Counting imaginary quadratic points via universal torsors

A conjecture of Manin predicts the distribution of rational points on Fano varieties.
We provide a framework for proofs of Manin’s conjecture for del Pezzo surfaces over
imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin’s conjecture over imaginary quadratic fields K for the quartic del Pezzo surface S of singularity type A3 with five lines given in P4 K by the equations x0x1 − x2x3 = x0x3 + x1x3 + x2x4 = 0.