Let ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ be the Hilbert scheme of closed 1-dimensional subschemes of degree 6 and arithmetic genus 4 in $\mathbb{P}^3$. Let $H$ be the component of ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ whose generic point corresponds to a canonical curve, that is, a complete intersection of a quadric and a cubic surface in $\mathbb{P}^3$. Let $F$ be the vector space of linear forms in the variables $z_1, z_2, z_3, z_4$. Denote by $F_d$ the vector space of homogeneous forms of degree $d$. Set $X = \{(f_2,f_3)\}$ where $f_2 \in \mathbb{P}(F_2)$ is a quadric surface, and $f_3 \in \mathbb{P}(F_3/f_2 \cdot F)$ is a cubic modulo $f_2$. We have a rational map, $\sigma : X \cdots \rightarrow H$ defined by $(f_2,f_3) \mapsto f_2 \cap f_3$. It fails to be regular along the locus where $f_2$ and $f_3$ acquire a common linear component. Our main result gives an explicit resolution of the indeterminacies of $\sigma$ as well as of the singularities of $H$.

2000 Mathematical Subject Classification: 14C05, 14N05, 14N10, 14N15.

A semilinear space S is ultrahomogeneous if each isomorphism between the semilinear structures induced on two finite subsets can be extended to an automorphism of S. We give a complete classification of all finite ultrahomogeneous semilinear spaces. Our theorem extends a result of A. Gardiner on graphs and a result of A. Devillers and J. Doyen on linear spaces.

2000 Mathematical Subject Classification: 05B25, 51E14, 20B25.

Let $\Phi=(f,g):({\mathbb C}^{n+1},{\bf 0}) \to({\mathbb C}^2,{\bf 0})$ be a pair of holomorphic germs with no blowing up in codimension 0. (Two examples are the following: $\Phi$ defines an isolated complete intersection singularity; $g=\ell^N$ where $\ell$ is a generic linear form with respect to $f$ and $N>0$.) We study how the Milnor fibrations of the germs $\varphi_{(\alpha:\beta)}=\alpha g-\beta f$ are related to each other when $(\alpha:\beta)$ varies in ${\mathbb P}^1$. More precisely, we construct isotopic subfibrations or subfibres of the Milnor fibrations of any two such germs. The proofs are based on the precise study of the subdiscs of complex lines meeting a fixed complex plane curve germ transversally, generalizing Lê's work on the Cerf diagram.

2000 Mathematical Subject Classification: 32S55, 32S15, 32S30.

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).

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 germs of $n$-parameter families of functions, that is, function-germs of the type $f : (\mathbb{R}^n \times \mathbb{R}, 0) \to (\mathbb{R}, 0)$ defined on the total space of the trivial bundle $ \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^n $. There is a natural notion of $V$-equivalence for such function-germs. We introduce the Young diagram of $n$-parameter families satisfying a non-degeneracy condition. We classify all such simple $n$-parameter families and give their versal deformations. This result has direct applications to contact and projective geometry.

We investigate categorical and amalgamation properties of the functor $\operatorname{Id_c}$ assigning to every partially ordered abelian group $G$ its $\langle \vee, 0 \rangle$-semilattice of compact ideals $\operatorname{Id_c} G$. Our main result is the following.

Theorem 1. Every diagram of finite Boolean semilattices indexed by a finite dismantlable partially ordered set can be lifted, with respect to the$\operatorname{Id_c}$functor, by a diagram of pseudo-simplicial vector spaces.

Pseudo-simplicial vector spaces are a special kind of finite-dimensional partially ordered vector spaces with interpolation over the field of rational numbers. The methods introduced also make it possible to prove the following ring-theoretical result.

Theorem 2. For any countable distributive$\langle \vee, 0 \rangle$-semilattices$S$and$T$and any field$K$, any$\langle \vee, 0 \rangle$-homomorphism$f \colon S \to T$can be lifted, with respect to the$\operatorname{Id_c}$functor on rings, by a homomorphism $f \colon A \to B$of$K$-algebras, for countably dimensional locally matricial algebras$A$and$B$over$K$.

We also state a lattice-theoretical analogue of Theorem 2 (with respect to the $\operatorname{Con_c}$ functor), and we provide counterexamples to various related statements. In particular, we prove that the result of Theorem 1 cannot be achieved with simplicial vector spaces alone.

Let $G$ denote a connected semisimple and simply connected algebraic group over an algebraically closed field $k$ of positive characteristic and let $g$ denote a regular element of $G$. Let $X$ denote any equivariant embedding of $G$. We prove that the closure of the conjugacy class of $g$ within $X$ is normal and Cohen–Macaulay. Moreover, when $X$ is smooth we prove that this closure is a local complete intersection. As a consequence, the closure of the unipotent variety within $X$ shares the same geometric properties.

We study the subgroup growth of profinite groups. We obtain a structure theorem for profinite groups of polynomial subgroup growth (PSG groups, for short), which essentially reduces their characterization to the case where the group is a cartesian product of finite simple groups. Analysing the growth behaviour of such cartesian products, we construct, for any real number $\alpha \ge 1$, a PSG profinite group whose degree is exactly $\alpha$. Applications to the behaviour of the abscissa of convergence of the associated zeta function $\sum a_n(G)n^{-s}$ are drawn. We also show that there is no gap between polynomial and non-polynomial subgroup growth by constructing non-PSG groups whose subgroup growth is arbitrarily slow. Our arguments rely heavily on the use of sieve methods in number theory. In particular, a Bombieri-type short intervals theorem and the so-called Fundamental lemma in sieve theory play an essential role in this paper.

1991 Mathematics Subject Classification: 20E07, 11N36.

Semisimple Banach algebras are well-known to have a unique (complete) algebra norm topology, but such uniqueness may fail if the radical is even one-dimensional. We obtain a necessary condition for uniqueness of norm when the algebra has finite-dimensional radical. In the case where the Banach algebra is separable, the condition is shown to be also sufficient for a large class of algebras, and in particular under various hypotheses of commutativity. Examples are given to show the limitations of the various sufficiency results, and these also give a good indication of where the difficulties lie in general. We conjecture that, at least in the separable case, our condition is both necessary and sufficient.

1991 Mathematics Subject Classification: 46H20, 46H40.

This paper introduces a new approach to the study of certain aspects of Galois module theory by combining ideas arising from the study of the Galois structure of torsors of finite group schemes with techniques coming from relative algebraic $K$-theory.

Let $H$ be the discrete Schrödinger operator

$$Hu(n):=u(n-1)+u(n+1)+v(n)u(n),\quad u(0)=0$$

acting on $l^2 (\mathbb{Z}^+)$, where the potential $v$ is real-valued and $v(n) \to 0$ as $n \to \infty$. Let <formula form="inline" disc="math" id="frm007"><formtex notation="AMSTeX">$P$ be the orthogonal projection onto a closed linear subspace $\Lambda \subset l^2 (\mathbb{Z}^+)$. In a recent paper E. B. Davies defines the second order spectrum ${\rm Spec}_2(H, \Lambda)$ of $H$ relative to $\Lambda$ as the set of $z \in \mathbb{C}$ such that the restriction to $\Lambda$ of the operator $P(H - z)^2P$ is not invertible within the space $\Lambda$. The purpose of this article is to investigate properties of ${\rm Spec}_2(H, \Lambda)$ when $\Lambda$ is large but finite dimensional. We explore in particular the connection between this set and the spectrum of $H$. Our main result provides sharp bounds in terms of the potential $v$ for the asymptotic behaviour of ${\rm Spec}_2(H, \Lambda)$ as $\Lambda$ increases towards $l^2(\mathbb{Z}^+)$.

Let $V$ be a vector space over the division ring $D$ of infinite dimension. We study locally finite, primitive groups $G$ of finitary linear automorphisms of $V$. We show that the derived group $G'$ of $G$ is infinite, simple, and lies in every non-trivial normal subgroup of $G$, and that $G' \le G \le {\rm Aut\,}G'$. Moreover if ${\rm char\,}D = 0$, then $G$ is either the finitary symmetric group or the alternating group on some infinite set. If $D$ is commutative, that is, if $D$ is a field, then all these results are known and are the consequence of the collective work of a number of people: in particular (in alphabetical order) V.~V.~Belyaev, J.~I.~Hall, F.~Leinen, U.~Meierfrankenfeld, R.~E.~Phillips, O.~Puglisi, A.~Radford and quite probably others. 2000 Mathematics Subject Classification: 20H25, 20H20, 20F50.

New criteria and Banach spaces are presented (for example, $GL$-spaces and Banach spaces with property $(\alpha)$) that ensure that the Boolean algebra generated by a pair of bounded, commuting Boolean algebras of projections is itself bounded. The notion of $R$-boundedness plays a fundamental role. It is shown that the strong operator closure of any $R$-bounded Boolean algebra of projections is necessarily Bade complete. Also, for a Dedekind $\sigma <formula form="inline" disc="math" id="frm006"><formtex notation="AMSTeX">$-complete Banach lattice $E$, the Boolean algebra consisting of all band projections in $E$ is $R$-bounded if and only if $E$ has finite cotype. In this situation, every bounded Boolean algebra of projections in $E$ is $R$-bounded and has a Bade complete strong closure.

2000 Mathematical Subject Classification: 10B10.

If $\mathbb{C}[G] \hookrightarrow \mathbb{C}[H]$ is an extension of Hopf domains of degree $d$, then $H \twoheadrightarrow G$ is an \'etale map. Equivalently, the variety $X_{\mathbb{C}[H]}$ of $d$-dimensional $\mathbb{C}[H]$-modules compatible with the trace map of the extension, is a smooth $\mbox{GL}_d$-variety with quotient $G$. If we replace $\mathbb{C}[H]$ by a non-commutative Hopf algebra $H$, we construct similarly a $\mbox{GL}_d$-variety and quotient map $\pi : X_H \twoheadrightarrow G$. The smooth locus of $H$ over $\mathbb{C}[G]$ is the set of points $g \in G$ such that $X_H$ is smooth along$\pi^{-1}(g)$. We relate this set to the separability locus of $H$ over $\mathbb{C}[G]$ as well as to the (ordinary) smooth locus of the commutative extension $\mathbb{C}[G] \hookrightarrow Z$ where $Z$ is the centre of $H$. In particular, we prove that the smooth locus coincides with the separability locus whenever $H$ is a reflexive Azumaya algebra. This implies that the quantum function algebras $O_{\epsilon}(G)$ and quantised enveloping algebras $U_{\epsilon}(\mathfrak{g})$ are as singular as possible. 1991 Mathematics Subject Classification: 16W30, 16R30.

We consider the Zariski closure of Thom–Boardman strata $\Sigma^{i, j}$ in the jet space for map-germs $(\mathbf{C}^n, 0)\to(\mathbf{C}^p, 0)$. We apply the geometric technique of calculating syzygies to the desingularization of $\Sigma^{i, j}$ constructed by Ronga. We investigate when the normalization of the coordinate ring of $\Sigma^{i, j}$ is rational in the cases $i = n - p + 1$ or $i = j = 1$. We also consider when these coordinate rings are normal with rational singularities.

Explicit quadratic Liapunov functions that provide necessary and sufficient conditions for the asymptotic stability of the system of linear difference equations $x (t + 1) = A x(t)$ are constructed by transforming the original systems to $y (t + 1) = G y(t)$ , where $G$ is a companion matrix associated with the characteristic polynomial of $A$. A necessary and sufficient condition for all roots of the characteristic polynomial to lie in the unit circle $|z| < 1$ on the complex plane is also derived.

We construct a family, $\{HM^{(k)}\}_{k\geq 2}$, of oriented topological equivalence invariants, for projective configurations of complex hyperplanes with fixed combinatorial pattern. We construct a family, $\{PM^{(k)}\}_{k \geq 2}$, of concordance invariants, for classical links with a fixed number of components. The constructions are carried out in the same manner, using the existence of natural additional structures on the nilpotent quotients of the fundamental groups of the complements. We give the explicit computation of the primary and secondary HM and PM invariants.

2000 Mathematical Subject Classification: primary 57M25, 32S50; secondary 20F10, 20F12.

For an arbitrary field $F$ we study the double scissors congruence groups $A_n(F)$ generated by admissible pairs of simplices in the projective $n$-space ${\mathbb P}_F^n$ over $F$. Let $(L, M)$ be an admissible pair of simplices whose faces are defined as hyperplanes in ${\mathbb P}_F^n$ such that they do not have common faces of the same dimension. When $L$ and $M$ are in general positions we define a linearly constructible motivic perverse sheaf producing a motivic cohomology whose Betti realization is the relative cohomology $H^{\ast}({\mathbb P}_F^n\setminus L, M\setminus L; {\mathbb Q})$. The motivic cohomology provides a simple explanation of why the generic part of $A_\bullet$ forms a Hopf algebra with well-defined coproduct. When $n = 2$ we define explicitly the coproduct on $A_2$ which simplifies the approach of Beilinson et al. We also complete the same task for $A_3$ and $A_4$ which enables us to develop further results in another paper connecting Aomoto trilogarithms to the classical ones.