On the unirationality of del Pezzo surfaces of degree 2

Among geometrically rational surfaces, del Pezzo surfaces of degree 2 over a field k containing at least one point are arguably the simplest that are not known to be unirational over k. Looking for k-rational curves on these surfaces, we extend some earlier work of Manin on this subject. We then focus on the case where k is a finite field, where we show that all except possibly three explicit del Pezzo surfaces of degree 2 are unirational over k.