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

TL;DR

This paper develops a new method to count lines on a degree 4 del Pezzo surface over various fields by overcoming orientability restrictions, extending enumerative geometry techniques.

Contribution

It introduces a quadratically enriched counting method that circumvents orientability constraints in enumerative problems on del Pezzo surfaces.

Findings

Successfully counts lines over different fields

Extends Kass-Wickelgren's approach to non-orientable cases

Provides a well-defined count on an open subset

Abstract

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.