Number/Representation Theory

Consider a rational elliptic surface over a field $k$ with characteristic $0$ given by $E: y^2 = x^3 + f(t)x + g(t)$, with $f,g\in k[t]$, $\deg(f) \leq 4$ and $\deg(g) \leq 6$. 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 the relationship between the existence of low genus multisections on these surfaces and the properties of $k$-unirationality and Zariski density of the $k$-rationals. More specifically, we study trisections, i.e., multisections of degree three, and use them to show Zariski density on a large family of surfaces. This is especially interesting since the results in this regard are partial for del Pezzo surfaces of degree one.