Eulogy for Andrew Wiles
The text of an oration to present Prof Andrew Wiles for the degree of Doctor of Science, Honoris Causa, at the University of Warwick
math.AG↗
arXiv subjects
Miles Reid
Publications and source records attributed to Miles Reid.
McKay correspondence
This is a rough write-up of my lecture at Kinosaki and two lectures at RIMS workshops in Dec 1996, on work in progress that has not yet reached any really worthwhile conclusion, but contains lots of fun calculations. History of Vafa's formula, how the McKay correspondence for finite subgroups of SL(n,C) relates to mirror symmetry. The main aim is to give numerical examples of how the 2 McKay correspondences (1) representations of G <--> cohomology of resolution (2) conjugacy classes of G <--> homology must work, and to restate my 1992 Conjecture as a tautology, like cohomology or K-theory of projective space. Another aim is to give an introduction to Nakamura's results on the Hilbert scheme of G-clusters, following his preprints and his many helpful explanations. This is partly based on joint work with Y. Ito, and has benefited from encouragement and invaluable suggestions of S. Mukai.
alg-geom↗
Chapters on algebraic surfaces
This is a first graduate course in algebraic geometry. It aims to give the student a lift up into the subject at the research level, with lots of interesting topics taken from the classification of surfaces, and a human-oriented discussion of some of the technical foundations, but with no pretence at an exhaustive treatment. The early chapters introduce topics that are useful throughout projective and algebraic geometry, make little demands, and lead to fun calculations. The intermediate chapters introduce elements of the technical language gradually, whereas the later chapters get into the substance of the classification of surfaces. Special features include the theory of minimal models of surfaces via Mori theory, a complete selfcontained proof of the theorems on classification of surfaces, and a clean treatment of the foundational results on rational and elliptic Gorenstein surface singularities. Contents: Chapter 1. The cubic surface p.4 Exercises to Chapter 1 p.12 Chapter 2. Rational scrolls p.14 Exercises to Chapter 2 p.23 Chapter A. Curves on surfaces and intersection numbers p.26 Exs to Ch A p.35 Chapter B. Sheaves and coherent cohomology p.37 Exercises to Chapter B p.47 Chapter C. Guide to the classification of surfaces 51 Chapter 3. K3s p.63 Exercises to Chapter 3 p.76 Chapter 4. Surfaces and singularities p.80 Exercises to Chapter 4 p.106 Chapter D. Minimal models of surfaces via Mori theory p.110 Chapter E. Proof of the classification of surfaces p.121 References p.146
alg-geom↗
The McKay correspondence for finite subgroups of SL(3,\C)
This is the final draft, containing very minor proof-reading corrections. Let G in SL(n,\C) be a finite subgroup and \fie: Y -> X = \C^n/G any resolution of singularities of the quotient space. We prove that crepant exceptional prime divisors of Y correspond one-to-one with ``junior'' conjugacy classes of G. When n = 2 this is a version of the McKay correspondence (with irreducible representations of G replaced by conjugacy classes). In the case n = 3, a resolution with K_Y = 0 is known to exist by work of Roan and others; we prove the existence of a basis of H^*(Y, \Q) by algebraic cycles in one-to-one correspondence with conjugacy classes of G. Our treatment leaves lots of open problems.
alg-geom↗
Akizuki's counterexample
Following Akizuki, I construct a Noetherian local integral domain C_M whose normalisation (integral closure) is not finite over C_M. My proof follows closely Akizuki's ingenious calculations.
alg-geom↗
Nonnormal del Pezzo surfaces
This paper studies reduced, connected, Gorenstein surfaces with ample -K, assumed to be reducible or nonnormal. The normalisation is a union of one or more standard surfaces (scrolls and Veronese surfaces), marked with a conic as double locus. The question is how to glue these together to get a Gorenstein scheme. In characteristic 0, the results amount to a classification of projective surfaces in the style of the 1880s. However, the methods involve a study of the dualising sheaf of a nonnormal variety in terms of Rosenlicht differentials, and there is a subtle pathology in characteristic p due to Mori and S. Goto.
alg-geom↗