Concurrent lines on del Pezzo surfaces of degree one

Abstract

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