Let X be a smooth projective complex curve of genus g ≥ 2, let Λ→X be a line bundle of degree d> 0, and let (E, φ) be a pair consisting of a vector bundle E →X such that Λ 2 E = Λ and a section φ ∈ H 0 (E) − 0. This paper will study the moduli theory of such pairs. However, it is by no means a routine generalization of the well-known theory of stable

Ever since the invention of geometric invariant theory, it has been understood that the quotient it constructs is not entirely canonical, but depends on a choice: the choice of a linearization of the group action. However, the founders of the subject never made a systematic study of this dependence. In light of its fundamental and elementary nature, this is a rather surprising gap, and this paper will attempt to fill it. In a way, this neglect is understandable, because the different quotients must be related by birational transformations, whose structure in higher dimensions is poorly understood. However, it has been considerably clarified in the last dozen years with the advent of Mori theory. In particular, the birational transformations that Mori called flips are ubiquitous in geometric invariant theory; indeed, one of our main results (3.3) describes the mild conditions under which the transformation between two quotients is given by a flip. This paper will not use any of the deep results of Mori theory, but the notion of a flip is central to it. The definition of a flip does not describe the birational transformation explicitly, but in the general case there is not much more to say. So to obtain more concrete results, hypotheses

Abstract. I prove the existence, and describe the structure, of moduli space of pairs (P, Θ) consisting of a projective variety P with semiabelian group action and an ample Cartier divisor on it satisfying a few simple conditions. Every connected component of this moduli space is proper. A component containing a projective toric variety is described by a configuration of several polytopes the main of which is the secondary polytope. On the other hand, the component containing a principally polarized abelian variety provides a moduli compactification of Ag. The main irreducible component of this compactification is described by an ”infinite periodic ” analog of secondary polytope and coincides with the

A fibration of graphs is a morphism that is a local isomorphism of in-neighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. This paper develops systematically the theory of graph fibrations, emphasizing in particular those results that recently found application in the theory of distributed systems.

We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n − 1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action G � X determines a collection of “peripheral ” subgroups H1,..., Hk ⊂ G so that the group pair (G, {H1,..., Hk}) is an n-dimensional Poincare duality pair. In particular, if G is a 2-dimensional 1-ended group of type F P2, and G � X is a free simplicial action on a coarse P D(3) space X, then G contains surface subgroups; if in addition X is simply connected, then we obtain a partial generalization of the Scott/Shalen compact core theorem to the setting of coarse P D(3) spaces. In the process we develop coarse topological language and a formulation of coarse Alexander duality which is suitable for applications involving quasi-isometries and geometric group theory.

1.1 Rankin-Selberg L-functions................... 2

Abstract. We construct modular Deligne-Mumford stacks Pd,g representable over Mg parametrizing Néron models of Jacobians as follows. Let B be a smooth curve and K its function field, let XK be a smooth genus-g curve over K admitting stable minimal model over B. The Néron model N(PicdXK) → B is then the base change of Pd,g via the moduli map B − → Mg of f, i.e.: N(PicdXK) ∼ = Pd,g × B. Moreover P Mg d,g is compactified by a Deligne-Mumford stack over Mg, giving a completion of Néron models naturally stratified in terms of Néron models.

The theory of characteristic classes of vector bundles and smooth manifolds plays an important role in the theory of smooth manifolds. An investigation of reasonable notions of characteristic classes of singular spaces started with a systematic study of singular spaces such as singular algebraic varieties. We give a quick survey of characteristic classes of singular varieties, mainly focusing on the functorial aspects of some important ones such as the singular versions of the Chern class, the Todd class and Thom–Hirzebruch’s L-class. Further we explain our recent “motivic” characteristic classes, which in a sense unify these three different theories of characteristic classes. We also discuss bivariant versions of them and characteristic classes of proalgebraic varieties, which are related to the motivic measures/integrations. Finally we explain some recent work on “stringy” versions of these theories, together with some references for “equivariant” counterparts.

We prove some results of the form “r residually irreducible and residually modular implies r is modular, ” where r is a suitable continuous odd 2-dimensional 2-adic representation of the absolute Galois group of Q. These results are analogous to those obtained by A. Wiles, R. Taylor, F. Diamond, and others for p-adic representations in the case when p is odd; some extra work is required to overcome the technical difficulties present in their methods when p = 2. The results are subject to the assumption that any choice of complex conjugation element acts nontrivially on the residual representation, and the results are also subject to an ordinariness hypothesis on the restriction of r to a decomposition group at 2. Our main theorem (Theorem 4) plays a major role in a programme initiated by Taylor to give a proof of Artin’s conjecture on the holomorphicity of L-functions for 2-dimensional icosahedral odd representations of the absolute Galois group of Q; some results of this programme are described in a paper that appears in this issue, jointly authored with K. Buzzard, N. Shepherd-Barron, and Taylor.

Abstract. Given a global field k and a geometrically integral algebraic k-scheme X, we relate a subgroup of the cohomological Brauer group of X to the Tate-Shafarevich group of the Gal(k/k)-module PicX. This generalizes the well-known relations that exist between Brauer groups of proper, smooth and geometrically integral curves over global fields and Tate-Shafarevich groups of Jacobians. As an application, we prove that the Hasse principle holds for the Brauer group of certain types of bundles of projective homogeneous varieties over number fields. 0. Introduction. M.Artin was the first to notice that there exist close relations between the Brauer group of a curve over a global field and the Tate-Shafarevich group of its Jacobian variety (see [34, §3]). The first precise results were obtained by this author in collaboration with J.Tate [op.cit]. Shortly afterwards, A.Grothendieck generalized