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Introduction to Algebraic Geometry

Steven Dale Cutkosky
Publisher: 
American Mathematical Society
Publication Date: 
2018
Number of Pages: 
484
Format: 
Hardcover
Series: 
Graduate Studies in Mathematics 188
Price: 
83.00
ISBN: 
9781470435189
Category: 
Textbook
[Reviewed by
Felipe Zaldivar
, on
07/30/2018
]

In learning graduate-level mathematics, some subjects have the good fortune that there is one book that stands above the others as a standard reference. Silverman’s book on the arithmetic of elliptic curves is one such example. On the other hand, algebraic geometry does not have one such peak, but a whole mountain range of classical books that introduce the subject to eager students. Shafarevich’s Basic Algebraic Geometry has kept its allure: a bridge between classical algebraic geometry of the early part of the 20th century and the neoclassical Grothendieck-style geometry of the second half of the past century. Mumford’s Red Book, re-edited with a long list of typographical errors by Springer and reincarnated in a new version as Algebraic Geometry II, with some additions and editing by T. Oka, still has that scent of a new exhilarating subject. Or Hartshorne’s Algebraic Geometry (Springer, 1977) textbook, dry on the text and alive in its exercises, which has been reprinted many times by the publisher, each new scan a little bit harder to read, fading to such a point that sometimes the text becomes almost illegible, reminding us that printing is no longer an art. And there are more, perhaps not as encyclopedic as the ones listed above, but still great books.

This new book is reader-friendly and well organized, with plenty of exercises, accessible and self-contained (26 pages of a breviary of commutative algebra with detailed references to well-appreciated texts on the subject). Cutkosky’s book could be your textbook of choice for an introductory course on algebraic geometry.

Quasi-projective varieties form the main subject, with just a glimpse of schemes and sheaf cohomology in two chapters. There are applications to the degree of a projective variety, the geometry of smooth projective curves and surfaces, including an introduction to enough intersection theory needed to formulate and prove the Riemann-Roch theorem for surfaces. The last two chapters consider the algebraic analogues of the implicit function theorem and covering maps. The highlights are a proof of the purity of the branch locus, the Abhyankar-Jung theorem on tamely ramified covers, and the two Bertini’s theorems, with a careful discussion of the differences when the ground field has positive characteristic.

 

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Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.

See the table of contents in the publisher's webpage.

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