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Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings (with an appendix by E. V. Flynn)

David Masser, Umberto Zannier (2015)

Journal of the European Mathematical Society

In recent papers we proved a special case of a variant of Pink’s Conjecture for a variety inside a semiabelian scheme: namely for any curve inside anything isogenous to a product of two elliptic schemes. Here we go beyond the elliptic situation by settling the crucial case of any simple abelian surface scheme defined over the field of algebraic numbers, thus confirming an earlier conjecture of Shou-Wu Zhang. This is of particular relevance in the topic, also in view of very recent counterexamples...

Torsors under tori and Néron models

Martin Bright (2011)

Journal de Théorie des Nombres de Bordeaux

Let R be a Henselian discrete valuation ring with field of fractions K . If X is a smooth variety over K and G a torus over K , then we consider X -torsors under G . If 𝒳 / R is a model of X then, using a result of Brahm, we show that X -torsors under G extend to 𝒳 -torsors under a Néron model of G if G is split by a tamely ramified extension of K . It follows that the evaluation map associated to such a torsor factors through reduction to the special fibre. In this way we can use the geometry of the special...

Toward Clemens' Conjecture in Degrees between 10 and 24

Johnsen, Trygve, Kleiman, Steven (1997)

Serdica Mathematical Journal

1 Supported in part by the Norwegian Research Council for Science and the Humanities. It is a pleasure for this author to thank the Department of Mathematics of the University of Sofia for organizing the remarkable conference in Zlatograd during the period August 28-September 2, 1995. It is also a pleasure to thank the M.I.T. Department of Mathematics for its hospitality from January 1 to July 31, 1993, when this work was started. 2Supported in part by NSF grant 9400918-DMS.We introduce and study...

Towards a Mori theory on compact Kähler threefolds III

Thomas Peternell (2001)

Bulletin de la Société Mathématique de France

Based on the results of the first two parts to this paper, we prove that the canonical bundle of a minimal Kähler threefold (i.e. K X is nef) is good,i.e.its Kodaira dimension equals the numerical Kodaira dimension, (in particular some multiple of K X is generated by global sections); unless X is simple. “Simple“ means that there is no compact subvariety through the very general point of X and X not Kummer. Moreover we show that a compact Kähler threefold with only terminal singularities whose canonical...

Towards the classification of weak Fano threefolds with ρ = 2

Joseph Cutrone, Nicholas Marshburn (2013)

Open Mathematics

In this paper, examples of type II Sarkisov links between smooth complex projective Fano threefolds with Picard number one are provided. To show examples of these links, we study smooth weak Fano threefolds X with Picard number two and with a divisorial extremal ray. We assume that the pluri-anticanonical morphism of X contracts only a finite number of curves. The numerical classification of these particular smooth weak Fano threefolds is completed and the geometric existence of some numerical cases...

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