The Manin-Peyre conjecture for three del Pezzo surfaces

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 $d\ge 3$.