Flynn, E. V. (2009) Homogeneous Spaces and Degree 4 del Pezzo Surfaces. Manuscripta Mathematica, 129 . pp. 369380. ISSN 00252611

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Official URL: http://www.springer.com/math/journal/229
Abstract
It is known that, given a genus 2 curve 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 H_delta for complete 2descent on the Jacobian of C, there is a V_delta (which we shall describe), which is a degree 4 del Pezzo surface defined over K, such that H_delta(K) nonempty implies V_delta(K) nonempty. 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 C and delta such that V = V_delta, up to a linear change in variable defined over K. We shall also apply this relationship to Hurlimann'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 Q, whose Jacobians have nontrivial members of the ShafarevichTate group. This example will differ from previous examples in the literature by having only two Qrational Weierstrass points.
Item Type:  Article 

Subjects:  H  N > Number theory 
Research Groups:  Number Theory Group 
ID Code:  803 
Deposited By:  E. Victor Flynn 
Deposited On:  01 Sep 2009 07:34 
Last Modified:  29 May 2015 18:29 
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