Titlebook: Algebraic Surfaces and Holomorphic Vector Bundles; Robert Friedman Textbook 1998 Springer Science+Business Media New York 1998 Blowing up.

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Introduction,aic geometry, for example Castelnuovo’s criterion for a surface to be rational (1895), the theorem of Enriques that a surface is rational or ruled if and only if . or . is zero (1905), and in general the role of the canonical divisor in the classification of surfaces. This theory was reworked from t

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Coherent Sheaves,rk in the analytic category, so that the reader can for the moment take . = ℂ{.,...,.} to be the ring of convergent power series at the origin if so desired. There are two paradigms for what a coherent shea . on . should look like:

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Birational Geometry,and the relationship between the invariants of the blown up surface . of ., we prove the Castelnuovo criterion for blowing down a curve. Using the Castelnuovo criterion, we show that every birational morphism between two smooth surfaces is a composition of blowups, and discuss various notions of min

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Vector Bundles over Ruled Surfaces,al, it seems a little difficult to obtain detailed information about moduli spaces when stability is defined with respect to an arbitrary ample divisor. We will need instead to consider ample divisors adapted to the geometry of the surfaces at hand. The examples we have in mind are all given as fibr

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An Introduction to Elliptic Surfaces, Kodaira’s classification of singular fibers of elliptic fibrations, and describe the canonical bundle and other basic invariants of an elliptic surface. We give a complete discussion of elliptic surfaces with a section and discuss, without proofs, the mechanism for obtaining all elliptic surfaces s

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Vector Bundles over Elliptic Surfaces,and time, we do not give complete proofs of all of the results, and this chapter is mainly intended as a sampler of various methods for studying bundles over .. For simplicity, we shall assume that . is simply connected, so that the base curve is ℙ. and shall concentrate on the case where . has a se

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