By Donu Arapura

ISBN-10: 1461418097

ISBN-13: 9781461418092

It is a fairly fast moving graduate point advent to advanced algebraic geometry, from the fundamentals to the frontier of the topic. It covers sheaf conception, cohomology, a few Hodge concept, in addition to a few of the extra algebraic points of algebraic geometry. the writer usually refers the reader if the therapy of a undeniable subject is quickly to be had somewhere else yet is going into huge aspect on subject matters for which his therapy places a twist or a extra obvious perspective. His instances of exploration and are selected very rigorously and intentionally. The textbook achieves its objective of taking new scholars of complicated algebraic geometry via this a deep but vast creation to an unlimited topic, ultimately bringing them to the leading edge of the subject through a non-intimidating type.

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Given an open set 30 2 Manifolds and Varieties via Sheaves U ⊂ X, a function F : U → k is regular if it is locally extendible to a regular function on an open set of An as defined above, that is, if every point of U has an open neighborhood V ⊂ Ank with a regular function G : V → k for which F = G|V ∩U . 4. Let X be an affine variety, and let OX (U) denote the set of regular functions on U. Then U → OX (U) is a sheaf of k-algebras and OX (X) ∼ = k[x0 , . . , xn ]/I(X). Proof. 15 and the previous lemma.

A rank-1 vector bundle will also be called a line bundle. The product X × kn is an example of a vector bundle, called the trivial bundle of rank n. A simple nontrivial example to keep in mind is the M¨obius strip, which is a real line bundle over the circle. The datum {(Ui , φi )} is called a local trivialization. Given a C∞ real vector bundle π : V → X, define the presheaf of sections V (U) = {s : U → π −1U | s is C∞ , π ◦ s = idU }. This is easily seen to be a sheaf. When V = X × Rn is the trivial vector bundle, a section is given by (x, f (x)), where f : X → Rn , so that V (X) is isomorphic to the space of vector-valued C∞ functions on X.

We study this further in the exercises. A big difference between algebraic varieties and manifolds is that the former can be very complicated, even locally. We want to say that a variety over an algebraically closed field k is nonsingular or smooth if it looks like affine space (very) locally, and in particular if it is a manifold when k = C. The implicit function suggests a way to make this condition more precise. Suppose that X ⊆ ANk is a closed subvariety defined by the ideal ( f1 , . . , fr ) and let x ∈ X.

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Algebraic Geometry over the Complex Numbers (Universitext) by Donu Arapura

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