Combinatorial Convexity and Algebraic Geometry

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Springer Science & Business Media, Oct 3, 1996 - Mathematics - 372 pages
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The aim of this book is to provide an introduction for students and nonspecialists to a fascinating relation between combinatorial geometry and algebraic geometry, as it has developed during the last two decades. This relation is known as the theory of toric varieties or sometimes as torus embeddings. Chapters I-IV provide a self-contained introduction to the theory of convex poly topes and polyhedral sets and can be used independently of any applications to algebraic geometry. Chapter V forms a link between the first and second part of the book. Though its material belongs to combinatorial convexity, its definitions and theorems are motivated by toric varieties. Often they simply translate algebraic geometric facts into combinatorial language. Chapters VI-VIII introduce toric va rieties in an elementary way, but one which may not, for specialists, be the most elegant. In considering toric varieties, many of the general notions of algebraic geometry occur and they can be dealt with in a concrete way. Therefore, Part 2 of the book may also serve as an introduction to algebraic geometry and preparation for farther reaching texts about this field. The prerequisites for both parts of the book are standard facts in linear algebra (including some facts on rings and fields) and calculus. Assuming those, all proofs in Chapters I-VII are complete with one exception (IV, Theorem 5.1). In Chapter VIII we use a few additional prerequisites with references from appropriate texts.
  

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Contents

III
3
IV
8
V
11
VI
14
VII
18
VIII
24
IX
29
X
35
XXXIV
179
XXXV
186
XXXVI
192
XXXVII
199
XXXVIII
214
XXXIX
224
XL
234
XLI
238

XI
40
XII
45
XIII
53
XIV
58
XV
65
XVI
70
XVII
78
XVIII
84
XIX
88
XX
92
XXI
96
XXII
103
XXIII
107
XXIV
115
XXV
120
XXVI
129
XXVII
135
XXVIII
138
XXIX
143
XXX
148
XXXI
154
XXXII
158
XXXIII
167
XLII
242
XLIII
248
XLIV
252
XLV
257
XLVI
259
XLVII
267
XLVIII
273
XLIX
281
L
287
LI
290
LII
296
LIII
303
LIV
307
LV
314
LVI
317
LVII
320
LVIII
324
LIX
329
LX
331
LXI
359
LXII
363
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