Algebraic surfaces and holomorphic vector bundles by Robert Friedman

By Robert Friedman

A singular function of the publication is its built-in method of algebraic floor idea and the learn of vector package thought on either curves and surfaces. whereas the 2 matters stay separate during the first few chapters, they develop into even more tightly interconnected because the ebook progresses. hence vector bundles over curves are studied to appreciate governed surfaces, after which reappear within the facts of Bogomolov's inequality for strong bundles, that is itself utilized to check canonical embeddings of surfaces through Reider's strategy. equally, governed and elliptic surfaces are mentioned intimately, sooner than the geometry of vector bundles over such surfaces is analysed. a few of the effects on vector bundles seem for the 1st time in booklet shape, sponsored via many examples, either one of surfaces and vector bundles, and over a hundred routines forming an essential component of the textual content. geared toward graduates with a radical first-year direction in algebraic geometry, in addition to extra complex scholars and researchers within the parts of algebraic geometry, gauge concept, or 4-manifold topology, some of the effects on vector bundles may also be of curiosity to physicists learning string thought.

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Let K ⊂ F be a finite normal extension of fields. Let A be a subring of K which is integrally closed in K and let B be the integral closure of A in F. Let G be the group of automorphisms of F which restrict to identity on K. a) For every σ ∈ G, one has σ(B) = B; b) For every point x ∈ Spec(A), the group G acts transitively on the fiber a ( φ)−1 (x) in Spec(B). Proof. — a) Let b ∈ B. Then σ(b) belongs to F and is integral over A. One thus has σ(b) ∈ B. This shows that σ(B) ⊂ B. Similarly, one has σ −1 (B) ⊂ B, hence B ⊂ σ(B).

Let F′ be a finite extension of F which is normal over K, let B′ be the integral closure of A in F′ . 4), there exists a chain q′0 ⊂ ⋅ ⋅ ⋅ ⊂ q′n of prime ideals of B′ such that pi = q′i ∩ A for every i. Let ̃ qn be a prime ideal of B′ such that ̃ qn ∩ B = qn . 2, there exists an automorphism σ of F′ such that σ∣K = id and σ(q′n ) = ̃ qn . For every integer i such that 0 ⩽ i ⩽ n − 1, let qi = σ(q′i ) ∩ B. Then q0 ⊂ ⋅ ⋅ ⋅ ⊂ qn is a chain of prime ideals of B. For every integer i, one has qi ∩ A = σ(q′i ) ∩ B ∩ A = σ(q′i ∩ A) = σ(pi ) = pi , hence the theorem.

Let a ∈ A be such that x n = ax n+1 . Since x ≠ 0 and A is an integral domain, we may simplify by x n , hence ax = 1. This shows that x is invertible. b) Let p be a prime ideal of A. Then, A/p is an artinian ring which is an integral domain. By part a), it is a field, hence p is a maximal ideal. c) Since every prime ideal of A is maximal, every point of Spec(A) is closed. If Spec(A) were infinite, there would exist an infinite sequence (x n ) of pairwise distinct points in Spec(A). The infinite sequence ∅ ⊂ {x1 } ⊂ {x1 , x2 } ⊂ .

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