Aspects of Quadratically Enriched Enumerative Geometry of Complete Intersections in Projective Space

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 Chapters of smooth vector bundles using the Poincar ́e-Hopf theo- rem in [KW21]. 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 in [LV21]. We also take a step toward giving an intrinsic geomet- ric interpretation of local indices for d-planes in complete intersections in projective space, identifying the local index with the determinant of a map defined entirely in terms of the intrinsic geometry of an embedding Pdk → Pnk .