Differential topology is the study of the (infinitesimal, local, and global) properties of structures on manifolds having no non-trivial local moduli, whereas differential geometry is the study of the (infinitesimal, local, and global) properties of structures on manifolds having non-trivial local moduli.
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The Coble variety is the moduli space of ordered sets of 6 points in the projective plane, and can be represented as a double cover of the projective 4-space branched over the Igusa quartic.