An overview of Manin's conjecture for del Pezzo surfaces

TL;DR

This paper surveys recent advances in understanding Manin's conjecture for del Pezzo surfaces, highlighting techniques and establishing an upper bound for a specific singular case.

Contribution

It provides a comprehensive overview of progress and demonstrates a new upper bound for a degree four singular del Pezzo surface.

Findings

Established an upper bound for a singular degree four del Pezzo surface

Summarized recent progress on Manin's conjecture for del Pezzo surfaces

Illustrated techniques used in proving bounds for these surfaces

Abstract

This paper surveys recent progress towards the Manin conjecture for (singular and non-singular) del Pezzo surfaces. To illustrate some of the techniques available, an upper bound of the expected order of magnitude is established for a singular del Pezzo surface of degree four.