Enriques surfaces and Jacobian elliptic K3 surfaces

This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with Jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptional behaviour. In a forthcoming paper, we will study arithmetic consequences of our construction.     

Stable pairs on elliptic K3 surfaces

We study semistable pairs on elliptic K3 surfaces with a section: we construct a family of moduli spaces of pairs, related by wall crossing phenomena, which can be studied to describe the birational correspondence between moduli spaces of sheaves of rank 2 and Hilbert schemes on the surface. In the 4-dimensional case, this can be used to get the isomorphism between the moduli space and the Hilbert scheme described by Friedman.     

Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces

We consider K3 surfaces which are double covers of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves on the conic bundle on the rational surface.     

Multiple supersingular elliptic fibers on elliptic surfaces

Elliptic surfaces over an algebraically closed field in characteristic p>0 with multiple supersingular elliptic fibers, that is, multiple fibers of a supersingular elliptic curve, are investigated. In particular, it is shown that for an elliptic surface with q=g+1 and a supersingular elliptic curve as a general fiber, where q is the dimension of an Albanese variety of the surface and g is the genus of the base curve, the multiplicities of the multiple supersingular elliptic fibers are not divisible by p². As an application of this result, the structure of false hyperelliptic surfaces is discussed on this basis.     

Density of positive rank fibers in elliptic fibrations

A question of Mazur asks whether for any non-constant elliptic fibration {Er}_r∈Q, the set {r∈Q:rank(Er(Q))>0}, if infinite, is dense in R (with respect to the Euclidean topology). This has been proved to be true for the family of quadratic twists of a fixed elliptic curve by a quadratic or a cubic polynomial. Here we settle Mazur's question affirmatively for the general quadratic and cubic fibrations. Moreover we show that our method works when Q is replaced by any real number field.     

Singular rational curves on elliptic K3 surfaces

We show that on every elliptic K3 surface there are rational curves ${(R_i)}_{i\in \mathbb{N}}$ such that $R_i^2\to \infty$, that is, of unbounded arithmetic genus. Moreover, we show that the union of the lifts of these curves to $\mathbb{P}({\mathrm{\Omega }}_X)$ is dense in the Zariski topology. As an application, we give a simple proof of a theorem of Kobayashi in the elliptic case, that is, there are no globally defined symmetric differential forms.     

The maximal singular fibres of elliptic K3 surfaces

We prove that the maximal singular fibres of an elliptic K3 surface have type I₁₉ and unless the characteristic of the ground field is 2. In characteristic 2, the maximal singular fibres are I₁₈ and . The paper supplements work of Shioda in [9] and [10]. Received: 23 September 2005     

Dynkin diagrams of rank 20 on supersingular K3 surfaces

We classify normal supersingular K 3 surfaces Y with total Milnor number 20 in characteristic p,where p is an odd prime that does not divide the discriminant of the Dynkin type of the rational double points on Y.This paper appeared in preprint form in the home page of the first author in the year 2005.     

The elliptic K3 surfaces with a maximal singular fibre

We give the defining equation of complex elliptic K3 surfaces with a maximal singular fibre. Then we study the reduction modulo p at a particularly interesting prime p. To cite this article: T. Shioda, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On symplectic quotients of K3 surfaces

In this note, we construct generalized Shioda-Inose structures on K3 surfaces using cyclic covers and almost functoriality of Shioda-Inose structures with respect to normal subgroups of a given group of symplectic automorphisms.     

On the Brill-Noether theory for K3 surfaces

Let (S, H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c ₁(E), H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for curves, we give a notion of Brill-Noether generic K3 surfaces.