Deformation theory / Robin Hartshorne

The book is divided into four chapters which, altogether comprise twenty-nine sections. The author's guiding principle is to focus on four standard situations: (A) Deformations of subschemes of a fixed ambient scheme X. (B) Deformations of line bundles on a fixed scheme X. (C) Deformations of coherent sheaves on a fixed scheme X. (D) Deformations of abstract schemes,including the local study of deformations of singularities as well as the global study of deformations of non-singular varieties (global moduli). For each of these particular situations, a number of typical problems is discussed, with the ultimate goal to establish a global parameter space classifying the isomorphism classes of the objects in question and, moreover, to describe its geometric properties. However, in this introductory text, the technically involved proofs of the existence of these global classifying spaces are not provided, as the author's primary goal is rather to lay the foundations of the respective deformation theory that allow to describe the local structure of the (assumed) global parameter space. (Zentralblatt)

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