Density of rational points on a family of del Pezzo surfaces of degree one

Let \(k\) be an infinite field of characteristic 0, and \(X\) a del Pezzo surface of degree \(d\) with at least one \(k\)-rational point. Various methods from algebraic geometry and arithmetic statistics have shown the Zariski density of the set \(X(k)\) of \(k\)-rational points in \(X\) for \(d\geq2\) (under an extra condition for \(d=2\)), but fail to work in generality when the degree of \(X\) is 1, leaving a large class of del Pezzo surfaces for which the question of density of rational points is still open. In this paper, we prove the Zariski density of \(X(k)\) when \(X\) has degree 1 and is represented in the weighted projective space \(\mathbb{P}(2,3,1,1)\) with coordinates \(x,y,z,w\) by an equation of the form \(y^2=x^3+az^6+bz^3w^3+cw^6\) for \(a,b,c\in k\) with \(a,c\) non-zero, under the condition that the elliptic surface obtained by blowing up the base point of the anticanonical linear system \(|-K_X|\) contains a smooth fiber above a point in \(\mathbb{P}^1\setminus\{(1:0),(0:1)\}\) with positive rank over \(k\). When \(k\) is of finite type over \(\mathbb{Q}\), this condition is sufficient and necessary.