Homogeneous Spaces and Degree 4 del Pezzo Surfaces.

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 2-descent 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 Shafarevich-Tate group. This example will differ from previous examples in the literature by having only two Q-rational Weierstrass points.