Homological stability and Manin's conjecture for rational curves on quartic del Pezzo surfaces

TL;DR

This paper proves versions of Manin's conjecture and the Cohen--Jones--Segal conjecture for rational curves on quartic del Pezzo surfaces over certain fields, using novel methods involving bar complexes and height zeta functions.

Contribution

It introduces a unified method based on bar complexes, dimension estimates, and virtual height zeta functions to prove conjectures for rational curves on quartic del Pezzo surfaces.

Findings

Validated Manin's conjecture over finite fields for large q.

Confirmed Cohen--Jones--Segal conjecture over complex numbers.

Developed new tools for counting rational curves using conic bundle structures.

Abstract

We prove a version of Manin's conjecture (over $\mathbb{F}_{q}$ for $q$ large) and the Cohen--Jones--Segal conjecture (over $\mathbb{C}$) for maps from rational curves to split quartic del Pezzo surfaces. The proofs share a common method which builds upon prior work of the first and fourth authors. The main ingredients of this method are (i) the construction of bar complexes formalizing the inclusion-exclusion principle and its point counting estimates, (ii) dimension estimates for spaces of rational curves using conic bundle structures, (iii) estimates of error terms using arguments of Sawin--Shusterman based on Katz's results, and (iv) a certain virtual height zeta function revealing the compatibility of bar complexes and Peyre's constant. Our argument substantiates the heuristic approach to Manin's conjecture over global function fields given by Batyrev and Ellenberg--Venkatesh in this case.