# Topics in geometry, fall 2007

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.
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.

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

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.