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 $X$ 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.

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.

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.

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 give a complete condition for any $n$ elements of ${\rm PSL}(2, {\mathbb R})$ to generate a Fuchsian group which is a Schottky group on that set of generators, and apply it to a question of Bowditch on representations of surface groups into ${\rm PSL}(2, {\mathbb R})$.

2000 Mathematical Subject Classification: 20H10, 32G15.

The set of normalizers between von Neumann (or, more generally, reflexive) algebras $\mathcal{A}$ and $\mathcal{B}$ (that is, the set of all operators $T$ such that $T \mathcal{A} T^{\ast} \subseteq \mathcal{B}$ and $T^{\ast} \mathcal{B} T \subseteq \mathcal{A}$) possesses ‘local linear structure’: it is a union of reflexive linear spaces. These spaces belong to the interesting class of normalizing linear spaces, namely, those linear spaces $\mathcal{U}$ of operators satisfying $\mathcal{UU}^{\ast} \mathcal{U} \subseteq \mathcal{U}$ (also known as ternary rings of operators). Such a space is reflexive whenever it is ultraweakly closed, and then it is of the form $\mathcal{U} = \{T : TL = \phi (L) T$ for all $L \in \mathcal{L}\}$ where $\mathcal{L}$ is a set of projections and $\phi$ a certain map defined on $\mathcal{L}$. A normalizing space consists of normalizers between appropriate von Neumann algebras $\mathcal{A}$ and $\mathcal{B}$. Necessary and sufficient conditions are found for a normalizing space to consist of normalizers between two reflexive algebras. Normalizing spaces which are bimodules over maximal abelian self-adjoint algebras consist of operators ‘supported’ on sets of the form $[f = g]$ where $f$ and $g$ are appropriate Borel functions. They also satisfy spectral synthesis in the sense of Arveson.

2000 Mathematical Subject Classification: 47L05 (primary), 47L35, 46L10 (secondary).

In this paper we adopt the hyperboloid in Minkowski space as the model of hyperbolic space. We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space. The hyperbolic Gauss map has been introduced by Ch. Epstein [J. Reine Angew. Math. 372 (1986) 96–135] in the Poincaré ball model, which is very useful for the study of constant mean curvature surfaces. However, it is very hard to perform the calculation because it has an intrinsic form. Here, we give an extrinsic definition and we study the singularities. In the study of the singularities of the hyperbolic Gauss map (indicatrix), we find that the hyperbolic Gauss indicatrix is much easier to calculate. We introduce the notion of hyperbolic Gauss–Kronecker curvature whose zero sets correspond to the singular set of the hyperbolic Gauss map (indicatrix). We also develop a local differential geometry of hypersurfaces concerning their contact with hyperhorospheres.

2000 Mathematical Subject Classification: 53A25, 53A05, 58C27.

In this paper the spaces of algebraic cycles on a real projective variety $X$ are studied as $\mathbb{Z}/2$-spaces under the action of the Galois group ${\rm Gal}(\mathbb{C}/\mathbb{R})$. In particular, the equivariant homotopy type of the group of algebraic $p$-cycles $\mathcal{Z}_p(\mathbb{P}_{\mathbb{C}}^n)$ is computed. A version of Lawson homology for real varieties is proposed. The real Lawson homology groups are computed for a class of real varieties.

2000 Mathematical Subject Classification: primary 55P91; secondary 14C05, 19L47, 55N91.