The band-edge exciton fine structure of wurtzite CdSe nanocrystals is investigated by a plane-wave pseudopotential method that includes spin-orbit coupling, screened electron-hole Coulomb interactions, and exchange interactions. Large-scale, systematic simulations have been carried out on quantum dots, nanorods, nanowires, and nanodisks. The size and shape dependence of the exciton fine structure is explored over the whole diameter-length configuration space and is explained by the interplay of quantum confinement, intrinsic crystal-field splitting, and electron-hole exchange interactions. Our results show that the band-edge exciton fine structure of CdSe nanocrystals is determined by the origin of their valence-band single-particle wave functions. Nanocrystals where the valence-band maximum originates from the bulk A band have a "dark" ground-state exciton. Nanocrystals where the valence-band maximum is derived from the bulk B band have a "quasi-bright" ground-state exciton. Thus, the diameter-length configuration map can be divided into two regions, corresponding to dark and quasi-bright ground-state excitons. We find that the dark/quasi-bright ground-state exciton crossover is not only diameter-dependent but also length-dependent, and it is characterized by a curve in the two-parameter space of diameter and length.