In SwinnertonDyer's charming paper "An application of computing to classfield theory", in CasselsFrohlich, he discusses the genesis of the Birch/SwinnertonDyer conjecture and numerical tests of it for the curves $y^2=x^3dx$. At the end of the paper, he speculates on higherdimensional analogues of the conjecture, linking Chow groups to Lfunctions, which in hindsight were essentially correct (BeilinsonBloch made them precise). Regarding these conjectures, he says that Bombieri and himself had found some evidence for them in the special case of cubic threefolds and the intersection of two quadric hypersurfaces. I know that in the case of cubic threefolds $X$, codimensiontwo cycles can be related to zerocycles in the Albanese $A_X$ of the Fano surface of lines in $X$, which (I assume) reduces the conjecture in this case to BSD for $A_X$. But what can be done for the intersection of two quadrics? Does anyone know what he was talking about here? Is this in print in more detail somewhere?

Maybe you already know this! But Miles Reid and Ron Donagi have shown that the intermediate Jacobian of the intersection of two quadrics is the Jacobian of a hyperelliptic curve. See Donagi's old paper where there is a beautiful generalization of the group law on an elliptic curve. Recent results on such questions all use Nori's general results or BlochSrinivas's paper on the diagonall see for example Nagel's paper or Voisin's paper. 

