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Over an algebraically closed field k, there are 16 lines on a degree 4 delPezzo surface, but for other fields the situation is more subtle. In order toimprove enumerative results over perfect fields, Kass and Wickelgren introducea method analogous to counting zeroes of sections of smooth vector bundlesusing the Poincare-Hopf theorem. However, the technique of Kass-Wickelgrenrequires the enumerative problem to satisfy a certain type of orientabilitycondition. The problem of counting lines on a degree 4 del Pezzo surface doesnot satisfy this orientability condition, so most of the work of this paper isdevoted to circumventing this problem. We do this by restricting to an open setwhere the orientability condition is satisfied, and checking that the countobtained is well-defined, similarly to an approach developed by Larson andVogt.

A quadratically enriched count of lines on a degree 4 del Pezzo surface