Description: 1 Curves on a Surface.- Invariants of a surface.- Divisors on a surface.- Adjunction 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 Birational 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 ?2.- Gieseker stability.- Unstable and semistable sheaves.- Change of polarization.- The differential geometry of stable vector bundles.- Exercises.- 5 Some Examples of Surfaces.- Rational ruledsurfaces.- General ruled surfaces.- Linear systems of cubics.- An introduction toK3 surfaces.- Exercises.- 6 Vector Bundles over Ruled Surfaces.- Suitable ample divisors.- Ruled surfaces.- A brief introduction to local and global moduli.- A Zariski open subset of the moduli space.- Exercises.- 7 An Introduction to Elliptic Surfaces.- Singular fibers.- Singular fibers of elliptic fibrations.- Invariants 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 elliptic surfaces.- A Zariski open subset of the moduli space.- An overview of Donaldson invariants.- The 2-dimensional invariant.- Moduli spaces via extensions.- Vector bundles with trivial determinant.- Even fiber degree and multiple fibers.- Exercises.- 9 Bogomolov's Inequality and Applications.- Statement ofthe theorem.- The theorems of Bombieri and Reider.- The proof of Bogomolov's theorem.- Symmetric powers of vector bundles on curves.- Restriction theorems.- Appendix: Galois descent theory.- Exercises.- 10 Classification of Algebraic Surfaces and of Stable.- Bundles.- Outline of the classification of surfaces.- Proof of Castelnuovo's theorem.- The Albanese map.- Proofs of the classification theorems for surfaces.- The Castelnuovo-deFranchis theorem.- Classification of threefolds.- Classification of vector bundles.- Exercises.- References.

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