The Manin-Peyre conjecture for three del Pezzo surfaces

TL;DR

This paper proves the Manin-Peyre conjecture for certain singular del Pezzo surfaces, improving error estimates and introducing a versatile method applicable to a broad class of such surfaces.

Contribution

It establishes the conjecture for specific singular del Pezzo surfaces and develops a general approach that enhances previous methods and error bounds.

Findings

Confirmed the Manin-Peyre conjecture for three specific singular del Pezzo surfaces.

Developed a unified method that improves error terms over previous approaches.

The method is applicable to most singular del Pezzo surfaces of degree at least 3.

Abstract

The Manin-Peyre conjecture is established for a split singular quintic del Pezzo surface with singularity type $\mathbf{A}_2$ and two split singular quartic del Pezzo surfaces with singularity types $\mathbf{A}_3+\mathbf{A}_1$ and $\mathbf{A}_4$ respectively. We use a unified and different slightly method from the previous and improve their error terms. Our method is general and can handle most of singular del Pezzo surfaces of degree $d\geq 3$.