代数几何V:FANO簇

出版时间：2009-1  出版社：科学出版社  作者：帕尔申  页数：247  

前言

　　要使我国的数学事业更好地发展起来，需要数学家淡泊名利并付出更艰苦地努力。另一方面，我们也要从客观上为数学家创造更有利的发展数学事业的外部环境，这主要是加强对数学事业的支持与投资力度，使数学家有较好的工作与生活条件，其中也包括改善与加强数学的出版工作。　　从出版方面来讲，除了较好较快地出版我们自己的成果外，引进国外的先进出版物无疑也是十分重要与必不可少的。从数学来说，施普林格（Springer）出版社至今仍然是世界上最具权威的出版社。科学出版社影印一批他们出版的好的新书，使我国广大数学家能以较低的价格购买，特别是在边远地区工作的数学家能普遍见到这些书，无疑是对推动我国数学的科研与教学十分有益的事。　　这次科学出版社购买了版权，一次影印了23本施普林格出版社出版的数学书，就是一件好事，也是值得继续做下去的事情。大体上分一下，这23本书中，包括基础数学书5本，应用数学书6本与计算数学书12本，其中有些书也具有交叉性质。这些书都是很新的，2000年以后出版的占绝大部分，共计16本，其余的也是1990年以后出版的。这些书可以使读者较快地了解数学某方面的前沿，例如基础数学中的数论、代数与拓扑三本，都是由该领域大数学家编著的“数学百科全书”的分册。对从事这方面研究的数学家了解该领域的前沿与全貌很有帮助。按照学科的特点，基础数学类的书以“经典”为主，应用和计算数学类的书以“前沿”为主。这些书的作者多数是国际知名的大数学家，例如《拓扑学》一书的作者诺维科夫是俄罗斯科学院的院士，曾获“菲尔兹奖”和“沃尔夫数学奖”。这些大数学家的著作无疑将会对我国的科研人员起到非常好的指导作用。　　当然，23本书只能涵盖数学的一部分，所以，这项工作还应该继续做下去。更进一步，有些读者面较广的好书还应该翻译成中文出版，使之有更大的读者群。　　总之，我对科学出版社影印施普林格出版社的部分数学著作这一举措表示热烈的支持，并盼望这一工作取得更大的成绩。

内容概要

The aim of this survey, written by V. A. lskovskikh and Yu. G.Prokhorov, is to provide an exposition of the structure theory of Fano varieties, i.e. algebraic varieties with an ample anticanonical divisor.Such varieties naturally appear in the birational classification of varieties of negative Kodaira dimension, and they are very close to rational ones. This EMS volume covers different approaches to the classification of Fano varieties such as the classical Fanolskovskikh"double projection"method and its modifications,the vector bundles method due to S. Mukai, and the method of extremal rays. The authors discuss uniruledness and rational connectedness as well as recent progress in rationality problems of Fano varieties. The appendix contains tables of some classes of Fano varieties.    This book will be very useful as a reference and research guide for researchers and graduate students in algebraic geometry.

书籍目录

IntroductionChapter 1. Preliminaries  1.1. Singularities  1.2. On Numerical Geometry of Cycles  1.3. On the Mori Minimal Model Program  1.4. Results on Minimal Models in Dimension ThreeChapter 2. Basic Properties of Fano Varieties  2.1. Definitions, Examples and the Simplest Properties  2.2. Some General Results  2.3. Existence of Good Divisors in the Fundamental Linear System  2.4. Base Points in the Fundamental Linear SystemChapter 3. Del Pezzo Varieties and Fano Varieties of Large Index  3.1. On Some Preliminary Results of Fujita  3.2. Del Pezzo Varieties. Definition and Preliminary Results  3.3. Nonsingular del Pezzo Varieties. Statement of the Main Theorem and Beginning of the Proof  3.4. Del Pezzo Varieties with Picard Number p = 1.  Continuation of the Proof of the Main Theorem  3.5. Del Pezzo Varieties with Picard Number p ≥ 2.  Conclusion of the Proof of the Main TheoremChapter 4. Fano Threefolds with p = 1  4.1. Elementary Rational Maps: Preliminary Results  4.2. Families of Lines and Conics on Fano Threefolds  4.3. Elementary Rational Maps with Center along a Line  4.4. Elementary Rational Maps with Center along a Conic  4.5. Elementary Rational Maps with Center at a Point  4.6. Some Other Rational MapsChapter 5. Fano Varieties of Coindex 3 with p = 1:The Vector Bundle Method  5.1. Fano Threefolds of Genus 6 and 8: Gushel's Approach  5.2. A Review of Mukai's Results on the Classification of Fano Manifolds of Coindex 3Chapter 6. Boundedness and Rational Connectedness of Fano Varieties  6.1. Uniruledness  6.2. Rational Connectedness of Fano VarietiesChapter 7. Fano Varieties with p ≥ 2  7.1. Fano Threefolds with Picard Number p ≥ 2 (Survey of Results of Mori and Mukai  7.2. A Survey of Results about Higher-dimensional Fano Varieties with Picard Number p ≥ 2Chapter 8. Rationality Questions for Fano Varieties I  8.1. Intermediate Jacobian and Prym Varieties  8.2. Intermediate Jacobian: the Abel-Jacobi Map  8.3. The Brauer Group as a Birational InvariantChapter 9. Rationality Questions for Fano Varieties II  9.1. Birational Automorphisms of Fano Varieties  9.2. Decomposition of Birational Maps in the Context of Mori TheoryChapter 10. Some General Constructions of Rationality and Unirationality  10.1. Some Constructions of Unirationality  10.2. Unirationality of Complete Intersections  10.3. Some General Constructions of RationalityChapter 11. Some Particular Results and Open Problems  11.1. On the Classification of Three-dimensional   -Fano Varieties  11.2. Generalizations  11.3. Some Particular Results  11.4. Some Open ProblemsChapter 12. Appendix: Tables  12.1. Del Pezzo Manifolds  12.2. Fano Threefolds with p = 1  12.3. Fano Threefolds with p = 2  12.4. Fano Threefolds with p = 3  12.5. Fano Threefolds with p = 4  12.6. Fano Threefolds with p ≥ 5  12.7. Fano Fourfolds of Index 2 with p ≥ 2  12.8. Toric Fano ThreefoldsReferencesIndex

用户评论 (总计3条)

•   代数几何Ⅴ,经典好书，得查字典呢
•   代数几何重要参考书，苏联学派的杰作
•   名家名著,好!精品.