Deformations of Algebraic Schemes

This book aims at giving an account with complete proofs of the results and techniques which are needed to understand the local deformation theory of algebraic schemes over an algebraically closed field, thus providing the tools needed for example in the local study of Hilbert schemes and moduli problems. We work exclusively in the category of locally noetherian schemes over a fixed algebraically closed field \k, avoiding to switch back and forth between the algebraic and the analytic category. The first two chapters give a self-contained treatment of formal deformation theory via the ``classical'' approach; cotangent complexes and functors are not introduced, nor the method of differential graded Lie algebras. Another chapter treats in more detail the most important deformation functors, with the single exception of vector bundles; this being motivated by reasons of space and because good monographs on the subject are already available. Although they are not the central issue of the book, we included a chapter on Hilbert schemes and Quot schemes, since it would be impossible to give meaningful examples and applications without them, and because of the lack of an appropriate reference. Deformation theory is closely tied with classical algebraic geometry because some of the issues which had remained controversial and unclear in the old language have found a natural explanation using the methods discussed here. We have included a section on plane curves which gives a good illustration of this point. In the Appendices we have collected several topics which are well known and standard but we felt it would be convenient for the reader to have them available here.