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.
Additional notes
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.
Homework grading
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.
- We will finish section II.1 and start with locally ringed spaces.
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.
Exercises.
- 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.
Exercises.
- 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.
Exercises.
- 6. March 13.
- We will finish our treatment of projective spaces and start with section II.2: schemes.
Exercises.
- 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.
Exercises.
- 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.
Exercises.
- 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.
Exercises.
- 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.
Exercises.
Deadline: May 22.
- 15. May 15.
- Topics from section II.5 will be discussed: sheaves of modules, pull-backs, tensor products and quasi-coherent sheaves.
Final exercise set.
Notes: there is no deadline, solution all problems have to be handed in and oral discussion of this set is required.
Johan Bosman <jgbosman@math.leidenuniv.nl>
Peter Bruin <pbruin@math.leidenuniv.nl>
Last modification: May 16, 2007.