  In this paper we study the topology of the spaces Hol(M,P{n},k) of (basepoint
preserving) holomorphic maps of a given degree k from a Riemann surface M of
genus g>0 into the n-th complex projective space P{n}, n>0. Using symmetric
products of the surface as well as the geometry of the associated Abel-Jacobi
maps, we construct an explicit topological compactification of Hol(M,P{n},k)
which allow us to construct spectral sequences with known E_1-terms that can be
used to determine the homology groups of Hol(M,P{n},k). Complete calculations
are given for the elliptic case (g=1). Complete rational calculations are also
given for hyperelliptic curves.
