The study of small and local deformations of algebraic varieties originates in the classical work of Kodaira and Spencer and its formalization by Grothendieck in the late 1950's. It has become increasingly important in algebraic geometry in every context where variational phenomena come into play, and in classification theory, e.g. the study of the local properties of moduli spaces.Today deformation theory is highly formalized and has ramified widely within mathematics. This self-contained account of deformation theory in classical algebraic geometry (over an algebraically closed field) brings together for the first time some results previously scattered in the literature, with proofs that are relatively little known, yet of everyday relevance to algebraic geometers. Based on Grothendieck's functorial approach it covers formal deformation theory, algebraization, isotriviality, Hilbert schemes, Quot schemes and flag Hilbert schemes. It includes applications to the construction and properties of Severi varieties of families of plane nodal curves, space curves, deformations of quotient singularities, Hilbert schemes of points, local Picard functors, etc. Many examples are provided. Most of the algebraic results needed are proved. The style of exposition is kept at a level amenable to graduate students with an average background in algebraic geometry.
From the reviews:
"One of the goals of Springer's Grundlehren series is to provide reliable and thorough accounts of certain portions of mathematics. This volume by Edoardo Sernesi does just that, and hence fits the series well. ? So this is a book for algebraic geometers; for them, it'll prove to be a useful resource and reference." (Fernando Q. Gouvêa, MathDL, August, 2006)
"Without any doubt, this is a masterly book on a highly advanced topic in algebraic geometry. ? The entire text is kept at a level that makes it suitable for graduate students ? . But even for experts and active researchers in algebraic geometry, this unique book on algebraic deformation theory offers a great deal of inspiration and new insights, too, and its future role as a standard source and reference book in the field can surely be taken for granted from now on." (Werner Kleinert, Zentralblatt MATH, Vol. 1102 (4), 2007)
"The book under review gives an introduction to classical deformation theory using modern language, and is apparently unique among textbooks in the recent literature in that it is largely self-contained and covers the main topics ? . It will be attractive for graduate students with a basic knowledge of commutative algebra and algebraic geometry as a base for advanced lectures. The need for such a book was evident for a long time; the reviewer is happy to have it on his bookshelf." (Marko Roczen, Mathematical Reviews, Issue 2008 e)
