本书由世界图书出版公司出版，主要内容包括：An Introduction to Elliptic Surfaces、Singular fibers、Singulex fibers of elliptic fibrations、lnvariants and the canonical bundle formula、Elliptic surfaces with a section and Weierstrass models、More general elliptic surfaces、The fundamental group等。

Preface

Introduction

1 Curves on a Surface

 Introduction

 Invariants of a surface

 Divisors on a surface

 Adjunetion and arithmetic genus

 The Riemann-Roch formula

 Algebraic proof of the Hodge index theorem

 Ample and nef divisors

 Exercises

2 Coherent Sheaves

 What is a coherent sheaf?

 A rapid review of Chern classes for projective varieties

 Rank 2 bundles and sub-line bundles

 Elementary modifications

 Singularities of coherent sheaves

 Torsion free and reflexive sheaves

 Double covers

 Appendix: some commutative algebra

 Exercises

3 B|ratlonal Geometry

 Blowing up

 The Castelnuovo criterion and factorization of birational morphisms

 Minimal models

 More general contractions

 Exercises

4 Stability

 Definition of Mumford-Takemoto stability

 Examples for curves

 Some examples of stable bundles on p2

 Gieseker stability

 Unstable and semlstable sheaves

 Change of polarization

 The differential geometry of stable vector bundles

 Exercises

5 Some Examples of Surfaces

 Rational ruled surfaces

 General ruled surfaces

 Linear systems of cubics

 An introduction to K3 surfaces

 Exercises

6 Vector Bundles over Ruled Surfaces

 Suitable ample divisors

 Ruled sur faces

 A brief introduction to local and global moduli

 A Zariski open subeet of the moduli space

 Exercises

7 An Introduction to Elliptic Surfaces

 Singular fibers

 Singulex fibers of elliptic fibrations

 lnvariants and the canonical bundle formula

 Elliptic surfaces with a section and Weierstrass models

 More general elliptic surfaces

 The fundamental group

 Exercises

8 Vector Bundles over Elliptic Surfaces

 Stable bundles on singular curves

 Stable bundles of odd fiber degree over elliptie surface

 A Zariski open subset of the modnii space

 An overview of Donaldson invariants

 The 2-dimensional invariant

 Moduli spaces via extensions

 Vector bundleswi thtrivial determinant

 Even fiber degree and multiple fibers

 Exercises

9 Bogomolov's Inequality and Applications

 Statementofththeorem

 The theorems of Bombieri and Reider

 Theproof of Bogomolov's theorem

 Symmetric powers of vector bundles on curves

 Restriction the orems

 Appendix:Galoisdes centtheory

 Exercises

10 Classification of Algebraic Surfacesand of Stable Bundles

 Outline of the classificationofs urfaces

 Proof of Castelnuovo's theorem

 The Albanese map

 Proofs of the classification the oremsf or surfaces

 The Castelnuovo-deFranchis theorem

 Classification of threefolds

 Classification of vector bundles

 Exercises

References

Index