Rational points on certain del Pezzo surfaces of degree one

Abstract

Let $f(z)=z^5+az^3+bz^2+cz+d \in \Z[z]$ and let us consider a del Pezzo surface of degree one given by the equation $\cal{E}_{f}: x^2-y^3-f(z)=0$. In this note we prove that if the set of rational points on the curve $E_{a, b}:Y^2=X^3+135(2a-15)X-1350(5a+2b-26)$ is infinite, then the set of rational points on the surface $\cal{E}_{f}$ is dense in the Zariski topology.