Please click here for the 2008 course.

Algebraic geometers study sets given by polynomial equations. Classically the coefficients of these equations were real or complex numbers, but later on coefficients were allowed from arbitrary fields, and even arbitrary rings. Taking an arbitrary algebraically closed field as the coefficient field, a rich and interesting theory appears going back at least to the end of the nineteenth century (Emmy Noether, David Hilbert,...). Importantly, it was shown that a very fruitful connection exists between geometry, on the one hand, and commutative algebra, on the other hand. This theme is still of the greatest importance in modern geometry. In the twentieth century the link with commutative algebra for example led to Grothendieck's theory of schemes. Several important results in mathematics are directly inspired by this theory; for example, think of the proof by Deligne of the Weil conjectures in the seventies, the proof by Faltings of the Mordell conjecture in the eighties, and the proof by Wiles of Fermat's Last Theorem in the nineties. For more, please read the Wikipedia lemma on Algebraic Geometry.

In this course we focus on algebraic geometry over an arbitrary algebraically closed field. As reading material we take Chapter I of the book `Algebraic Geometry' by Robin Hartshorne [HAG]. We intend to cover topics like: affine varieties, Hilbert's Nullstellensatz, projective varieties, morphisms, function fields, rational maps, dimension, tangent spaces and singularities, intersection theory in projective space, the abstract notion of a variety. Most of the commutative algebra we need will be discussed in class and/or in the exercise sessions.

For more reading material, I recommend:

[RdBk] D. Mumford, `The Red Book of Varieties and Schemes', Lecture Notes in
Mathematics 1358,

[AM] M.F. Atiyah and I.G. MacDonald, `Introduction to Commutative Algebra' and

[Eis] D. Eisenbud, `Commutative Algebra with a View Toward Algebraic Geometry'.

Classes will be on Mondays and Thursdays, from 13:45 -- 15:30 (i.e. slots 5/6), starting on September 3. No courses are scheduled in week 43. The Thursday sessions are ususally devoted to exercises and are led by Johan Bosman, room 229, email: jgbosman (at math.leidenuniv.nl). On Thursdays homework problems are given. All homework should be given to Johan Bosman.

The final grading of this course will be based on the marks for the homework problems that will be given during the semester (50 %), as well as on the marks for a final, somewhat more substantial set of exercises (50 %). The rules for homework are as follows. Homework has to be handed in within a week. If the homework is handed in later, but within two weeks, the maximum grade is 6 out of 10. If the homework is handed in later than two weeks, or not at all, the grade is 0. One can get at most 9 points for the problems. The remaining 1 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).

Here is a set of additional exercises (last update: August 28).

Here is what we did.

Week 36:

Class: Introduction, affine n-space over an algebraically closed field,
points, coordinates, algebraic sets, the algebraic sets in A^n are the closed
sets of a topology on A^n: the Zariski topology; on the affine line we get the
cofinite topology; hypersurfaces, Hilbert Basissatz, the ideal of a subset of
A^n, basic properties, radicals, Hilbert Nullstellensatz, 1-1 correspondence
between radical ideals of k[x_1,...,x_n] and algebraic subsets of A^n, Weak
Nullstellensatz, deduction of HNS from WNS, noetherian spaces, A^n is a
noetherian space.

Exercise session: additional exercises 1,2 and 7.

Homework (graded): (1) Deduce the Weak Nullstellensatz from the Hilbert
Nullstellensatz. (2) Do exercise I.1.4 from [HAG].

Reading in [HAG]: section I.1, up to and including Example 1.4.4, although we
did not treat irreducibility yet.

Week 37:

Class (lecture given by Bas Edixhoven): irreducible spaces, decomposition of algebraic sets into irreducible
components, irreducible algebraic sets correspond to prime ideals, affine
varieties, coordinate rings, examples, quasi-affine varieties, noetherian rings,
proof of the Hilbert Basissatz.

Exercise session: additional exercises 4, 6 and [HAG] I.1.7.

Homework (graded): additional exercises 3, 5 and [HAG] I.1.6.

Reading in [HAG]: section I.1 until `dimension'.

Week 38:

Class: integrality in rings, Noether's normalisation lemma, proof of the Weak
Nulstellensatz, projective n-space, homogeneous coordinates,
affine coordinate charts, homogeneous polynomials, homogeneous ideals, algebraic
sets in projective space, the Zariski topology, on the projective line we get
the cofinite topology, the affine charts are open.

Exercise session: additional exercises 9, 10, 11, 12.

Homework (graded): [HAG], I.1.1(a),(b), I.1.2 (you may skip the question about
dimension if you want), additional exercise 8.

Reading in [HAG]: section I.2 until the definition of projective variety. The
proof we gave of the Weak Nullstellensatz can be found in [RdBk], pp. 1--4.

Week 39:

Class: the cone over a projective algebraic set, basic properties of projective
algebraic sets: a non-empty
projective algebraic set is irreducible if it is given by a homogeneous
prime ideal; decomposition into components; homogeneous Nullstellensatz;
identifications of the affine coordinate charts with affine space are
homeomorphisms: homogenising and dehomogenising polynomials; every projective
algebraic set is covered by open subsets which are themselves affine algebraic
sets; conics; classifying spaces of conics with special geometric properties:
incidence properties and degeneration properties, projective and
quasi-projective varieties, the general notion of a variety.

Exercise session: additional exercises 12--15 and [HAG] Exercise I.2.14.

Homework (graded): [HAG] Exercise I.2.2, I.2.9 and additional exercise 22.

Reading in [HAG]: section I.2.

Week 40:

Class: motivation of morphisms, regular
functions, definition of a morphism, isomorphism, the category of
varieties over k, the local ring at a point, the ring of regular functions
of an
affine variety naturally contains its coordinate ring,
the function field, a proposition making the ring of regular functions, the
local ring at a point and the function field explicit for an affine variety
(proof later).

Reading in [HAG]: section I.3 until the statement of Theorem 3.2, and
Proposition 3.3. Please read already the
statements of the theorems and propositions (without the proofs)
later in this section.

Exercise session: additional exercises
19, 20, 21, 23, 24 and [HAG] Exercise I.3.1.

Homework (graded): [HAG] Exercises I.2.12 and 2.15.

Week 41:

Class: review of ring of regular functions, local rings, and function fields;
relation with the coordinate ring in the case of affine varieties;
characterization of morphisms from a given variety to an affine variety;
equivalence of categories between affine varieties on the one hand, and finitely
generated k-algebras which are also a domain, on the other; calculation of the
ring of regular functions on a projective variety.

Reading in [HAG]: section I.3.

Exercise session: [HAG] Exercises I.3.4, 3.5, 3.6 and additional
exercises 25, 26.

Homework (graded): additional exercise 34 and [HAG],
Exercises I.3.2, 3.7 and 3.9. Note: a `curve in P^2' means: `a (projective)
hypersurface in P^2'.

Week 42:

Class: rational maps, domain, birational map, birational equivalence, Cremona
transformation, dominant rational maps, functorial bijection from the set of
dominant rational maps from X to Y to the set of k-algebra homs from K(Y) to
K(X), thus: X,Y birationally equivalent iff K(X) and K(Y) isomorphic as
k-algebras; every variety X has a basis for the topology consisting of affine
varieties; wish list for dimension; definition of dimension of X as the
transcendence degree of K(X); proof that the dimension of a closed subvariety of
X is strictly smaller than the dimension of X, modulo an algebraic lemma.

Reading in [HAG]: Section I.4.

Exercise session: [HAG] Exercises I.4.1, 4.2, 4.3, 4.4 and
4.5, as well as additional exercise 40.

Homework (graded): [HAG] Exercise I.3.14. Read the first sentence as: let H be a
hyperplane in P^{n+1} (i.e. a projective hypersurface given by a linear equation, cf.
Exercise I.2.11), and let P in P^{n+1} - H be a point. In the rest, replace P^n
by H. Hint for (i): work in coordinates, e.g. give an explicit parametrisation of the
line connecting P and Q. Or do (ii) first and see the general pattern.

Week 44:

Class: review of dimension, basic properties, Krull's Hauptidealsatz in its
geometric incarnation, topological characterization of dimension; products:
definition, existence of the product of affine varieties, product of two
projective spaces via the Segre-embedding, existence of product of
quasi-projective varieties, Hausdorff axiom, the line with the double origin is
not a variety.

Reading in [HAG]: the text in [HAG] Section I.1 following Corollary 1.6;
Exercises [HAG] I.3.15 and 3.16; [HAG] Lemma I.4.1.

Exercise session: [HAG], Exercise I.4.9, some old stuff, Groebner bases and
elimination theory.

Homework (graded): [HAG], Exercise I.1.10 (take the definition of dimension of a
topological space from [HAG], p.5 below), and: let U,V be non-empty open affine
subvarieties of a variety X. Prove that the intersection of U and V is again an
affine variety.

Week 45:

Class: Hausdorff property and consequences: the graph of a morphism is closed;
main theorem of elimination theory (proof omitted); image of a projective
variety under a morphism to P^n is again a projective variety; blowing-up of the
plane in the origin: definition and properties; strict transform of some curves.

Reading in [HAG]: Section I.4, `Blowing up'; the main theorem of elimination
theory is stated (in an equivalent, algebraic way) in [HAG], Theorem 5.7A.

Exercise session: [HAG] Exercise I.3.19(a), 3.21 and additional exercises
38, 39.

Homework (graded): [HAG] Exercise I.4.10 and: let X be an affine variety and let
X -> A^n be a morphism. Is the image of X closed in A^n?

Week 46:

Class: the tangent space: intuitive definition in affine space using linear parts,
intrinsic definition (Zariski tangent space), this gives back the intuitive
definition in affine space, singularity and non-singularity, Jacobi criterion,
a local ring on a non-singular curve is a discrete valuation ring, order of
vanishing and pole order of rational functions.

Reading in [HAG]: Section I.5.

Exercise session: [HAG] Exercises I.5.1 and 5.8; additional exercises 42, 44,
46.

Homework (graded): [HAG] Exercise I.5.2 and additional exercise 43.

Week 47:

Class: introduction to intersection theory in projective space; the affine and
projective dimension theorem; lemma on numerical polynomials; graded modules
over the graded polynomial ring k[x_0,...,x_n]; Hilbert function;
Hilbert's theorem; Hilbert polynomial; examples.

Reading in [HAG]: Section I.7 up until Prop. 7.6. Skip Prop. 7.4 for now,
if you want.

Exercise session: [HAG], Exercises I.5.4, I.7.1, I.7.2.

Homework (graded): additional exercise 47.

Week 48:

Class: examples taken up again; degree of a non-empty projective algebraic set;
why dimension is given by the degree of the Hilbert polynomial; main theorem
about intersecting a projective variety with a hypersurface; Bezout's theorem,
examples.

Reading in [HAG]: Section I.7. Read the statement of Prop. 7.4.

Exercise session: additional exercises 28, 29, 49 and [HAG] I.7.3--7.8.

Homework: no homework.

Week 49:

Exercise session (on Monday 3rd!): see week 48.

Class: brief introduction into (non-singular projective) curves: function fields
of transcendence degree 1, divisors, Riemann-Roch spaces, differentials,
genus, theorem of Riemann-Roch, applications, a non-singular projective curve of
genus 1 with a given point on it has a canonical group structure.

Reading: parts of [HAG] I.6 and IV.1; alternatively, J. Silverman, The
arithmetic of elliptic curves, Chapter II.

The DEADLINE for the weekly exercises is December 6th. The deadline for the final exercise set is Friday February 1st. This final exercise set determines one half of the final grade. The solutions of the final problem set will be discussed between the student and the instructor before the final grade is given. The student is expected to make an appointment with the instructor for this final discussion.