# Topics in geometry, fall 2008

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 (David Hilbert, Emmy Noether, ...). 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 the following topics: affine varieties, Hilbert's Nullstellensatz, projective varieties, theory of conics, morphisms, function fields, rational maps, dimension, tangent spaces, singularities and intersection theory in projective space. Most of the commutative algebra we need will be discussed in class.

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 Thursdays, from 13:45 -- 15:30 (i.e. slots 5/6), starting on September 4. There will be no lectures in week 43. In total there will be 13 sessions.

NOTE: the starting time of the lectures has changed to 15:00.

During the semester several exam exercises will be given. Each student has to bring her/his solutions to the exam exercises to the exam, where a discussion of the solutions takes place. An appointment for the exam has to be made individually. The possible dates for the exam lie between now and February 1st 2009. The exam exercises are: [HAG] I.2.12, I.3.7 (note: a `curve in P^2' here means: `a hypersurface in P^2'), I.4.10, I.5.11, additional exercises 3, 4, 34, 47, and the exercises from this exercise sheet.

Here is a set of additional exercises.

Here's what we did:

Lecture 1:
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.
Useful exercises: additional exercise 2; [HAG] I.1.4; determine all pythagorean triples, i.e. all solutions in the integers of the equation x^2+y^2=z^2.
Reading in [HAG]: Section I.1, up to and including Corollary 1.4.

Lecture 2:
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 based on weak Nullstellensatz, irreducible topological spaces, decomposition of algebraic sets into irreducible components, irreducible algebraic sets correspond to prime ideals, affine varieties, coordinate rings, quasi-affine varieities, distinguished open sets D(f) on an affine variety, they form a basis for the Zariski topology, affine varieties are quasi-compact.
Useful exercises: deduce the weak Nullstellensatz from the Hilbert Nullstellensatz; [HAG] I.1.3, 1.6, additional exercise 5.
Reading in [HAG]: Section I.1 until Corollary 1.6.

Lecture 3:
Noetherian rings, proof of Hilbert Basissatz, integrality in rings, Noether Normalisation Lemma, proof of the Weak Nullstellensatz, projective space of a vector space, projective space over a field, affine coordinate charts, homogeneous polynomials and their zero sets.
Useful exercises: additional exercises 6, 9, 10, 11; [HAG] I.1.1.b, 1.2.
Reading in [HAG]: Section I.2, up to the first Definition.

Lecture 4:
Homogeneous polynomials, homogeneous ideals, algebraic sets in projective space, they form the closed sets of a topology: Zariski topology, cones of a projective algebraic set, coordinate charts are homeomorphic to affine space, homogenizing and dehomogenizing polynomials, conics, classifying space of conics and special subsets defined by conditions on conics.
Useful exercises: [HAG] I.2.1, 2.2, 2.3, 2.9, additional exercises 22 and 24.

Lecture 5:
Pascal's theorem, projective and quasi-projective varieties, general notion of a variety, 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).
Useful exercises: [HAG] I.2.9, I.2.15, I.3.2; additional exercises 12, 19, 20.
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.

Lecture 6 (given by Lenny Taelman):
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.
Useful exercises: [HAG] I.3.5, 3.6.

Lecture 7:
Examples of bijective morphisms of affine varieties which are not isomorphisms, homogeneous coordinate ring and function field of a projective variety, rational maps, dominant rational maps, examples, birational equivalence, Cremona transformations, the natural association: dominant rational maps from X to Y ----> k-algebra homomorphisms from K(Y) to K(X) given by pullback, is a bijection.
Useful exercises: additional exercise 15, [HAG] I.4.1, 4.2, 4.3.
Reading in [HAG]: I.4, up to and including the proof of Corollary 4.5.

Lecture 8:
Products: product of two affine varieties, product of two projective spaces via the Segre embedding, product of two quasi-projective varieties, the Hausdorff axiom, corollaries: if two morphisms agree on a non-empty open subset, then they agree everywhere, a graph of a morphism is closed; statement of the main theorem of elimination theory, image of a projective variety under a morphism is closed.
Useful exercises: [HAG] I.2.14, 2.15, 2.16, 3.14, 3.15(a)-(c), 3.16, 3.19(a); additional exercises 30, 35.
Reading in [HAG]: read once again I.1--I.4 up to and including the proof of Corollary 4.5.

Lecture 9:
Blowing-up of the plane in the origin: definition and properties; strict transform of some curves; 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, Krull's Hauptidealsatz in its geometric incarnation, topological characterization of dimension.
Useful exercises: additional exercise 41, [HAG] I.4.4, I.4.5.
Reading in [HAG]: Section I.1, from Definition on bottom of page on; Section I.4, pages 28--30.

Lecture 10:
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; introduction to intersection theory in projective space; the affine and projective dimension theorem.
Useful exercises: additional exercises 28, 29, 42, 43; [HAG] I.5.1, 5.2, 5.3, 5.8, 5.9.
Reading in [HAG]: Section I.5 and beginning of I.7.

Lecture 11:
Intersection theory via graded S-modules. Numerical polynomials, Hilbert function, Hilbert's theorem on finitely generated graded S-modules, Hilbert polynomial, Hilbert polynomial of a non-empty projective algebraic set, examples, degree of an algebraic set, equality of deg(Hilbert polynomial) and dim(algebraic set), degree of a hypersurface, arithmetic genus g of a projective curve, g=(d-1)(d-2)/2 for a curve of degree d in the plane.
Useful exercises: additional exercises 26, 28, 29, 49.
Reading in [HAG]: Section I.7 up to and including the proof of Prop. 7.6.

Lecture 12:
Hilbert polynomial of a non-empty projective algebraic set, examples, equality of deg(Hilbert polynomial) and dim(algebraic set) - now with proof, degree, degree of a hypersurface, properties of the degree, proof of Bezout with the right definition of intersection multiplicity, using composition series; examples. Beginning of a discussion of the theory of non-singular projective curves; a rational map on a non-singular curve extends to a morphism.
Useful exercises: [HAG] I.7.1, I.7.2.