# Concurrent lines on del Pezzo surfaces of degree one

## TL;DR

This paper investigates the maximum number of exceptional curves passing through points on del Pezzo surfaces of degree one, revealing characteristic-dependent bounds and demonstrating their sharpness in most cases.

## Contribution

It establishes precise upper bounds on the number of exceptional curves through points on del Pezzo surfaces of degree one, depending on the point's location and the characteristic.

## Key findings

- Maximum of 16 exceptional curves through points on the ramification curve in characteristic 2.
- Maximum of 10 exceptional curves through points on the ramification curve in other characteristics.
- Maximum of 12 exceptional curves through points outside the ramification curve in characteristic 3.

## Abstract

Let $X$ be a del Pezzo surface of degree one over an algebraically closed field $k$, and let $K_X$ be its canonical divisor. The morphism $\varphi$ induced by the linear system $|-2K_X|$ realizes $X$ as a double cover of a cone in $\mathbb{P}^3$ that is ramified over a smooth curve of degree 6. The surface $X$ contains 240 curves with negative self-intersection, called exceptional curves. We prove that for a point~$P$ on the ramification curve of $\varphi$, at most sixteen exceptional curves go through~$P$ in characteristic $2$, and at most ten in all other characteristics. Moreover, we prove that for a point $Q$ outside the ramification curve of $\varphi$, at most twelve exceptional curves go through $Q$ in characteristic $3$, and at most ten in all other characteristics. We show that these upper bounds are sharp in all cases except possibly in characteristic 5 outside the ramification curve.