Concurrent lines on del Pezzo surfaces of degree one

Abstract:

Let $X$ be a del Pezzo surface of degree one over an algebraically closed field, and $K_X$ its canonical divisor. The morphism $\varphi$ induced by $|-2K_X|$ realizes $X$ as a double cover of a cone in $\mathbb {P}^3$, ramified over a smooth sextic curve. The surface $X$ contains 240 exceptional curves. We prove the following statements. For a point $P$ on the ramification curve of $\varphi$, at most sixteen exceptional curves contain $P$ in characteristic $2$, and at most ten in all other characteristics. Moreover, for a point $Q$ outside the ramification curve, at most twelve exceptional curves contain $Q$ in characteristic $3$, and at most ten in all other characteristics. We show that these upper bounds are sharp, except possibly in characteristic 5 outside the ramification curve.