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

TL;DR

This paper investigates the relationship between points on degree one del Pezzo surfaces and torsion properties of their associated elliptic fibers, establishing thresholds for when points are torsion and providing counterexamples.

Contribution

It extends the understanding of exceptional curve configurations on degree one del Pezzo surfaces and identifies the minimum number of curves needed for torsion points, building on prior work for degree two.

Findings

Points in 9 or more exceptional curves are torsion on the elliptic fiber.

Counterexamples exist with non-torsion points in 7 intersecting curves.

Partial results are provided for the case of 8 intersecting curves.

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.