Torsion points and concurrent exceptional curves on del Pezzo surfaces of degree one

Abstract

The blow-up of the anticanonical base point on a del Pezzo surface $S$ of degree 1 gives rise to a rational elliptic surface $\mathscr{E}$ with only irreducible fibers. The sections of minimal height of $\mathscr{E}$ are in correspondence with the $240$ exceptional curves on $S$. A natural question arises when studying the configuration of these curves: if a point on $S$ is contained in 'many' exceptional curves, it is torsion on its fiber on $\mathscr{E}$? In 2005, Kuwata proved for the analogous question on del Pezzo surfaces of degree $2$, where there are 56 exceptional curves, that if 'many' equals $4$ or more, the answer is yes. In this paper, we prove that for del Pezzo surfaces of degree 1, the answer is yes if 'many' equals $9$ or more. Moreover, we give counterexamples where a \textsl{non}-torsion point lies in the intersection of $7$ exceptional curves. We give partial results for the still open case of 8 intersecting exceptional curves.