[UAG] M. Reid, Undergraduate algebraic geometry, CUP. For other books
see the references in the PYDC entry or below
Timetable for 2007-08:
Tue 2-3, MS.04
Wed 10-11, B3.01
Thu 3-4, MS.05
Arrangements for 2007-08:
I will lecture the first and last two weeks of the course, replacing
some of the material of Sections~2 and 6-7 of the book with appropriate
new stuff (see below). The core sections, corresponding to Sections 3-5
of the book, will be lectured together jointly with Sarah Davis
<S.E.Davis@warwick.ac.uk>, or may be given in part as a self-help
seminar if students volunteer.
Content
The course will consist of a core of theoretical material from the
central Sections 3-5 of my book [UAG], Undergraduate algebraic
geometry, as a first introduction to the categorical framework of
algebraic geometry: the Nullstellensatz, the definition of affine and
projective varieties, morphisms and rational maps between them. In
addition, the course will cover new material directed towards areas of
current research, including:
(1) Varieties in projective space such as quadrics, Grassmannians,
Segre and Veronese embeddings and other homogeneous and
quasi-homogeneous varieties.
(2) Some introductory material in toric geometry illustrating
properties of affine and projective algebraic varieties.
(3) Group actions, their quotients and resolutions.
(4) The Riemann-Roch theorem for curves.
Students may also be interested in looking in on the Algebraic
Geometry reading seminar (Wed from 2:00 this term) or the MSc seminar
module MA505 Algebraic Geometry. Please give your
name and e-mail to Sarah to be included on the mailing list.
Assignment sheets:
There will be approximately 8 compulsory assignments counting for 30\%
of the credit; students may gain credit as an alternative by lecturing
part of the material, or by producing lecture notes of sections
containing new material. Some help with the assessed work will be given
if required.
If you want to be kept in touch, please write to Sarah to be included on
the mailing list.
Last year's assessed work as pdf file.
Prerequisites and suitability for 3rd year MMathers
The course uses basic definitions from most 1st and 2nd year courses,
including algebra, geometry, analysis. More than technical
prerequisites, the main requirement is the sophistication to work
simultaneously with ideas from several areas of mathematics: projective
geometry, rings and fields, basic topology and analysis. I would expect
the course to be harder than most for third year students, but not
impossible for those who are well motivated and can rise to a challenge.
The reward consists of access to 4th year and MSc/PhD courses and
projects in algebraic geometry.
The PYDC entry is
here.
This is a first introduction to algebraic geometry at the MSc level. One
of the central topics is the relation between the commutative algebra of
graded rings and the geometry of projective varieties and singularities.
The emphasis is on concrete examples of curves, surfaces, projective
arieties and singularities, as illustrated in my Park City lecture notes
[Chapters] and in the book in progress [More Chapters]. Technical
foundational material, for example on commutative algebra and affine
schemes, on sheaves and coherent cohomology, will be introduced as
required, without detailed proofs. Students may wish to study more
material in a parallel self-help seminar (I can try to arrange exam
credit for this as necessary).
Prerequisites
This is a second course in Algebraic Geometry, and it assumes
background knowledge at the level of the undergraduate course MA4A5 Algebraic Geometry or the material of [UAG] or
the first two chapters of Shafarevich, Basic algebraic geometry.
Timetable for 2006-07, Term 2:
Tue 2-3, B3.01
Thu 10-11, B3.01
Fri 11-12, B3.01
Several beginning postgraduates and visiting students will be
studying the subject at some level, and should take keep a look-out for
seminars at Warwick and elsewhere. (e.g. the COW and Calf that meets in
Warwick, Oxford, London, Bath, etc. Google "COW seminar").
Books:
M. Reid, Undergraduate commutative algebra, CUP
M. Reid, Chapters on algebraic surfaces, in Complex algebraic
geometry (Park City, 1993), Amer. Math. Soc., Providence, RI, 1997,
3--159. The preprint version is available from my website + Surfaces
M. Reid, More chapters (book in preparation), see my website +
Surfaces
J.-P. Serre, Faisceaux algébriques cohérents, Ann. of Math.
61 (1955)
A. Borel and J.-P. Serre, Le théorème de Riemann-Roch, Bull.
Soc. Math. France 86 (1958) 97--136
I.R. Shafarevich, Basic algebraic geometry I, Springer, 1994
R. Hartshorne, Algebraic geometry, Springer Graduate Texts, 1977
H. Matsumura, Commutative ring theory, CUP
Assessment:
Three-hour written examination
Other material at the beginning graduate level
See the first
chapters of Chapters on algebraic surfaces, Park City 1993
index
and from More chapters (book in preparation):
Cyclic
surface quotient singularities
Graded
rings
See the
Homework.
See also my Surface
links
My front page
Google