Algebraic Geometry

This is a first course in commutative algebraic geometry, for students who have some background in basic algebra, and in particular in the theory of rings and fields. The first part covers some basic commutative algebra, building towards the basic theory of affine and projective algebraic varieties.

Learning Outcomes:On successful completion of this module, the student will:
1. be familiar with the basic concepts, methods and results of elementary Algebraic Geometry and elementary Commutative Algebra.
2. be able to recognise affine and projective algebraic varieties, describe them them in the language of Commutative Algebra
and analyse their basic properties using the methods and tools of the later.
3. be able to describe, construct and analyze affine and projective algebraic varieties, including their singular points.
3. be able to apply the techniques of ideal theory to basic problems in the theory of affine and projective algebraic curves and surfaces.
4. be able to apply the techniques of basic elimination theory to problems in elementary algebraic geometry
5. be familiar with the basic concepts of classical intersection theory.

Core Textbooks:

Elementary: I. Dolgacev, Introduction to Algebraic Geometry
Advanced:   D. Mumford, The Red Book of Varieties and Schemes (we will only cover parts of this somewhat advanced text, given that it requires more commutative algebra than I can assume known).

For basic algebra and commutative algebra:

Intermediate:
Atiyah & McDonald: "An introduction to commutative algebra".
Serge Lang:         "Algebra".

Bourbaki:         "Algebra I", "Algebra II" and "Commutative Algebra"
H. Matsumura:  "Commutative Ring Theory".

Online resources:

General:

Donu Arapura's Notes on Basic Algebraic Geometry
Various online texts.
A directory of online resources.

Galois Connections:

Topics covered in class:

1. Rings and ideals; definition of prime and maximal ideals; multiplicative sets; quotient of a ring by an ideal; integral domains and principal ideal domains (P.I.D.s).
2. Ideal generated by a set; ideal intersections and ideal sums.
3. Characterizations of prime and maximal ideals. The prime and maximal spectra of a ring. Krull's theorem; the Krull dimension of a ring. Noetherian rings. Ideal-theoretic characterization of fields and integral domains.
4. Basic theory of semilattices and order lattices. The lattice ($J(R),+,\cap)$ of ideals of a ring. The product of ideals. The semiring $(J(R),+,\cdot)$.
5. Radical of an ideal; roots of an element; nilpotent elements; the nilradical of a ring.  Reduced rings. Radical ideals. The quotient criterion for radical ideals. Prime ideals are radical. The reduction of a ring. I-order of elements and order of nilpotency. The radicalization map as a closure operator. The semilattice of radical ideals.
6. Closure and kernel operators.
7. Modules and algebras over a commutative ring; morphism and isomorphism of algebras; monoid algebras and polynomial algebras; basic  terminology and conventions for polynomials. Z-modules are the same as Abelian groups; Z-algebras are the same as commutative rings. First isomorphism theorem for commutative R-algebras; application to commutative rings.
8. Finitely generated algebras. Presentation of a finitely-generated algebra through generators and relations; polynomial algebras are the free commutative algebras on finite sets of generators; the universality property of polynomial algebras.
9. Galois correspondences; examples.
10.
Hilbert's basis theorem. The case of coefficients in a field.
10. Hilbert's weal and hard nullstellensatz. The (affine) ideal variety-correspondence.

11. Affine varieties; the Zarsiski tangent space; singular points.
12. Projective space; homogeneous polynomials; projective varieties and projectivisation.

Tutorials:

Tutorial 1: The ideals of the ring of integers --part 1:
-The ring of integers is a P.I.D.
-Divisibility as a partial order relation; connection with the inclusion relation on ideals
-Ideal intersections and sums correspond to the lcm and gcd.

Tutorial 2: The ideals of the ring of integers -- part 2:
-The lattice J(Z) and the spectrum Spec(Z). Meaning of Noetherianness of Z.
-Irreducibility versus primeness. Primality. Primality implies that all irreducibles are prime.
-The Euclidean algorithm and the unique factorization theorem (the fundamental theorem of arithmetic).
-Basics on generalizations: Arithmetic in Euclidean Rings, P.I.Ds, Bezout domains, UFD's, GCDs, (pre-)Schreier domains etc.
Tutorial 3: Square-free content of a natural number; radicalization map and radical ideals of Z; the nilradical of the rings Z_n;
description of the nilpotent elements of Z_n.
Tutorial 4: Viete's relations.
Tutorial 5: Resultants and discriminants.

Revision Tutorial: Affine curves; computing intersection points through resultants; some examples of intersection multiplicity; ordinary double points and cusps for affine real and complex curves.

Some lecture notes
(with apologies for any typos):

Notes on closure and contraction operators
Notes on Galois Correspondences
Notes on projective spaces

Problem sets:

Problem Set 1
Problem Set 2
Problem Set 3
Problem Set 4

Exam format:

The exam will consist of 4 questions (each worth 25 points), of which you have to answer 3 questions correctly in order to get a maximum score (a score of 75 points will be renormalized to 100%). For those questions which have sub-questions, each sub-question will be numbered and contain an indication of how many points it is worth. For example:

Question 1 [25 p] ....

a. [10 p]

b. [8 p]

c. [7p]

Question 2 [25 p] ...