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Generalized nuclei of complex surfaces

For eachg 1, we study a familyYg(n) of complex surfaces which admit a singular bration over CP 1 by complex curves of genus g. By examining a handlebody description for Yg(n), we show that these complex surfaces can be smoothly decomposed as the Milnor ber of a Brieskorn homology 3-sphere union a small submanifold, termed a \nucleus". This description generalizes known decompositions for elliptic surfaces.