In this chapter we apply the methods of infinitesimal and formal deformations from Chapters 1, 2, and 3 to global questions. The foremost question in every situation is whether there exists a global “moduli space” parametrizing isomorphism classes of the objects in question. To make this question precise, we introduce the functorial language in Section 23, we define the notions of coarse moduli space and fine moduli space, and mention various properties of a functor that help determine whether it may be representable. The “easy” cases—that is, easy to state, though the proofs are not easy—are the cases of closed subschemes and invertible sheaves, Situations A and B, where the functor is representable respectively by the Hilbert scheme and the Picard scheme (Section 24).
KeywordsModulus Space Vector Bundle Elliptic Curve Elliptic Curf Hilbert Scheme
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