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

2025/06/20 by Das, Ronno, Lehmann, Brian, Tanimoto, Sho +1 · 2 citations
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2506.17071

Abstract

We prove a version of Manin's conjecture (over \mathbbF_q for q large) and the Cohen--Jones--Segal conjecture (over 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.

Cited by

• Homological stability and weak approximation
• The spaces of rational curves on del Pezzo surfaces via conic bundles

Related

• The spaces of rational curves on del Pezzo surfaces via conic bundles
• Manin's conjecture for quintic del Pezzo surfaces with a conic bundle structure
• Classifying sections of del Pezzo fibrations, II
• Universal torsors over Del Pezzo surfaces and rational points
• The split torsor method for Manin's conjecture