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≥2 (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 P(2,3,1,1) with coordinates x,y,z,w by an equation of the form y2=x3+az6+bz3w3+cw6 for a,b,c∈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 |−KX| contains a smooth fiber above a point in P1∖{(1:0),(0:1)} with positive rank over k. When k is of finite type over Q, this condition is sufficient and necessary.