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

TL;DR

This paper computes quadratic counts of lines on smooth del Pezzo surfaces of degrees 2 and 4 using Chow-Witt groups, revealing that the quadratic count coincides with the classical count in this setting.

Contribution

It provides explicit computations of Euler classes in Chow-Witt groups for these surfaces and shows the quadratic count aligns with the classical count due to the Chow-Witt group structure.

Findings

Quadratic counts are not enriched in this setting.

Chow-Witt groups of certain projective bundles are described explicitly.

The quadratic count matches the classical count for these surfaces.

Abstract

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.