RATIONAL POINTS ON CERTAIN DEL PEZZO SURFACES OF DEGREE ONE

Abstract

AbstractLet$f(z)=z^5+az^3+bz^2+cz+d \in \Z[z]$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 paper we prove that if the set of rational points on the curveEa,b:Y2=X3+ 135(2a−15)X−1350(5a+ 2b− 26) is infinite then the set of rational points on the surface ϵfis dense in the Zariski topology.