Motivic counting of curves on split quintic del Pezzo surfaces

TL;DR

This paper proves a comprehensive version of the Batyrev--Manin--Peyre conjecture for split quintic del Pezzo surfaces, addressing both rational points over function fields and a motivic perspective.

Contribution

It establishes the 'all-the-heights' conjecture for these surfaces in both arithmetic and motivic contexts, extending previous results.

Findings

Confirmed the conjecture for rational points over global function fields.

Extended the conjecture to a motivic framework over general base fields.

Provided new insights into counting curves on del Pezzo surfaces.

Abstract

We prove the "all-the-heights'' version of the Batyrev--Manin--Peyre conjecture for split quintic del Pezzo surfaces, both for counting rational points over global function fields in positive characteristic and for the motivic version over a general base field.