A Quadratically Enriched Count Of Lines In Smooth Del Pezzo Surfaces Of Degree 2 And 4

We give a computation of some Euler classes in Chow-Witt groups associated to the count of lines of smooth del Pezzo surfaces of degree 2 and 4. The description of Chow-Witt groups of projective bundles over Grassmannians for vector bundles that are not relatively orientable is the main part of the article. We show that the quadratic count is not enriched as the Chow-Witt group is isomorphic to the Chow group. In this setting, we give an expression of the classes of even rank in the Chow-Witt group as multiples of the hyperbolic element h. A direct application of this construction is the count for the del Pezzo surfaces.