Title: Homological sieve and Manin’s conjecture

Abstract: I present our 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. We call this method as homological sieve method. 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 verifies the heuristic approach to Manin's conjecture over global function fields given by Batyrev and Ellenberg-Venkatesh, and it is a nice combination of various tools from algebraic geometry (birational geometry of moduli spaces of rational curves), arithmetic geometry (simplicial schemes, their homotopy theory, and Grothendieck—Lefschetz trace formula), algebraic topology (the inclusion-exclusion principle and Vassiliev type method of the bar complexes) and some elementary analytic number theory. This is joint work with Ronno Das, Brian Lehmann, and Phil Tosteson with a help by Will Sawin and Mark Shusterman.