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 birationa] 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 …… 9 Bogomolov's Inequality and Applications 10 Classification of Algebraic Surfaces and of Stable Bundles References Index
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