# Topics in geometry, fall 2006

Algebraic geometers study sets given by polynomial equations. Classically the coefficients of these equations used to be real or complex, but later on coefficients were allowed from arbitrary fields, and even arbitrary rings. Taking an arbitrary algebraically closed field as the ground field, a rich and interesting theory appears going back at least to the times of Emmy Noether and David Hilbert. Importantly, they showed 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'.

Here is a set of additional exercises.

The final grading of this course will be based on the marks for the homework problems that will be given during the semester, as well as on the marks for a final set of exercises. The deadline for the weekly exercises is December 22nd. The deadline for the final set of exercises is February 2nd. Once you are finished with these, please make an appointment with the instructor in order to discuss them.

Classes will be on Thursdays, from 11:15 -- 13:00 (i.e. slots 3/4) and from 13:45 -- 15:30 (i.e. slots 5/6), starting on September 7. Exercise sessions will in general be in slots 3/4. Exceptions will be mentioned below. No courses are scheduled in week 43. Weekly exercise sessions are included in this schedule. The exercise sessions will be led by Johan Bosman, room 229, email: jgbosman (at math.leidenuniv.nl). Homework should be given to him.

Here is what we did.
September 7:
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, proof of Hilbert Nullstellensatz from the Weak Nullstellensatz.
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.

September 14:
Exercise session: additional exercises 1, 16, 2, 5.
Class: 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.
Reading in [HAG]: section I.1 until `dimension'.

September 21:
Exercise session: additional exercises 5, 15, 16, [HAG] I.1.1(a), (b).
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.
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.

September 28:
Exercise session: additional exercises 18, 20, 21.
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.

October 5th:
Exercise session: additional exercises 6--9, and [HAG], Exercise I.2.14.
Homework (graded): [HAG] I.2.12, 2.15, 2.16.
Class: definition of a projective variety, motivation of morphisms, regular functions, varieties, 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 is naturally isomorphic to its coordinate ring (proof later), the ring of regular functions on a projective variety is just the ring of constant functions (proof later), equivalence of categories between affine varieties on the one hand and finitely generated k-algebras which are also domains on the other (proof later).
Reading in [HAG]: section I.3 until local ring at a point. Also read already the statements of the theorems and propositions later in this section.

October 12th: (extended exercise session led by Jan Schepers)
Exercise session: additional exercises 22, 23, 24, 26, and [HAG], Exercises I.3.1, 3.4, 3.5, 3.6.
Homework (graded): additional exercises 27, 33, 34 en [HAG] I. 3.2 en I.3.7(b).

October 19th:
Exercise session: additional exercise 19 and [HAG] I.3.5, 3.6.
Homework (graded): [HAG] I.3.8, 3.9 and 3.14.
Class: function field of a variety, proofs of the various propositions of October 5th.

November 2nd:
Exercise session: [HAG] I.4.3, 4.4a,b, 4.6 and additional exercises 30, 31.
Homework (graded): [HAG] I.4.4c and 4.5.
Class: rational maps, birational maps, birational equivalence, Cremona transformations, dominant rational maps, a dominant rational map f : X -> Y induces by composition a k-algebra homomorphism f^* : K(Y) -> K(X), the association f -> f^* is a functorial bijection from dominant rational maps X -> Y to k-algebra homomorphisms K(Y) -> K(X), X and Y are birationally equivalent iff they have isomorphic function fields, useful lemma: if X is a variety and p a point on X then p has an affine open neighbourhood. Start with products.
Reading in [HAG]: section I.4 up to and including the proof of Corollary 4.5.

November 9th:
Homework (graded): prove that A^1 is not complete.
Class: products, product of two affine varieties, product of two quasi-projective varieties using the Segre-embedding, Hausdorff property: the diagonal in a self-product of a variety is closed, morphisms to a variety that are equal on a dense open subset are equal, the line with the double origin is not a variety, main theorem of elimination theory.
Reading in [HAG]: exercises I.3.15 and 3.16 and Theorem I.5.7A. For a discussion of elimination theory and complete varieties see also [RdBk], Section I.9.

November 16th:
Exercise session: additional exercises 10 and 37--39.
Class: the blowing-up of the affine plane in a point, resolving plane curve singularities by blowing-up, start with dimension.
Reading in [HAG]: pp. 28--30. Our treatment of dimension follows [RdBk], Section I.7.

November 23rd:
Exercise session: more blowing-up, and [HAG] I.1.10.
Homework: no homework.
Class: dimension, dimension becomes strictly smaller on a proper closed subvariety, Krull's Hauptidealsatz, topological characterisation of dimension, 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.
Reading in [HAG]: section I.5 until completion.

November 30th:
Exercise session: additional exercises 41--44 and [HAG] I.5.1 and 5.8.
Class: affine and projective dimension theorem, intersection theory in projective space, lemma on numerical polynomials, graded S-modules, Hilbert function, Hilbert polynomial, Hilbert polynomial of a projective algebraic set.
Reading in [HAG]: Section I.7 up to and including Theorem 7.5.

December 7th:
Exercise session: additional exercises 45--47 and [HAG] I.5.3.
Homework: no homework.
Class: intersection theory in projective space: degree, intersection multiplicity, main theorem, many examples, link with cohomology.