In the previous two chapters we have studied infinitesimal deformations of scheme and sheaves in preparation for the study of global moduli questions. The ideal or gold standard for a moduli space is the model represented by the Hilbert scheme (1.1): there is a scheme whose points parametrize the objects in question, there is a universal family over this scheme, and any other family is obtained by a unique base extension from the universal family. In general, moduli problems do not have such a nice answer, and even if they do, the existence may be quite subtle.
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