Rational points on certain del Pezzo surfaces of degree one

Original · EN

Let f(z)=z⁵+az³+bz²+cz+d ∈ [z] and let us consider a del Pezzo surface of degree one given by the equation Ef: x²-y³-f(z)=0. In this note we prove that if the set of rational points on the curve Eₐ, b:Y²=X³+135(2a-15)X-1350(5a+2b-26) is infinite, then the set of rational points on the surface Ef is dense in the Zariski topology.