The Manin-Peyre conjecture for three del Pezzo surfaces

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 $d\geq 3$.