Density of rational points on Enriques surfaces

Let $X$ be an Enriques surface defined over a number field $K$. Then there exists a finite extension $K'/K$ such that the set of $K'$-rational points of $X$ is Zariski dense.Comment: 8 pages, LaTe

Rationality of quotients by linear actions of affine groups

Let G be the (special) affine group, semidirect product of SL_n and C^n. In this paper we study the representation theory of G and in particular the question of rationality for V/G where V is a generically free G-representation. We show that the answer to this question is positive if the dimension of V is sufficiently large and V is indecomposable. We have a more precise theorem if V is a two-step extension 0 -> S -> V -> Q -> 0 with S, Q completely reducible.Comment: 18 pages; dedicated to Fabrizio Catanese on the occasion of his 60th birthda

Hyperelliptic Szpiro inequality

We generalize the classical Szpiro inequality to the case of a semistable family of hyperelliptic curves. We show that for a semistable symplectic Lefschetz fibration of hyperelliptic curves of genus $g$, the number $N$ of non-separating vanishing cycles and the number $D$ of singular fibers satisfy the inequality $N \leq (4g+2)D$.Comment: LaTeX2e, 27 page

Symplectic Lefschetz fibrations with arbitrary fundamental groups

In this paper we give an explicit construction of a symplectic Lefschetz fibration whose total space is a smooth compact four dimensional manifold with a prescribed fundamental group. We also study the numerical properties of the sections in symplectic Lefschetz fibrations and their relation to the structure of the monodromy group.Comment: 45 pages, LaTeX2e. Minor mistakes corrected. New appendix by Ivan Smith added, proving the non-existence of SLF with monodromy contained in the Torelli grou

Geometric properties of curves defined over number fields

The article contains a detailed proof of the famous Belyi theorem on geometry of complex algebraic curves defined over number fields. It also includes the discussion of several constructions and conjectures inspired by Belyi’s result which where brought up by the first author during his colloquium talks at different universities within the period from 1979 to 1984

Luttinger-liquid-like transport in long InSb nanowires

Long nanowires of degenerate semiconductor InSb in asbestos matrix (wire diameter is around 50 \AA, length 0.1 - 1 mm) were prepared. Electrical conduction of these nanowires is studied over a temperature range 1.5 - 350 K. It is found that a zero-field electrical conduction is a power function of the temperature $G\propto T^\alpha$ with the typical exponent $\alpha \approx 4$. Current-voltage characteristics of such nanowires are found to be nonlinear and at sufficiently low temperatures follows the power law $I\propto V^\beta$. It is shown that the electrical conduction of these nanowires cannot be accounted for in terms of ordinary single-electron theories and exhibits features expected for impure Luttinger liquid. For a simple approximation of impure LL as a pure one broken into drops by weak links, the estimated weak-link density is around $10^3-10^4$ per cm.Comment: 5 pages, 2 figure

Endomorphisms of abelian varieties, cyclotomic extensions and Lie algebras

We prove an analogue of the Tate conjecture on homomorphisms of abelian varieties over infinite cyclotomic extensions of finitely generated fields of characteristic zero.Comment: 9 page

Spitsbergen volume : Frontiers of Rationality

This volume contains 20 papers related to the workshop Frontiers of Rationality that was held in Longyearbyen, Spitsbergen, in July 2014

Remarks on endomorphisms and rational points

Let X be a variety over a number field and let f: X --> X be an "interesting" rational self-map with a fixed point q. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold (originally obtained by Claire Voisin and the first author in 2007).Comment: LaTeX, 22 pages. v2: some minor observations added, misprints corrected, appendix modified