Algebraic Geometry

A first characterization of Algebraic Geometry is: the study of the common zero sets of a collection of polynomial  equations in a given number of variables x1,...,xn. These polynomials were originally assumed to have real coefficients so that the zero set  would lie in real n-space. After the advent of complex numbers,  it was soon noticed that the theory became much more manageable if these coefficients were taken to be complex (for instance a polynomial in a single real variable of degree d has at most d real solutions,  but if the coefficients are complex, then the number of complex solutions is exactly d, if we are willing to count solutions with multiplicity). It had also been observed that things became even better if we also include “solutions at infinity.” This means looking  for solutions in a somewhat bigger space then complex n-space, namely complex projective n-space. This was followed by the observation that  for much of the theory the only property of the complex numbers that intervened was that they make up an algebraically closed field. We thus arrive at what is called Projective Algebraic Geometry: the study of common zero sets of systems of homogeneous polynomials in n+1 variables x0,...,xn with coefficients in an algebraically closed field.

However, if it so happens that the polynomials have their coefficients in a smaller field that is not algebraically closed such as the field of rational numbers, then it makes sense (and there may be good reason) to ask for solutions with coefficients in that field. This can be a subtle issue which usually involves Galois theory (this also explains why it was not a good idea to start working real coefficients). Things become even more complicated if the algebraically closed field is replaced by a ring like the integers. Such questions are by no means uninteresting, for many natural problems in number theory can be stated that way.  In the 1950ies it was recognized that in order to accommodate this kind of generality, Algebraic Geometry had to be rebuilt from the ground up. This foundational work was carried out in a relatively short period (1958-1970) under  the supervision of A. Grothendieck. The tools and language developed by him (with his notion  of scheme taking the place of an algebraic variety) and the underlying way of looking at things are now commonly accepted  as the best framework to work in. At the same time, experience has taught us that the scheme setting is ill-suited for  a first acquaintance with algebraic geometry, and this is why most of this course is concerned with Algebraic Geometry over an algebraically closed field. Yet it is our aim to make contact with the modern approach.

The final grade will be based on both the handed in Answers to Problems and the Final Exam,  the details of this will  be made known at the beginning of the course.
Date re-sit: August 23, 2010

Apart from a set of Lecture notes (which I will revise the  as we go along, so do not print this yet in full), it is advised you get hold of the book (which is available in a somewhat cheaper soft cover edition) R. Hartshorne: Algebraic Geometry, Springer Verlag GTM 52

in spite of the fact that we will  only cover a small part of it (Ch. 1 and a part of Ch. 2).