Abstract
We start by introducing the Hilbert scheme, which will be a model for the other situations, and which will provide us with examples as we go along. Then in Section 2 we discuss deformations over the dual numbers for Situations A, B, and C. In Section 3 we introduce the cotangent complex and the T i functors, which are needed to discuss deformations of abstract schemes (Situation D) in Section 5. In Section 4 we examine the special role of nonsingular varieties, using the infini-tesimal lifting property and the T i functors. We also show that the relative notion of a smooth morphism is characterized by the vanishing of the relative T 1 functors.
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© 2010 Robin Hartshorne
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Hartshorne, R. (2010). First-Order Deformations. In: Deformation Theory. Graduate Texts in Mathematics, vol 257. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1596-2_2
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DOI: https://doi.org/10.1007/978-1-4419-1596-2_2
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Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-1595-5
Online ISBN: 978-1-4419-1596-2
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