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

Abstract

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.