By Robert Friedman

This publication covers the idea of algebraic surfaces and holomorphic vector bundles in an built-in demeanour. it's aimed toward graduate scholars who've had an intensive first-year direction in algebraic geometry (at the extent of Hartshorne's Algebraic Geometry), in addition to extra complex graduate scholars and researchers within the components of algebraic geometry, gauge conception, or 4-manifold topology. some of the effects on vector bundles also needs to be of curiosity to physicists learning string conception. a unique characteristic of the publication is its built-in method of algebraic floor idea and the research of vector package deal concept on either curves and surfaces. whereas the 2 matters stay separate during the first few chapters, and are studied in exchange chapters, they turn into even more tightly interconnected because the e-book progresses. therefore vector bundles over curves are studied to appreciate governed surfaces, after which reappear within the facts of Bogomolov's inequality for strong bundles, that's itself utilized to check canonical embeddings of surfaces through Reider's technique. equally, governed and elliptic surfaces are mentioned intimately, after which the geometry of vector bundles over such surfaces is analyzed. a few of the effects on vector bundles seem for the 1st time in e-book shape, compatible for graduate scholars. The e-book additionally has a robust emphasis on examples, either one of surfaces and vector bundles. There are over a hundred routines which shape a vital part of the textual content.

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**Extra resources for Algebraic Surfaces and Holomorphic Vector Bundles**

**Example text**

Let R be a graded k-algebra such that char(k) = and there is a k-derivation D: R ----+ R of weight -1 and an element t E RI with the property that D(t) = 1. If A := {r E R I D(r) = O}, then A is a graded k-subalgebra of R, t is transcendental over A and R = A[t]. Moreover, D = -it on A[t]. Proof. The fact that A is a graded k-subalgebra of R is immediate. Let A[T] be the polynomial A-algebra in the indeterminate T. Grade A[T] by deg(aTm) = deg( a) + m for every a E A homogeneous and m 2: 0. Consider the homomorphism of graded k-algebras rp: A[T] ----+ R such that rplA = idA and rp(T) = t.

Replacing h, ... ,1s by h, h - adl' fs - ash, we may therefore assume that

This shows that (X, L) is a smooth polarized variety of dimension 2:: 3 (which is not a cone), such that HO(X, Tx ® L -1) i= 0. ° A Counterexample in Characteristic 3 Let k be an algebraically closed field of characteristic 3. In JP>3 consider the surface X of equation f = T5 + T1Ti + T2T; + T3 T * f. £ We have = 0, = Ti + 2T1 T 3 = Ti - T1T3, T'f - T 2T 3· The closed subvariety in]p>3 of equations *k = Tj- T1T2, Tl- T2 T 3 = Ti - T1 T 3 = T;- T1T2 = * ° is (at least set-theoretically) the line as one can easily see.

### Algebraic Surfaces and Holomorphic Vector Bundles by Robert Friedman

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