代數(shù)幾何V:FANO簇

前言

　　要使我國的數(shù)學(xué)事業(yè)更好地發(fā)展起來，需要數(shù)學(xué)家淡泊名利并付出更艱苦地努力。另一方面，我們也要從客觀上為數(shù)學(xué)家創(chuàng)造更有利的發(fā)展數(shù)學(xué)事業(yè)的外部環(huán)境，這主要是加強對數(shù)學(xué)事業(yè)的支持與投資力度，使數(shù)學(xué)家有較好的工作與生活條件，其中也包括改善與加強數(shù)學(xué)的出版工作?！　某霭娣矫鎭碇v，除了較好較快地出版我們自己的成果外，引進(jìn)國外的先進(jìn)出版物無疑也是十分重要與必不可少的。從數(shù)學(xué)來說，施普林格（Springer）出版社至今仍然是世界上最具權(quán)威的出版社。科學(xué)出版社影印一批他們出版的好的新書，使我國廣大數(shù)學(xué)家能以較低的價格購買，特別是在邊遠(yuǎn)地區(qū)工作的數(shù)學(xué)家能普遍見到這些書，無疑是對推動我國數(shù)學(xué)的科研與教學(xué)十分有益的事?！　∵@次科學(xué)出版社購買了版權(quán)，一次影印了23本施普林格出版社出版的數(shù)學(xué)書，就是一件好事，也是值得繼續(xù)做下去的事情。大體上分一下，這23本書中，包括基礎(chǔ)數(shù)學(xué)書5本，應(yīng)用數(shù)學(xué)書6本與計算數(shù)學(xué)書12本，其中有些書也具有交叉性質(zhì)。這些書都是很新的，2000年以后出版的占絕大部分，共計16本，其余的也是1990年以后出版的。這些書可以使讀者較快地了解數(shù)學(xué)某方面的前沿，例如基礎(chǔ)數(shù)學(xué)中的數(shù)論、代數(shù)與拓?fù)淙荆际怯稍擃I(lǐng)域大數(shù)學(xué)家編著的“數(shù)學(xué)百科全書”的分冊。對從事這方面研究的數(shù)學(xué)家了解該領(lǐng)域的前沿與全貌很有幫助。按照學(xué)科的特點，基礎(chǔ)數(shù)學(xué)類的書以“經(jīng)典”為主，應(yīng)用和計算數(shù)學(xué)類的書以“前沿”為主。這些書的作者多數(shù)是國際知名的大數(shù)學(xué)家，例如《拓?fù)鋵W(xué)》一書的作者諾維科夫是俄羅斯科學(xué)院的院士，曾獲“菲爾茲獎”和“沃爾夫數(shù)學(xué)獎”。這些大數(shù)學(xué)家的著作無疑將會對我國的科研人員起到非常好的指導(dǎo)作用?！　‘?dāng)然，23本書只能涵蓋數(shù)學(xué)的一部分，所以，這項工作還應(yīng)該繼續(xù)做下去。更進(jìn)一步，有些讀者面較廣的好書還應(yīng)該翻譯成中文出版，使之有更大的讀者群?！　】傊覍茖W(xué)出版社影印施普林格出版社的部分?jǐn)?shù)學(xué)著作這一舉措表示熱烈的支持，并盼望這一工作取得更大的成績。

內(nèi)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.

書籍目錄

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

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