Although the theory of singularities of curves - resolution, classification, numerical invariants - goes through with comparatively little change in finite characteristic, pencils of curves are more difficult. Bertini's theorem only holds in a much weaker form, and it is convenient to restrict to pencils such that, when all base points are resolved, the general member of the pencil becomes non-singular. Even here, the usual rule for calculating the Euler characteristic of the resolved surface has to be modified by a term measuring wild ramification.

We begin by describing this background, then proceed to discuss the exceptional members of a pencil. In characteristic 0 it was shown by Há and Lê and by Lê and Weber, using topological reasoning, that exceptional members can be characterised by their Euler characteristics. We present a combinatorial argument giving a corresponding result in characteristic p. We first treat pencils with no base points, and then reduce the remaining case to this.

2000 Mathematical Subject Classification: primary 14H20; secondary 14D05, 14E22, 14F20.

The controlled wild algebra is introduced, a covering criterion for a finite-dimensional algebra to be controlled wild is given, and this criterion is applied to the algebras with radical square zero, algebras with zero relations, local algebras and finite p-group algebras.

2000 Mathematical Subject Classification: 16G20, 16G60.

The main part of the paper finds necessary conditions for solubility of a pencil of curves of genus 1, each of which is a 2-covering of an elliptic curve with at least one 2-division point. As in previous work, these are proved subject to Schinzel's Hypothesis and to the finiteness of the Tate-\u{S}afarevi\u{c} group of elliptic curves defined over a number field. It thus generalizes earlier work of Colliot-Thélène, Skorobogatov and the second author.

The final section gives necessary conditions (though of a rather ugly nature) for the solubility of a Del Pezzo surface of degree 4.

2000 Mathematical Subject Classification: 11D25.

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.

We consider a Hamiltonian setup $(\mathcal M,\omega,H,\mathfrak L,\Gamma,\mathcal P)$, where $(\mathcal M,\omega)$ is a symplectic manifold, $\mathfrak L$ is a distribution of Lagrangian subspaces in $\mathcal M$, $\mathcal P$ is a Lagrangian submanifold of $\mathcal M$, $H$ is a smooth time-dependent Hamiltonian function on $\mathcal M$, and $\Gamma:[a,b]\to\mathcal M$ is an integral curve of the Hamiltonian flow $\vec H$ starting at $\mathcal P$. We do not require any convexity property of the Hamiltonian function $H$. Under the assumption that $\Gamma(b)$ is not $\mathcal P$-focal, we introduce the Maslov index $\mathrm i_{\mathrm{maslov}}(\Gamma)$ of $\Gamma$ given in terms of the first relative homology group of the Lagrangian Grassmannian; under generic circumstances $\mathrm i_{\mathrm{maslov}}(\Gamma)$ is computed as a sort of algebraic count of the $\mathcal P$-focal points along $\Gamma$. We prove the following version of the Index Theorem: under suitable hypotheses, the Morse index of the Lagrangian action functional restricted to suitable variations of $\Gamma$ is equal to the sum of $\mathrm i_{\mathrm{maslov}}(\Gamma)$ and a convexity term of the Hamiltonian $H$ relative to the submanifold $\mathcal P$. When the result is applied to the case of the cotangent bundle $\mathcal M=TM^*$ of a semi-Riemannian manifold $(M,g)$ and to the geodesic Hamiltonian $H(q,p)=\frac12 g^{-1}(p,p)$, we obtain a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics with variable endpoints in Riemannian geometry.

2000 Mathematical Subject Classification: 37J05, 53C22, 53C50, 53D12, 70H05.

The Hardy operator $T_a$ on a tree $\Gamma$ is defined by \[ (T_af)(x):=v(x) \int^x_a f(t)u(t)\,dt \quad \mbox{for } a, x\in \Gamma. \] Properties of $T_a$ as a map from $L^p(\Gamma)$ into itself are established for $1\le p \le \infty$. The main result is that, with appropriate assumptions on $u$ and $v$, the approximation numbers $a_n(T_a)$ of $T_a$ satisfy \begin{equation*} \tag{$*$} \lim_{n\rightarrow \infty} na_n(T_a) = \alpha_p\int_{\Gamma}|uv|\,dt \end{equation*} for a specified constant $\alpha_p$ and $1 p<\infty$. This extends results of Naimark, Newman and Solomyak for $p=2$. Hitherto, for $p\neq 2$, $(*)$ was unknown even when $\Gamma$ is an interval. Also, upper and lower estimates for the $l^q$ and weak-$l^q$ norms of $\{a_n(T_a)\}$ are determined.

2000 Mathematical Subject Classification: 47G10, 47B10.

We relate the brace products of a fibration with section to the differentials in its Serre spectral sequence. In the particular case of free loop fibrations, we establish a link between these differentials and Browder operations in the fiber. Applications and several calculations (for the particular cases of spheres and wedges of spheres) are given.

2000 Mathematical Subject Classification: 55P35, 55Q15, 55R05, 55R20.

Given a fibered 3-manifold M, we investigate exactly which boundary slopes can be realized by perturbing fibrations along product discs. Since these perturbed fibrations cap off to give taut foliations in the corresponding surgery manifolds, we obtain surgery information. For example, recall that a knot k is said to satisfy Property P if no finite surgery along k yields a simply-connected 3-manifold. We show that if a non-trivial fibered knot $k\subset S^3$ fails to satisfy Property P, then necessarily k is hyperbolic with degeneracy slope $\pm\frac{2}{1}$. When k is hyperbolic and $d(k)=\frac{2}{1}$ (respectively, $-\frac{2}{1}$), we show that the only candidate for a counterexample to Property P is surgery coefficient $\frac{1}{1}$ (respectively, $-\frac{1}{1}$).

2000 Mathematical Subject Classification: primary 57M25; secondary 57R30.

Let R be a Seifert surface obtained by applying Seifert's algorithm to a connected diagram D for a link L. In this paper, letting D be almost alternating, we give a practical algorithm to determine whether L is a fibered link and R is a fiber surface. We further show that L is a fibered link and R is a fiber surface for L if and only if R is a Hopf plumbing, that is, a successive plumbing of a finite number of Hopf bands. It has been known for some time that this is true if D is alternating, and we show that it is not always true if D is 2-almost alternating. In the appendix, we partially answer C. Adams's open question concerning almost alternating diagrams.

2000 Mathematical Subject Classification: 57M25.

We determine explicitly all the stable manifolds for the stochastic flows on certain homogeneous spaces of a semisimple Lie group of non-compact type induced by Lévy processes on the Lie group.

2000 Mathematical Subject Classification: 58J65.