Let C be a smooth projective curve and X be a smooth projective variety. We will consider the space of degree d holomorphic maps from C to X. When X is a projective space, Segal discovered a remarkable phenomenon: as the degree increases, the homology of the space of holomorphic maps approximates the homology of the space of all continuous maps. Ellenberg-Venkatesh observed that this phenomenon is related to Manin's conjectures about rational points on Fano varieties, suggesting it holds more generally. I will talk about joint work with Ronno Das considering the case where X is a blowup of a projective space at finitely many points (including the case of del Pezzo surfaces).