Hilbert series and graded rings
	===============================

The methods discussed in these lectures are used to construct and to study
interesting examples of algebraic varieties.  To put it another way, they
generate systems of homogeneous polynomials in several variables with
prescribed properties.

This area is ideal for study and experiment using computer algebra systems.
At the same time, the calculations can easily get out of hand, so human
choices are essential.  The subject matter could be studied using any of
the standard systems such as Macaulay2.  I use the system Magma myself
so I stick to that.  Indeed, one thing I hope to do is to explain the
new geometrical graded ring packages in Magma.

Aim
---
I will talk about
    -- homogeneous polynomials and graded rings
    -- Hilbert series
    -- methods for making big, complicated systems of equations from
	small, hopefully simpler ones: this is sometimes called Gorenstein
	projection and unprojection, and relates to elimination of variables
    -- explicit use of the Magma computer algebra system for solving
    	problems in this area.

In more detail, I will
    -- give examples of systems of homogeneous polynomials, and their
    	meaning as weighted projective algebraic varieties; an important
	part of the course is to recognise favourite equation 'formats'
    -- give examples of Hilbert series; outline the basic theory of
	Hilbert series and its connections with free resolutions
    -- relate the Riemann--Roch theorem and Hilbert series
    -- discuss principles of constructing a ring with a given Hilbert
	series, with emphasis on the geometric properties of the ring
	(including a very brief motivational discussion of Hilbert schemes)
    -- discuss the recent lists of curves, surfaces and other varieties
	with a view to making new lists
    -- examine elementary polynomial manoeuvres one can make to a graded
    	ring, and discuss their prominent role in recent work in
	algebraic geometry.

The emphasis will be on making the mathematical technology work in
practice, and not on the proofs behind it.


Prerequisites
-------------
The minimal prerequisite is a working knowledge of polynomials in several
variables.  For example, knowledge of Groebner basis, especially experience
using it in examples, would be fine.

Some commutative algebra --- rings, ideals --- would be an advantage, but
we can talk in very explicit terms about systems of polynomials, so it is
not essential.

It will be a bonus if you know about free resolutions of ideals or about
syzygies.  Again, you may know this from Groebner basis.  In any case, I
will discuss free resolutions a little in the lectures.

And of course, some algebraic geometry would be useful: if I named one thing,
then familiarity with projective space and homogeneous polynomials; also
experience of curves or surfaces in projective space would be an advantage.


Relevant reading
----------------
For introductory material there are two standard books:
    Cox, Little, O'Shea,  Ideals, Varieties and Algorithms,
    Cox, Little, O'Shea,  Using Algebraic Geometry.
Both books cover a large amount of material.  I think it is right to say
that you will benefit from anything in them, especially the examples, that
you follow.  They do a good job of explaining the connection between
algebra --- both in the form of systems of polynomials, and rings and
ideals --- and geometry.

If you have a Groebner basis background, you may enjoy looking at
    Bayer and Stillman,
    Computation of Hilbert functions.
    J. Symbolic Comput. 14 (1992), no. 1, 31--50.

I will have lecture notes to hand out for the course, and hope to have
them prepared in advance.

A beginning algebraic geometry book that I know well is
    Miles Reid,
    Undergraduate Algebraic Geometry,
    LMS Student Texts 12
Working out some of the examples from that, or from any textbook, will
certainly be useful.

You may like to look at Magma's webpage
    http://magma.maths.usyd.edu.au/magma/
or the (currently rather sparse) pages discussing the graded ring
material of the course
    http://www.maths.warwick.ac.uk/~gavinb/grdb.html

I mention a little advanced reading, it may only be relevant during or
after the course.  For example, the paper
    S.\ Alt{\i}nok, G.\ Brown, M.\ Reid,
    Fano 3-folds, K3 surfaces and graded rings,
    in Topology and Geometry: Commemorating SISTAG,
    A.J. Berrick et~al.\ Eds, Contemp.  Math. {\bf 314} AMS, 2002
contains an account of the methods in some contexts.  It has introductory
material too, and is firmly aimed at UK algebraic geometry graduate students.

If you are happy with ordinary projective space, you may like to look
at the calculations of
    Chapter 2, Rational scrolls,
    of Chapters on Algebraic Surfaces, Miles Reid
    in Complex Algebraic Geometry, J\'anos Koll\'ar Editor,
    AMS/IAS Park City Mathematics Series Volume 3, 1997
which you can also find on the web at
http://www.maths.warwick.ac.uk/~miles/surf/ParkC/ch2.ps