Homological stability and Manin’s conjecture

I present our ongoing proofs for a version of Manin’s conjecture over F_q for q large and Cohen—Jones—Segal conjecture over C for rational curves on split quartic del Pezzo surfaces. The proofs share a common method which builds upon prior work of Das—Tosteson. 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 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 verifies the heuristic approach to Manin’s conjecture over global function fields given by Batyrev and Ellenberg–Venkatesh. This is joint work with Ronno Das, Brian Lehmann, and Phil Tosteson with a help by Will Sawin.