Construction and Applications of Trisections of Low Genus on Del Pezzo Surfaces of Degree One

Abstract

Consider a rational elliptic surface over a field $k$ with characteristic $0$ given by $\mathcal{E}: y^2 = x^3 + f(t)x + g(t)$, with $f,g\in k[t]$, $\text{deg}(f) \leq 4$ and $\text{deg}(g) \leq 6$. If all the bad fibres are irreducible, such a surface comes from the blow-up of a del Pezzo surface of degree one. We are interested in studying multisections, curves which intersect each fibre a fixed number of times, specifically, trisections (three times). Many configurations of singularities on a trisection lead to a lower genus. Here, we focus on of several them: by specifying conditions on the coefficients $f,g$ of the surface $\mathcal{E}$, and looking at trisections which pass through a given point three times, we obtain a pencil of cubics on such surfaces. Our construction allows us to prove in several cases the Zariski density of the rational points. This is especially interesting since the results in this regard are partial for del Pezzo surfaces of degree one.