Brian Lehmann (BC)

 Suppose X is a Fano variety.  Manin’s Conjecture (over a finite field) predicts the asymptotic behavior of the number of rational curves on X as the degree increases.  Analogously, the Cohen-Jones-Segal conjecture (over the complex numbers) predicts the asymptotic behavior of the homology groups of the moduli space of rational curves as the degree increases.

I will discuss 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.