Manin's conjecture for two quartic del Pezzo surfaces with 3A_1 and A_1+A_2 singularity types

TL;DR

This paper proves Manin's conjecture for two specific quartic del Pezzo surfaces with particular singularity types by analyzing divisor functions and employing Weil's bound for Kloosterman sums.

Contribution

It establishes the conjecture for these surfaces, utilizing new techniques involving divisor functions and equidistribution results.

Findings

Manin's conjecture verified for two quartic del Pezzo surfaces

Use of Weil's bound for Kloosterman sums is crucial

Analysis of restricted divisor functions in the proof

Abstract

We prove Manin's conjecture for two del Pezzo surfaces of degree four which are split over Q and whose singularity types are respectively 3A_1 and A_1+A_2. For this, we study a certain restricted divisor function and use a result about the equidistribution of its values in arithmetic progressions. In this task, Weil's bound for Kloosterman sums plays a key role.