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Deformations of Algebraic Schemes (Grundlehren der mathematischen Wissenschaften)
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About Deformations Of Algebraic Schemes
Product Description This account of deformation theory in classical algebraic geometry over an algebraically closed field presents for the first time some results previously scattered in the literature, with proofs that are relatively little known, yet relevant to algebraic geometers. 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 Publishers Weekly 代数的多様体の小さく局所的な変形の研究は、1950年代の終わりの小平とスペンサーの古典的な仕事およびグロタンディークによるその定式化に端を発している.今日では、変分的な現象が関与してくるあらゆる状況における代数幾何学、および分類理論においてますます重要になってきている.また、変形理論は高度に定式化され、幅広く枝分かれしている. 本書は、(代数的閉体上の)古典的代数幾何学における変形理論を1冊にまとめている.いろいろな文献に散見されていた以前のいくつかの結果が、ほとんど知られていない証明とともに初めて一堂に会している. ノードを持つ平面曲線族のSeveri多様体の構造と特性、空間曲線、商特異点の変形、点のヒルベルトスキーム、局所Picard関手などへの応用も含まれている.本書の詳細な説明は、大学院生レベルで理解でき、また、多くの例が含まれている. Copyright© Reed Business Information, a division of Reed Elsevier Inc. All rights reserved. Review 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) From the Back Cover 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. About the Author Edoardo Sernesi - vita Present position:Professore ordinario di Geometria, Facoltà di Scienze MFN, Università Roma Tre Education:- Laurea in Matematica- Univers
