Homological stability for rational curves on quartic del Pezzo surfaces

The moduli space of rational curves on a Fano variety is expected to exhibit “motivic” stability. Both Manin’s conjecture (over a finite field) and the Cohen-Jones-Segal conjecture (over the complex numbers) are instances of this meta-conjecture.

I will discuss ongoing joint work with Ronno Das, Sho Tanimoto, and Philip Tosteson in which we prove versions of these two conjectures for degree 4 del Pezzo surfaces. The proofs share a common method, demonstrating the compatibility of these conjectures in this special case. Our work builds upon a new technique developed previously by Das-Tosteson using additional arguments from algebraic geometry, topology, and number theory.