  We consider the space of holomorphic maps from a compact Riemann surface to a
projective space blown up at finitely many points. We show that the homology of
this mapping space equals that of the space of continuous maps that intersect
the exceptional divisors positively, once the degree of the maps is
sufficiently positive compared to the degree of homology. The proof uses a
version of Vassiliev's method of simplicial resolution. As a consequence, we
obtain a homological stability result for rational curves on the degree $5$ del
Pezzo surface, which is analogous to a case of the Batyrev--Manin conjectures
on rational point counts.
