Homogeneous spaces and degree 4 del Pezzo surfaces

It is known that, given a genus 2 curve C : y2 = f(x), where f(x) is quintic and defined over a field K, of characteristic different from 2, and given a homogeneous space ℋδ for complete 2-descent on the Jacobian of C, there is a Vδ (which we shall describe), which is a degree 4 del Pezzo surface defined over K, such that ℋδ (K) ≠ Ø ⇒ Vδ (K) ≠ Ø. 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, C and δ such that V = V 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 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. © 2009 Springer-Verlag.