CONCURRENT LINES ON DEL PEZZO SURFACES OF DEGREE ONE

Let X be a del Pezzo surface of degree one over an algebraically closed field, and K-X its canonical divisor. The morphism phi induced by |-2K(X)| realizes X as a double cover of a cone in 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 phi, 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.