Miles Reid

The McKay correspondence as an equivalence of derived categories

Let G be a finite group of automorphisms of a nonsingular complex threefold M such that the canonical bundle omega_M is locally trivial as a G-sheaf. We prove that the Hilbert scheme Y=GHilb M parametrising G-clusters in M is a crepant resolution of X=M/G and that there is a derived equivalence (Fourier- Mukai transform) between coherent sheaves on Y and coherent G-sheaves on M. This identifies the K theory of Y with the equivariant K theory of M, and thus generalises the classical McKay correspondence. Some higher dimensional extensions are possible.

Decomposition of Toric Morphisms

(0.1) This paper applies the ideas of Mori theory [4] to toric varieties. Let X be a projective tonic variety (over any field) constructed from a simplicial fan F. The cone of effective 1-cycles NE(X) is polyhedral (1.7), spanned by the 1-strata l w ⊂ X; the condition that a 1-stratum l w gives an extremal ray R = Q + l w of NE(X) has a nice interpretation (2.10) in terms of the geometry of F around the wall w.

Explicit Birational Geometry of 3-Folds: Fano 3-fold hypersurfaces

We study the birational geometry of the 95 families of Fano 3-fold weighted hypersurfaces X = Xd ⊂ P(1, a1, a2, a3, a4), corresponding to the famous 95 families of K3 surfaces Xd ⊂ P(a1, a2, a3, a4) of Reid and Fletcher ([C3-f, §4] and [Fl, 13.3]). Our main aim is to prove a rigidity theorem for the general Xd in each family, by analogy with the famous theorem of Iskovskikh and Manin on the rigidity of the quartic 3-fold; on the way, we derive as much instruction and amusement as possible on topics in biregular and birational geometry of Fano 3-folds and Mori fibre spaces. While this paper uncovers an amazing wealth of new phenomena and methods of calculation, many of the basic questions remain open, and we spell some of these out.

Kustin–Miller unprojection without complexes

A main ingredient for Kustin–Miller unprojection, as developed in [PR], is the module HomR(I, ωR), where R is a local Gorenstein ring and I a codimension one ideal with R/I Gorenstein. We prove a method of calculating it in a relative setting using resolutions. We give three applications. In the first we generalise a result of [CFHR]. The second and the third are about Tom and Jerry, two families of Gorenstein codimension four rings with 9 × 16 resolutions.

Embeddings of curves and surfaces

We prove a general embedding theorem for Cohen-Macaulay curves (possibly nonreduced), and deduce a cheap proof of the standard results on pluricanonical embeddings of surfaces, assuming vanishing H 1 (2 K X ) = 0.

Explicit birational geometry of 3-folds

Foreword 1. One parameter families containing three dimensional toric Gorenstein singularities K. Altmann 2. Nonrational covers of CPm x CPn J. Kollar 3. Essentials of the method of maximal singularities A. V. Pukhlikov 4. Working with weighted complete intersections A. R. Iano-Fletcher 5. Fano 3-fold hypersurfaces A. Corti, A. V. Pukhlikov and M. Reid 6. Singularities of linear systems and 3-fold birational geometry A. Corti 7. Twenty five years of 3-folds - an old persons view M. Reid.

Fano 3-folds in codimension 4, Tom and Jerry. Part I

We introduce a strategy based on Kustin–Miller unprojection that allows us to construct many hundreds of Gorenstein codimension 4 ideals with 9×16 resolutions (that is, nine equations and sixteen first syzygies). Our two basic games are called Tom and Jerry; the main application is the biregular construction of most of the anticanonically polarised Mori Fano 3-folds of Altinok’s thesis. There are 115 cases whose numerical data (in effect, the Hilbert series) allow a Type I projection. In every case, at least one Tom and one Jerry construction works, providing at least two deformation families of quasismooth Fano 3-folds having the same numerics but different topology.

Ice cream and orbifold Riemann-Roch

We give an orbifold Riemann-Roch formula in closed form for the Hilbert series of a quasismooth polarized n-fold (X,D), under the assumption that X is projectively Gorenstein with only isolated orbifold points. Our formula is a sum of parts each of which is integral and Gorenstein symmetric of the same canonical weight; the orbifold parts are called ice cream functions. This form of the Hilbert series is particularly useful for computer algebra, and we illustrate it on examples of K3 surfaces and Calabi-Yau 3-folds. These results apply also with higher dimensional orbifold strata (see [1] and [2]), although the precise statements are considerably trickier. We expect to return to this in future publications.

Constructing Algebraic Varieties via Commutative Algebra

Problems on the existence and moduli of abstract varieties in the classification of varieties can often be studied by embedding the variety X into projective space, preferably in terms of an intrinsically determined ample line bundle L such as the (anti-) canonical class or its submultiples. A comparatively modern twist on this old story is to study the graded coordinate ring R(X,L) = ⊕ n≥0 H0(X,L⊗n), which in interesting cases is a Gorenstein ring; this makes available theoretical and computations tools from commutative algebra and computer algebra. The varieties of interest are curves, surfaces, 3-folds, and historical results of Enriques, Fano and others are sometimes available to serve as a guide. This has been a prominent area of work within European algebraic geometry in recent decades, and the lecture will present the current state of knowledge, together with some recent examples.

Diptych varieties, I

We present a new class of affine Gorenstein 6-folds obtained by smoothing the 1-dimensional singular locus of a reducible affine toric surface; their existence is established using explicit methods in toric geometry and serial use of Kustin–Miller Gorenstein unprojection. These varieties have applications as key varieties in constructing other varieties, including local models of Mori flips of Type A.