By Donu Arapura

ISBN-10: 1461418097

ISBN-13: 9781461418092

This is a comparatively fast-paced graduate point advent to complicated algebraic geometry, from the fundamentals to the frontier of the topic. It covers sheaf thought, cohomology, a few Hodge concept, in addition to a number of the extra algebraic elements of algebraic geometry. the writer often refers the reader if the therapy of a undeniable subject is quickly on hand in different places yet is going into massive element on issues for which his therapy places a twist or a extra obvious standpoint. His instances of exploration and are selected very conscientiously and intentionally. The textbook achieves its objective of taking new scholars of advanced algebraic geometry via this a deep but large advent to an unlimited topic, finally bringing them to the leading edge of the subject through a non-intimidating style.

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**Example text**

Given a vector bundle π : V → X over a manifold and a C∞ map f : Y → X, show that the set f ∗V = {(y, v) ∈ Y × V | π (v) = f (y)} with its ﬁrst projection to Y determines a vector bundle. 12. Given a vector bundle V → X with local trivialization φi : V |Ui → Ui × kn , check that the matrix-valued functions gi j = φi−1 ◦ φ j on Ui ∩U j satisfy the cocycle identity gik = gi j g jk on Ui ∩U j ∩Uk . Conversely, check that any collection gi j satisfying this identity arises from a vector bundle. 13.

This is an ideal, and it sufﬁces to show that 1 ∈ J. By the Nullstellensatz, it is enough to check that Z(J ) = 0, / where J ⊂ k[x0 , . . , xn ] is the preimage of J. By assumption, for any a ∈ X there exist polynomials f , g such that g(a) = 0 and F(x) = f (x)/g(x) for all x in a neighborhood of a. We have g¯ ∈ J, where g¯ is the image of g in S. Therefore a ∈ / Z(J ). Thus an afﬁne variety gives rise to a k-space (X, OX ). The ring of global regular functions O(X) = OX (X) is an integral domain called the coordinate ring of X.

Show that O(X) ∼ = k[x1 , . . , xn ]. 10. 14. Verify that the image of Segre’s embedding Pn × Pm ⊂ P(n+1)(m+1)−1 is Zariski closed, and the diagonal Δ is closed in the product when m = n. 15. Prove that O(Pnk ) = k. Deduce that Pnk is not afﬁne unless n = 0. 16. Fix an integer d > 0 and let N = n+d − 1. The dth Veronese map vd : d Pnk → PNk is given by sending [x0 , . . , xn ] to [v], where v is the vector of degree-d monomials listed in some order. Show that this map is a morphism and that the image is Zariski closed.

### Algebraic Geometry over the Complex Numbers by Donu Arapura

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