Density of rational points on del Pezzo surfaces of degree one

The Segre-Manin Theorem implies that if a del Pezzo surface  S of degree at least three,
defined over the rational numbers, has a  rational point, then the rational points are Zariski
dense in S. A  result of Manin yields the same for degree two, as long as the initial  point
does not lie on a certain divisor. Similar results for del Pezzo  surfaces of degree one are
meager: they either depend on conjectures,  or they are restricted to small families of surfaces.

Every del Pezzo surface of degree one has a natural elliptic  fibration. We will show that
if S is a del Pezzo surface of degree one  defined over a number field k and P, a point in
S(k), is distinct from  the unique base point of the elliptic fibration on S and has order at 
least six on the fibre where it lies, then there is a (genus at most  one) non-torsion
multisection through P. The existence of infinitely  many rational points in this multisection
implies that the set of  k-rational points is dense in S. (This is work in progress with Ronald 
van Luijk).