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Title: Introduction to Toric Varieties. (AM-131) by William Fulton ISBN: 0-691-00049-2 Publisher: Princeton University Press Pub. Date: 12 July, 1993 Format: Paperback Volumes: 1 List Price(USD): $35.00 |
Average Customer Rating: 4 (1 review)
Rating: 4
Summary: Good introduction to the subject
Comment: Anyone who needs a concrete set of examples from the set of general algebraic varieites will find them in toric varieties. The definitions and resulting constructions of toric varieties satisfy the need for an intuitive understanding of varieties. In addition, toric varieties are the easiest collection of varieties to manipulate from the standpoint of computational-geometric algorithms. Toric varieties also have applications to various areas of mathematical physics, such as in mirror symmetry. Indeed, the case of toric varieties is one of the few examples where an explicit mirror can be found.
Fulton gives an excellent overview of toric varieties in this short book, and the reading is fairly easy going. He introduces toric varieties in the first chapter as objects originating from compactification studies, with projective n-space the natural example as a compactification of complex n-space. It is their definition in terms of fans in lattices however that permeates chapter one. The author's treatment is very understandable, and he does not hesitate to use many diagrams and figures to illustrate the concepts. This is followed by a consideration of compactness and resolution of singularities. The example given of the resolution of a two-dimensional toric singularity is done, interestingly, via Hirzebruch-Jung continued fractions. A taste of the algebraic topology of toric varieties is given in the next chapter, where the fundamental groups and Euler characteristics are calculated, along with the cohomology of line bundles over toric varieties. More of this follows in the next chapter, where a statement and proof of Serre duality is given, along with a calculation of Bott numbers. The most interesting results are in the last chapter of the book on intersection theory. Because of the intuitive nature of toric varieties, one can see the very abstract constructions of algebraic geometry take on a concrete form. I think one can appreciate the more abstract constructions in algebraic geometry if the more concrete examples are studied first. This is especially true for those seeking to apply these ideas, for example physicists, who must grasp them quickly and efficiently.
This book should give readers sufficient insight into the subject to move on to applications or to more advanced treatments of toric varieties or algebraic geometry.
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Title: Intersection Theory by W. Fulton ISBN: 0387985492 Publisher: Springer-Verlag Pub. Date: 01 November, 1998 List Price(USD): $39.95 |
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Title: Hodge Theory and Complex Algebraic Geometry Volume 1 by Claire Voisin, B. Bollobas, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, Leila Schneps ISBN: 0521802601 Publisher: Cambridge University Press Pub. Date: 15 September, 2002 List Price(USD): $90.00 |
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Title: Principles of Algebraic Geometry by Phillip Griffiths, Joseph Harris ISBN: 0471050598 Publisher: Wiley-Interscience Pub. Date: 02 August, 1994 List Price(USD): $94.50 |
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Title: Lectures on Curves on an Algebraic Surface. (AM-59) (Annals of Mathematics Studies) by David Mumford ISBN: 0691079935 Publisher: Princeton University Press Pub. Date: 01 August, 1966 List Price(USD): $30.00 |
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Title: An Invitation to Algebraic Geometry by Karen E. Smith, Lauri Kahanpaa, Pekka Kekaelaeinen, William N. Traves ISBN: 0387989803 Publisher: Springer-Verlag Pub. Date: 31 October, 2000 List Price(USD): $49.95 |
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