A precise description of a subset $\mathcal{A}$ of $\mathbb{Z} / n \mathbb{Z}$ satisfying

$$ | \mathcal{A} + \mathcal{A} | \leq 2.04 | \mathcal{A} | $$

is given. Basically, there exists a subgroup $\mathcal{H}$ of $\mathbb{Z} / n \mathbb{Z}$ such that $\mathcal{A}$ is included in an arithmetic progression of $\ell$ cosets modulo $\mathcal{H}$ and

$$(\ell - 1) | \mathcal{H} | \leq | \mathcal{A} + \mathcal{A} | - | \mathcal{A} |.$$

2000 Mathematical Subject Classification: 11B50, 11B83, 20E34.

We consider quadratic diophantine equations of the shape

$$ Q(x_1, \ldots, x_s) = 0 \qquad (1)$$

for a polynomial $Q(X_1, \ldots, X_s) \in \mathbf{Z}[X_1, \ldots, X_s]$ of degree $2$. Let $H$ be an upper bound for the absolute values of the coefficients of $Q$, and assume that the homogeneous quadratic part of $Q$ is non-singular. We prove, for all $s \ge 3$, the existence of a polynomial bound $\Lambda_s(H)$ with the following property: if equation (1) has a solution $\mathbf{x} \in \mathbf{Z}^s$ at all, then it has one satisfying

$$ |x_i| \le \Lambda_s(H) \quad (1 \le i \le s). $$

For $s = 3$ and $s = 4$ no polynomial bounds $\Lambda_s(H)$ were previously known, and for $s \ge 5$ we have been able to improve existing bounds quite significantly.

A cover of a hypergraph is a collection of edges whose union contains all vertices. Let $H = (V, E)$ be a $k$-uniform, $D$-regular hypergraph on $n$ vertices, in which no two vertices are contained in more than $o(D / e^{2k} \log D)$ edges as $D$ tends to infinity. Our results include the fact that if $k = o(\log D)$, then there is a cover of $(1 + o(1)) n / k$ edges, extending the known result that this holds for fixed $k$. On the other hand, if $k \geq 4 \log D $ then there are $k$-uniform, $D$-regular hypergraphs on $n$ vertices in which no two vertices are contained in more than one edge, and yet the smallest cover has at least $\Omega (\frac {n}{k} \log (\frac {k}{\log D} ))$ edges. Several extensions and variants are also obtained, as well as the following geometric application. The minimum number of lines required to separate $n$ random points in the unit square is, almost surely, $\Theta (n^{2/3} / (\log n)^{1/3}).$

2000 Mathematical Subject Classification: 05C65, 05D15, 60D05.

A graph $H$ is said to divide a graph $G$ if there exists a set $S$ of subgraphs of $G$, all isomorphic to $H$, such that the edge set of $G$ is partitioned by the edge sets of the subgraphs in $S$. Thus, a graph $G$ is a common multiple of two graphs if each of the two graphs divides $G$.

This paper considers common multiples of a complete graph of order $m$ and a complete graph of order $n$. The complete graph of order $n$ is denoted $K_n$. In particular, for all positive integers $n$, the set of integers $q$ for which there exists a common multiple of $K_3$ and $K_n$ having precisely $q$ edges is determined.

It is shown that there exists a common multiple of $K_3$ and $K_n$ having $q$ edges if and only if $q \equiv 0 \, ({\rm mod}\, 3)$, $q \equiv 0 \, ({\rm mod}\, \binom n2)$ and

(1) $q \neq 3 \binom n2$ when $n \equiv 5 \, ({\rm mod}\, 6)$;

(2) $q \geq (n + 1) \binom n2$ when $n$ is even;

(3) $q \notin \{36, 42, 48\}$ when $n = 4$.

The proof of this result uses a variety of techniques including the use of Johnson graphs, Skolem and Langford sequences, and equitable partial Steiner triple systems.

2000 Mathematical Subject Classification: 05C70, 05B30, 05B07.

Let $A$ be any one of the three elliptic curves over $\mathbb{Q}$ with conductor 11. We show that $A$ has Mordell–Weil rank zero over its field of 5-division points. In each case we also compute the 5-primary part of the Tate–Shafarevich group. Our calculations make use of the Galois equivariance of the Cassels–Tate pairing.

In this paper we give a realization of crystal bases for quantum affine algebras using some new combinatorial objects which we call the Young walls. The Young walls consist of colored blocks with various shapes that are built on a given ground-state wall and can be viewed as generalizations of Young diagrams. The rules for building Young walls and the action of Kashiwara operators are given explicitly in terms of combinatorics of Young walls. The crystal graph of a basic representation is characterized as the set of all reduced proper Young walls. The character of a basic representation can be computed easily by counting the number of colored blocks that have been added to the ground-state wall.

2000 Mathematical Subject Classification: 17B37, 17B65, 81R50, 82B23.

Let $X$ be a closed subscheme embedded in a scheme $W$, smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I} (X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We prove that there exists a proper, birational morphism, $\pi : W_r \longrightarrow W$, obtained as a composition of monoidal transformations, so that if $X_r \subset W_r$ denotes the strict transform of $X \subset W$ then:

(1) the morphism $\pi : W_r \longrightarrow W$ is an embedded desingularization of $X$ (as in Hironaka's Theorem);

(2) the total transform of ${\mathcal I} (X)$ in ${\mathcal O}_{W_r}$ factors as a product of an invertible sheaf of ideals ${\mathcal L}$ supported on the exceptional locus, and the sheaf of ideals defining the strict transform of $X$ (that is, ${\mathcal I}(X){\mathcal O}_{W_r} = {\mathcal L} \cdot {\mathcal I}(X_r)$).

Thus (2) asserts that we can obtain, in a simple manner, the equations defining the desingularization of $X$.

2000 Mathematical Subject Classification: 14E15.

Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parameterized family of deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a module over the ‘rational Cherednik algebra’, and a natural contravariant form on this module. In the case of the imprimitive complex reflection groups $G(m, p, N)$, the set of singular parameters in the parameterized family of these structures is described explicitly, using the theory of non-symmetric Jack polynomials.

2000 Mathematical Subject Classification: 20F55 (primary), 52C35, 05E05, 33C08 (secondary).

Given an algorithm for resolution of singularities that satisfies certain conditions (‘a good algorithm’), natural notions of simultaneous algorithmic resolution, and of equi-resolution, for families of embedded schemes (parametrized by a reduced scheme $T$) are defined. It is proved that these notions are equivalent. Something similar is done for families of sheaves of ideals, where the goal is algorithmic simultaneous principalization. A consequence is that given a family of embedded schemes over a reduced $T$, this parameter scheme can be naturally expressed as a disjoint union of locally closed sets $T_j$, such that the induced family on each part $T_j$ is equi-resolvable. In particular, this can be applied to the Hilbert scheme of a smooth projective variety; in fact, our result shows that, in characteristic zero, the underlying topological space of any Hilbert scheme parametrizing embedded schemes can be naturally stratified in equi-resolvable families.

We introduce orbifold Euler numbers for normal surfaces with boundary $\mathbb{Q}$-divisors. These numbers behave multiplicatively under finite maps and in the log canonical case we prove that they satisfy the Bogomolov–Miyaoka–Yau type inequality. Existence of such a generalization was earlier conjectured by G. Megyesi [Proc. London Math. Soc. (3) 78 (1999) 241–282]. Most of the paper is devoted to properties of local orbifold Euler numbers and to their computation.

As a first application we show that our results imply a generalized version of R. Holzapfel's ‘proportionality theorem’ [Ball and surface arithmetics, Aspects of Mathematics E29 (Vieweg, Braunschweig, 1998)]. Then we show a simple proof of a necessary condition for the logarithmic comparison theorem which recovers an earlier result by F. Calderón-Moreno, F. Castro-Jiménez, D. Mond and L. Narváez-Macarro [Comment. Math. Helv. 77 (2002) 24–38].

Then we prove effective versions of Bogomolov's result on boundedness of rational curves in some surfaces of general type (conjectured by G. Tian [Springer Lecture Notes in Mathematics 1646 (1996) 143–185)]. Finally, we give some applications to singularities of plane curves; for example, we improve F. Hirzebruch's bound on the maximal number of cusps of a plane curve.

2000 Mathematical Subject Classification: 14J17, 14J29, 14C17.

Let $R$ be a ring and $M_R$ be an $R$-module. We characterize the existence of ${\rm Add} M$-covers and ${\rm Add} M$-envelopes in terms of finiteness conditions of $M$ over its endomorphism ring. We then present some applications related to the existence of well-behaved direct sum decompositions for direct products of copies of $M$. Our results can be viewed as natural extensions of classical theorems of Bass and Chase on coherent and perfect rings.

Broué's abelian defect conjecture suggests a deep link between the module categories of a block of a group algebra and its Brauer correspondent, viz. that they should be derived equivalent. We are able to verify Broué's conjecture for the Hall–Janko group, even its double cover $2.J_2$, as well as for $U_3(4)$ and ${\rm Sp}_4(4)$. In fact we verify Rickard's refinement to Broué's conjecture and show that the derived equivalence can be chosen to be a splendid equivalence for these examples.

2000 Mathematical Subject Classification: 20C20, 20C34.

The principal 3-block of a Chevalley group $G_2(q)$ with $q$ a power of 2 satisfying $q \equiv 2$ or 5 mod 9 and the principal 3-block of $G_2(2)$ are Morita equivalent.

2000 Mathematical Subject Classification: 20C05, 20C20, 20C33.

We prove the following: if a group $\Gamma$ is torsion-free, and relatively hyperbolic (with the Bounded Coset Penetration property), relative to a subgroup admitting a finite classifying space, then $\Gamma$ admits a finite classifying space. In this case, if the subgroup admits a boundary in the sense of $\mathcal{Z}$-structures, we prove that $\Gamma$ admits a boundary. This extends classical results of Rips, and of Bestvina and Mess to the relative case.

Let $D$ be a bounded strongly convex domain and let $f$ be a holomorphic self-map of $D$. In this paper we introduce and study the dilatation $\alpha (f)$ of $f$ defined, if $f$ has no fixed points in $D$, as the usual boundary dilatation coefficient of $f$ at its Wolff point, or, if $f$ has some fixed points in $D$, as the ratio of shrinking of the Kobayashi balls around a fixed point of $f$. In particular, we show that the map $\alpha$, defined as $\alpha : f \mapsto \alpha (f) \in [0,1]$, is lower semicontinuous. Among other things, this allows us to study the limits of a family of holomorphic self-maps of $D$. In the case of an inner fixed point, the dilatation is an intrinsic measure of the order of contact of $f(D)$ to $\partial D$.

Finally, using complex geodesics, we define and study a directional dilatation, which is a measure of the shrinking of the domain along a given direction. Again, results of semicontinuity are given and applied to a family of holomorphic self-maps.

2000 Mathematical Subject Classification: primary 32H99; secondary 30F99, 32H15.

The conjecture that Radon measures in Euclidean space with positive finite density are rectifiable was a central problem in Geometric Measure Theory for fifty years. This conjecture was positively resolved by Preiss in 1986, using methods entirely dependent on the symmetry of the Euclidean unit ball. Since then, due to reasons of isometric immersion of metric spaces into $l_{\infty}$ and the uncommon nature of the sup norm even in finite dimensions, a popular model problem for generalising this result to non-Euclidean spaces has been the study of 2-uniform measures in $l^{3}_{\infty}$. The rectifiability or otherwise of these measures has been a well-known question.

In this paper the stronger result that locally 2-uniform measures in $l^{3}_{\infty}$ are rectifiable is proved. This is the first result that proves rectifiability, from an initial condition about densities, for general Radon measures of dimension greater than 1 outside Euclidean space.

2000 Mathematical Subject Classification: 28A75.

The statement (and the proof) of Theorem 4.1.1 of the paper ‘Mackey formula in type A’ [Proc. London. Math. Soc. (3) 80 (2000) 545–574] is false. In this corrigendum we provide a correct statement (and a correct proof). As a consequence, we see that Corollary 4.1.2 of the paper holds. So all the other statements in the paper are correct (up to minor misprints…).

2000 Mathematical Subject Classification: 20G05, 20G40.

We study Rouquier blocks of symmetric groups and Schur algebras in detail, and obtain explicit descriptions for the radical layers of the principal indecomposable, Weyl, Young and Specht modules of these blocks. At the same time, the Jantzen filtrations of the Weyl modules are shown to coincide with their radical filtrations. We also address the conjectures of Martin, Lascoux–Leclerc–Thibon–Rouquier and James for these blocks.

We classify smooth ${\rm SO}(n)$-equivariant principal bundles over $S^n$ in terms of their isotropy representations over the north and south poles. This is an example of a general result classifying equivariant $(\Pi, G)$-bundles over manifolds with cohomogeneity 1.

2000 Mathematical Subject Classification: 55R91.

The definition of Thom polynomials for Lagrange, Legendre and critical point function singularities is given. The approach is based on the notion of classifying space of singularities. This approach provides a universal method for computing Thom polynomials. Characteristic classes of complex Lagrange and Legendre singularities of small codimension are computed. Two independent methods for these computations are presented. The first is the direct one based on the resolution of singularities, the Gysin homomorphism, and the notion of adjacency exponents. The second uses the existence theorem for Thom polynomials and is based on Rimányi's idea of using symmetries. The expressions for the resulting polynomials reduced modulo 2 agree with those obtained by Vassiliev for the real case.