Homology of spaces of curves on blowups

TL;DR

This paper studies the homology of spaces of holomorphic maps from Riemann surfaces to blown-up projective spaces, establishing a stability result and connecting it to rational curves on del Pezzo surfaces.

Contribution

It proves that the homology of these mapping spaces matches that of certain continuous maps under positivity conditions, using Vassiliev's method.

Findings

Homology of holomorphic map spaces equals that of positive intersection continuous maps.

Homological stability for rational curves on degree 5 del Pezzo surface.

Connection to Batyrev--Manin conjectures on rational points.

Abstract

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.