The spaces of rational curves on del Pezzo surfaces via conic bundles

TL;DR

This paper proves Manin's conjecture for certain del Pezzo surfaces over function fields using a homological sieve method and establishes bounds for counting rational curves on these surfaces.

Contribution

It applies a homological sieve approach to confirm Peyre's height conjecture for split quintic del Pezzo surfaces over $ ext{F}_q(t)$ and provides bounds for rational curves over finite fields.

Findings

Proves Peyre's all height approach for specific del Pezzo surfaces.

Establishes lower bounds for rational curve counts over finite fields.

Utilizes the homological sieve method in the proof.

Abstract

Using the homological sieve method developed by Das--Lehmann--Tosteson and the author, we prove Peyre's all height approach to Manin's conjecture for split quintic del Pezzo surfaces defined over $\mathbb F_q(t)$ assuming $q$ is sufficiently large. We also establish lower bounds of correct magnitude for the counting function of rational curves on split low degree del Pezzo surfaces defined over $\mathbb F_q$ assuming $q$ is large.