The Albanese map of a 3-fold of general type whose canonical map is composed with a pencil

Received: 15 October 1999; in final form: 13 June 2000 / Published online: 29 April 2002     

Twelve points on the projective line, branched covers, and rational elliptic fibrations

The following divisors in the space of twelve points on are actually the same: the possible locus of the twelve nodal fibers in a rational elliptic fibration (i.e. a pencil of plane cubic curves); degree 12 binary forms that can be expressed as a cube plus a square; the locus of the twelve tangents to a smooth plane quartic from a general point of the plane; the branch locus of a degree 4 map from a hyperelliptic genus 3 curve to ; the branch locus of a degree 3 map from a genus 4 curve to induced by a theta-characteristic; and several more. The corresponding moduli spaces are smooth, but they are not all isomorphic; some are finite étale covers of others. We describe the web of interconnections among these spaces, and give monodromy, rationality, and Prym-related consequences. Enumerative consequences include: (i) the degree of this locus is 3762 (e.g. there are 3762 rational elliptic fibrations with nodes above 11 given general points of the base); (ii) if is a cover as in , then there are 135 different such covers branched at the same points; (iii) the general set of 12 tangent lines that arise in turn up in 120 essentially different ways. Some parts of this story are well known, and some other parts were known classically (to Zeuthen, Zariski, Coble, Mumford, and others). The unified picture is surprisingly intricate and connects many beautiful constructions, including Recillas' trigonal construction and Shioda's -Mordell-Weil lattice. Received November 3, 1999 / Published online February 5, 2001     

Global holomorphic 2-forms and pluricanonical systems on threefolds

The change of zero locus of a global holomorphic 2-form on a threefold under birational transformations is investigated. It is proved that existence of 2-forms with certain conditions on their zero loci on a threefold of nonnegative Kodaira dimension limits types of terminal singularities appearing on its minimal models. As a result of the restriction on the types of terminal singularities and Reid's Riemann-Roch formula, a universal bound N is found such that the linear system NK defines a birational map from a threefold of general type admitting those 2-forms, where K is the canonical bundle of the threefold. Received March 10, 2000 / Published online October 11, 2000     

Automorphism groups of real algebraic curves which are double covers of the real projective plane

For each integer g≥ 3 we give the complete list of groups acting as full automorphism groups of real algebraic curves of genus $g$ which are double covers of the real projective plane. Explicit polynomial equations of such curves and the formulae of their automorphisms are also given. Received: 29 April 1999 / Revised version: 26 November 1999     

Mumford curves in a specialized pencil of sextics

We discuss Mumford curves in the pencil on a Del Pezzo quintic surface constructed by Edge [Ed1]. The abstract group structures of the normalizer of the corresponding Schottky groups are described, which give us some knowledges on Mumford loci in moduli space of curves. Received: 5 July 2000 / Accepted: 23 October 2000     

Isolated rational curves on K3-fibered Calabi–Yau threefolds

In this paper we study 16 complete intersection K3-fibered Calabi--Yau variety types in biprojective space ℙ ^{n 1}}×ℙ¹. These are all the CICY-types that are K3 fibered by the projection on the second factor. We prove existence of isolated rational curves of bidegree (d,0) for every positive integer d on a general Calabi–Yau variety of these types. The proof depends heavily on existence theorems for curves on K3-surfaces proved by S. Mori and K. Oguiso. Some of these varieties are related to Calabi–Yau varieties in projective space by a determinantal contraction, and we use this to prove existence of rational curves of every degree for a general Calabi–Yau variety in projective space. Received: 14 October 1997 / Revised version: 18 January 1998     

Compactification of the universal Picard over the moduli of stable curves

This article provides two different, but closely related, moduli problems, which in characteristic zero provide a type of compactification of the universal Picard over the moduli of stable curves. Although neither is of finite type, both are limits of a sequence of stacks, each of which is a separated algebraic stack of finite type. We discuss relations to previous compactifications and partial compactifications, give a number of examples related to this compactification, and work out the structure of its fibres over certain fixed curves. Some applications are also discussed. Received January 5, 1998; in final form April 1, 1999 / Published online July 3, 2000     

Pentes en cohomologie rigide et F-isocristaux unipotents

We study the slopes of Frobenius on the rigid cohomology and the rigid cohomology with compact support of an algebraic variety over a perfect field of positive characteristic. We then prove that any unipotent overconvergent F-isocrystal on a smooth variety has a slope filtration whose graded parts are pure. Received: 23 December 1998 / Revised version: 5 July 1999     

Intersection of independent regenerative sets

We investigate the nature of the intersection of two independent regenerative sets. The approach combines Bochners subordination and potential theory for a pair of Markov processes in duality. Received: 21 November 1997 / Revised version: 31 August 1998     

Fonction L unité d'un groupe de Barsotti-Tate

The purpose of this article is to give a cohomological formula for the unit-root part of the L-function associated to a Barsotti-Tate group G on a scheme S over a field of characteristic p when G extends to some compactification of S. This is an analogue of a part of a conjecture of Katz according to wich the L-function of an F-crystal should be expressed in terms of the p-adic etale sheaf corresponding to the unit-root part of the crystal. In order to carry out this project, we use the technics of [E-LS II] wich require in our case an extension of the Dieudonné crystalline theory ([B-B-M]) to “crystal of level mG” in the sense of Berthelot. We show that the unit-root L-function of the Dieudonné crystal associated to G can be expressed in terms of the syntomic cohomology of the Ext group of G by the constant sheaf.

Received: 24 March 1997 / Revised version: 6 January 1998     

Irregularity of certain algebraic fiber spaces

Let X, Y be smooth complex projective varieties, and be a fiber space whose general fiber is a curve of genus g. Denote by q _f the relative irregularity of f. It is proved that , if f is not generically trivial; moreover, if either a) f is non-constant and the general fiber is either hyperelliptic or bielliptic or b) q(Y)= 0, then , and the bound is best possible. A classification of fiber surfaces of genus 3 with q _f = 2 is also given in this note. Received: 19 March 1997 / Revised version: 29 October 1997     

Generalized Shioda–Inose structures on K3 surfaces

In this note, we study the action of finite groups of symplectic automorphisms on K3 surfaces which yield quotients birational to generalized Kummer surfaces. For each possible group, we determine the Picard number of the K3 surface admitting such an action and for singular K3 surfaces we show the uniqueness of the associated abelian surface. Received: 9 April 1998 / Revised version: 17 July 1998     

Representing links in 3-manifolds¶ by branched coverings of S3

We introduce a planar coloured-diagram representation of links in 3-manifolds given as branched coverings of the 3-sphere. We also prove an equivalence theorem based on local moves and the existence of a universal configuration for such representation. As an application we give unified proofs of two different results on existence of fibered links in 3-manifolds. Received: 7 April 1997     

Triades et familles de courbes gauches

Sans résumé Received: 3 April 1998 / in final form: 16 December 1998