  Over an algebraically closed field k, there are 16 lines on a degree 4 del
Pezzo surface, but for other fields the situation is more subtle. In order to
improve enumerative results over perfect fields, Kass and Wickelgren introduce
a method analogous to counting zeroes of sections of smooth vector bundles
using the Poincare-Hopf theorem. However, the technique of Kass-Wickelgren
requires the enumerative problem to satisfy a certain type of orientability
condition. The problem of counting lines on a degree 4 del Pezzo surface does
not satisfy this orientability condition, so most of the work of this paper is
devoted to circumventing this problem. We do this by restricting to an open set
where the orientability condition is satisfied, and checking that the count
obtained is well-defined, similarly to an approach developed by Larson and
Vogt.
