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.