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manuscripta mathematica

, Volume 129, Issue 3, pp 369–380 | Cite as

Homogeneous spaces and degree 4 del Pezzo surfaces

  • E. V. FlynnEmail author
Article
  • 79 Downloads

Abstract

It is known that, given a genus 2 curve \({{\mathcal C} : {y^2 = f(x)}}\) , where f(x) is quintic and defined over a field K, of characteristic different from 2, and given a homogeneous space \({{\mathcal H}_\delta}\) for complete 2-descent on the Jacobian of \({{\mathcal C}}\) , there is a V δ (which we shall describe), which is a degree 4 del Pezzo surface defined over K, such that \({{\mathcal H}_\delta(K) \not= \emptyset \implies V_\delta(K) \not= \emptyset}\) . We shall prove that every degree 4 del Pezzo surface V, defined over K, arises in this way; furthermore, we shall show explicitly how, given V, to find \({{\mathcal C}}\) and δ such that VV δ , up to a linear change in variable defined over K. We shall also apply this relationship to Hürlimann’s example of a degree 4 del Pezzo surface violating the Hasse principle, and derive an explicit parametrised infinite family of genus 2 curves, defined over \({{\mathbb Q}}\) , whose Jacobians have nontrivial members of the Shafarevich-Tate group. This example will differ from previous examples in the literature by having only two \({{\mathbb Q}}\) -rational Weierstrass points.

Mathematics Subject Classification (2000)

Primary 11G30 Secondary 11G10 14H40 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

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