.3.
11. April 17.
We will start in chapter IV of Hartshorne's book: curves. In particular we will treat the equivalence between regular projective irreducible curves and function fields.
12. April 24.
More about curves: the Riemann–Roch formula, differentials, hyperelliptic curves.

Exercises. Since we didn't get far enough in class, the deadline for this assignment is postponed to May 15.

13. May 1.
No course.

14. May 8.
We will continue the treatment of curves from last time. The Hurwitz formula is one of today's topics.