In Chapter 1 we studied deformations of a structure over the dual numbers. These are the first-order infinitesimal deformations. In this chapter we discuss deformations over arbitrary Artin rings, deformations of higher order. A new feature of these is that having a deformation over one Artin ring, it is not always possible to extend it to a larger Artin ring: there may be obstructions. So in each particular case we investigate the obstructions, and when extensions do exist, try to enumerate them. For closed subschemes, invertible sheaves, and vector bundles, that is, Situations A, B, C, we do this in Sections 6, 7. For abstract schemes, Situation D, the results are in Section 10. In Sections 8, 9 we deal with three special cases, namely Cohen–Macaulay subschemes of codimension 2, locally complete intersection schemes, and Gorenstein subschemes in codimension 3. In each of these cases we can track the deformations explicitly and show that they are unobstructed. In Section 11 we show how an obstruction theory affects the local ring of the corresponding para-meter space, and in Section 12 we apply this to prove a classical bound on the dimension of the Hilbert scheme of curves in P3. In Section 13 we describe one of Mumford’s examples of “pathologies” in algebraic geometry, a family of nonsingular curves in P3 whose Hilbert scheme is generically nonreduced.
KeywordsExact Sequence Irreducible Component Local Ring Complete Intersection Hilbert Scheme
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