Let $C$ be a compact Riemann surface, and $X$ be smooth projective variety. We will consider the space of holomorphic maps $C\to X$.

When $X={\mathbb{P}}^n$, Segal demonstrated a remarkable stabilization phenomenon: as $d$ increases, the homology of the component of degree $d$ holomorphic maps converges to homology of the component of degree $d$ continuous maps $C\to X$. Ellenberg-Venkatesh and others have observed that this phenomenon is related to arithmetic conjectures about rational points on Fano varieties due to Batyrev and Manin. This suggests that this stabilization phenomenon may hold more generally.

I will talk about joint work with Ronno Das using the Vassiliev method to study the case of blowups of projective space at finitely many points (in particular del Pezzo surfaces).