Rational Functions, Labelled Configurations, and Hilbert Schemes

In this paper, we continue the study of the homotopy type of spaces of rational functions from S 2 to C P n begun in [ 3 , 4 ]. We prove that, for n > 1, Rat k (C P n ) is homotopy equivalent to C k (R 2 , S 2 n −1 ), the configuration space of distinct points in R 2 with labels in S 2 n −1 of length at most k . This desuspends the stable homotopy theoretic theorems of [ 3 , 4 ]. We also give direct homotopy equivalences between C k (R 2 , S 2 n −1 ) and the Hilbert scheme moduli space for Rat k (C P n ) defined by Atiyah and Hitchin [ 1 ]. When n = 1, these results no longer hold in general, and, as an illustration, we determine the homotopy types of Rat 2 (C P 1 ) and C 2 (R 2 , S 1 ) and show how they differ.