The Geometry of the Space of Holomorphic Maps from a Riemann Surface into Complex Projective Space

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In this paper we study the topology of the spaces Hol(M,Pn,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 Pn, 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,Pn,k) which allow us to construct spectral sequences with known E₁-terms that can be used to determine the homology groups of Hol(M,Pn,k). Complete calculations are given for the elliptic case (g=1). Complete rational calculations are also given for hyperelliptic curves.