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.     

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.     

2-Descent on elliptic curves and rational points on certain Kummer surfaces

The first part of this paper further refines the methodology for 2-descents on elliptic curves with rational 2-division points which was introduced in [J.-L. Colliot-Thélène, A.N. Skorobogatov, Peter Swinnerton-Dyer, Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, Invent. Math. 134 (1998) 579-650]. To describe the rest, let E⁽¹⁾ and E⁽²⁾ be elliptic curves, D⁽¹⁾ and D⁽²⁾ their respective 2-coverings, and X be the Kummer surface attached to D⁽¹⁾×D⁽²⁾. In the appendix we study the Brauer-Manin obstruction to the existence of rational points on X. In the second part of the paper, in which we further assume that the two elliptic curves have all their 2-division points rational, we obtain sufficient conditions for X to contain rational points; and we consider how these conditions are related to Brauer-Manin obstructions. This second part depends on the hypothesis that the relevent Tate-Shafarevich group is finite, but it does not require Schinzel's Hypothesis.     

Exceptional curves on rational surfaces having K2⩾0

We characterize the rational surfaces X which have a finite number of (?1)-curves under the assumption that ?K_X is nef, where K_X is a canonical divisor on X, and has self-intersection zero. We prove also that if ?K_X is not nef and has self-intersection zero, then X has a finite number of (?1)-curves. To cite this article: M. Lahyane, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

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.     

Automorphism groups of rational elliptic surfaces with section and constant J-map

In this paper, the automorphism groups of relatively minimal rational elliptic surfaces with section which have constant J-maps are classified. The ground field is ?. The automorphism group of such a surface β: B → ?¹, denoted by Au t(B), consists of all biholomorphic maps on the complex manifold B. The group Au t(B) is isomorphic to the semi-direct product MW(B) ? Aut _σ (B) of the Mordell-Weil groupMW(B) (the group of sections of B), and the subgroup Aut _σ (B) of the automorphisms preserving a fixed section σ of B which is called the zero section on B. The Mordell-Weil group MW(B) is determined by the configuration of singular fibers on the elliptic surface B due to Oguiso and Shioda [9]. In this work, the subgroup Aut _σ (B) is determined with respect to the configuration of singular fibers of B. Together with a previous paper [4] where the case with non-constant J-maps was considered, this completes the classification of automorphism groups of relatively minimal rational elliptic surfaces with section.     

Seshadri constants on ruled surfaces: the rational and the elliptic cases

We study the Seshadri constants on geometrically ruled surfaces. The unstable case is completely solved. Moreover, we give some bounds for the stable case. We apply these results to compute the Seshadri constant of the rational and elliptic ruled surfaces. Both cases are completely determined. The elliptic case provides an interesting picture of how particular is the behavior of the Seshadri constants.     

Density of rational curves on K3 surfaces

Using the dynamics of self rational maps of elliptic $K3$ surfaces together with deformation theory, we prove that the union of rational curves is dense on a very general $K3$ surface and that the union of elliptic curves is dense in the 1st jet space of a very general $K3$ surface, both in the strong topology.     

Configurations of singular fibres on rational elliptic surfaces in characteristic two

Antimatter domains are defined to be the integral domains which do not have any atoms. It is proved that each integral domain can be em-bedded as a subring of some antimatter domain which is not a field. Any fragmented domain is an antimatter domain, but the converse fails in each positive Krull dimension. A detailed study is made of the passage of the“an-timatter”property between the partners within an overring extension. Special attention is given to characterizing antimatter domains in classes of valuation domains, pseudo-valuation domains, and various types of pullbacks.     

Artificial boundary conditions for elliptic systems on polyhedral truncation surfaces

For a sufficiently broad class of formally selfadjoint boundary value problems in the domains with conical outlets to infinity, including exterior boundary value problems, we suggest an algorithm for constructing some artificial boundary conditions on polyhedral truncation surfaces that guarantees higher precision of approximation as the infinite domain is replaced with a large but bounded domain. The errors are estimated. Anisotropic 3-dimensional elasticity problems are discussed as an example.     

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).