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Ewald G. — Combinatorial convexity and algebraic geometry
Ewald G. — Combinatorial convexity and algebraic geometry

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Название: Combinatorial convexity and algebraic geometry

Автор: Ewald G.

Аннотация:

This text provides an introduction to the theory of convex polytopes and polyhedral sets, to algebraic geometry and to the fascinating connections between these fields: the theory of toric varieties (or torus embeddings). The fist part of the book contains an introduction to the theory of polytopes - one of the most important parts of classical geometry in n-dimensional Euclidean space. Since the discussion here is independent of any applications to algebraic geometry, it would also be suitable for a course in geometry. This part also provides large parts of the mathematical background of linear optimization and of the geometrical aspects in Computer Science. The second part introduces toric varieties in an elementary way, building on the concepts of combinatorial geometry introduced in the first part. Many of the general concepts of algebraic geometry arise in this treatment and can be dealt with concretely. This part of the book can thus serve for a one-semester introduction to algebraic geometry, with the first part serving as a reference for combinatorial geometry.


Язык: en

Рубрика: Математика/Алгебра/Комбинаторика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1996

Количество страниц: 372

Добавлена в каталог: 22.11.2004

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\lambda$-parallel body      107
$\mathcal{F}$-polytope      276
$\mathcal{O}(X_{\small{\sum}})$-module of global sections      263
(n-l)-diagram      85
Action, torus      239
Affine algebraic set      200
Affine algebraic variety      203
Affine combination      4
Affine dependency      49
Affine hull      4
Affine toric morphism      220
Affine toric subvariety, invariant      234
Affine toric variety      215
Affine toric variety, $\mathbb{Q}$-factorial      232
Affine toric variety, nonsingular      232
Affine toric variety, quasi-smooth      232
Affine toric variety, regular      232
Affine toric variety, smooth      232
Affine transform      52
Affine variety      203
Affinely dependent      4
AIon      96
Alexandrov — Fenchel’s inequality      129
Algebra, monomial      215
Algebraic set. affine      200
Algebraic subset      200 274
Algebraic torus      216
Algebraic variety, affine      203
Algebraic variety, irreducible      203
Ample sheaf      281
Angle, outer      120
Angle, outer (Q,P)      124
Angle, outer Q-      124
Angularr measure      120
Anticanonical divisor of toric variety      303
Apex      6
Associated polytope      174
Barnette sphere      87
Barycentric coordinates      4
Barycentric subdivision      74
Base space      246
Base, of the (n-1)-diagram      85
Big torus      242
Bipyramid with basis Q      41
Blaschke’s selection theorem      112
Blowdown      248
Blowdown equivariant      248
Blowup      248
Blowup equivariant      248
Body, $\lambda$-parallel      107
Body, convex      6
Body, polar      24
Boundary      308
Boundary operation      309
Boundary, complex      34
Boundary, local shadow      124
Boundary, reduced      34
Boundary, sharp local shadow      124
Boundary, sharp M-shadow      122
Bounding chain      309
Bruckner      86
Bruckner sphere      87
Bruggesser — Mani shelling      93
Bundle, fiber      246
Cartier divisor      264
Cartier divisor, invariant      263
Cech cohomology group      319
Cell, complex      66
Cell, k-      65
Cell, spherical      65
Cell, straight      65
Center      23
Centrally symmetric      23
Chain, bounding      309
Chain, complex      311
Chain, group      308
Chain, homologous chains      309
Chain, mapping      311
Characteristic, Euler - of a sheaf      324
Chern class      324
Chow ring      287
Class, Chern      324
Class, Todd      325
Closed star      67
Coboundary operation      311
Cochain group      319
Coface      51
Coface, complex      51
Cohomology group      311
Cohomology group, Cech      319
Collapse process      97
Collapse, elementary (s, m)-      97
Collapse, process      97
Combination, affine      4
Combination, convex      4
Combinatorial, equivalence      34
Combinatorial, isomorphism      34
Combinatorial, Picard group      170
Combinatorial, Picard number      173
Combinatorially, equivalent      34
Combinatorially, isomorphic      34
Combinatorially, isomorphic cones      152
Combinatorially, isomorphic polytopes      34
Compact factor      230
Complete fan      67
Complete fan, polyhedral      186
Complete fan, rational      186
Complete toric variety      251
Complex, boundary      34
Complex, cell      66
Complex, chain      311
Complex, coface      51
Complex, face      51
Complex, isomorphic complexes      66
Complex, quotient      81
Complex, reduced boundary      34
Complex, simplicial      66
Complex, spherical      66
Complex, subcomplex      67
Complex, topology      221
Cone      6
Cone of normals      16
Cone, combinatorially isomorphic cones      152
Cone, dual      16
Cone, lattice      144
Cone, normal cone at x      16
Cone, of normals      16
Cone, polyhedral      6
Cone, quotient      152
Cone, rational polyhedral      144
Cone, regular lattice      146
Cone, simplex      146
Cone, simplicial      146
Cone, with apex      6
Convex      3
Convex, body      6
Convex, combination      4
Convex, function      19
Convex, hull      4 10
Convex, polytope      5
Convex, strictly      27
Coordinate, functions      218
Coordinate, ring      202
Coordinate, transformation      221
Coordinates, barycentric      4
Cospan      146
Critical point      301
Critical point, index of a      301
Crosspolytope, n-      42
Culminate      95
Curve, moment      42
Cyclic polytope      42 44
De Concini — Procesi’s Theorem      252
Dehn — Sommerville equations      82 83
Del Pezzo, polytope      193
Del Pezzo, variety      304
Dependency, affine      49
Dependency, space $\mathcal{L}(X)$ of linear dependencies      49
Dependent, affinely      4
Diagram, (n-l)      85
Diagram, Gale      58
Diagram, Schlegel      85
Differentiable convex body      27
DIMENSION      6 211
Dimension, Krull      211
Direct sum      105
Distance function      21 23
Divisor, anticanonical - of a toric variety      303
Divisor, Carrier      264
Divisor, effective      263
Divisor, invariant      263
Divisor, invariant Cartier      265
Divisor, invariant prime      263
Divisor, locally principal      264
Divisor, prime      263
Divisor, principal      264
Divisors, Weil      263
Dual cone      16
Dual face      152
Dual shelling      95
EDGE      14
Effective divisor      263
Ehrhart’s Theorem      137
Element, face element representing a toric subvariety      289
Element, polytope      175
Elementary (s, m)-collapse      97
Elementary subdivision      70
Elementary subdivision, inverse      70
Embeddable      89
Embeddable, polyhedrally      89
Embedded torus      242
Embedding, polyhedral      89
Embedding, star-shaped      90
Embedding, torus      226
Equivalence, combinatorial      34
Equivalent, combinatorially      34
Equivariant line bundle      286
Equivariant morphism      244
Equivariant, blowdown      248
Equivariant, blowup      248
Equivariant, map      244
Equivariantly projective toric variety      276
Euler characteristic of a sheaf      324
Euler — Poincare’s theorem      78
f-vector      35
Face      14
Face elerrisnt representing a toric subvariety      289
Face, complex      51
Face, dual      152
Face, figure      38
Face, free      97
Face, improper      14
Face, k-      14
Face, polar      37
Facet      14
Fan      67
Fan, complete      67
Fan, map of fans      158
Fan, minimal regular complete      187
Fan, of K      17
Fan, perpendicular projection      246
Fan, polyhedral      67
Fan, polyhedral complete      186
Fan, quotient      81
Fan, rational complete      186
Fan, regular      165
Fan, simplicia]      67
Fan, strongly polytopal      159 175
Fan, unimodular equivalent fans      186
Fan-like sphere      91
Fano, polytope      192
Fano, toric-variety      304
Farey’s lemma      180
Fiber bundle      246
Fiber, typical      246
Fibration      246
Fibration, trivial      246
Figure, face      38
Figure, vertex      38
Finitely generated monoid      154
Free face      97
Function, convex      19
Function, distance      22
Function, Morse      301
Function, positive homogeneous      20
Function, rational      208
Function, regular      208
Function, sheaf of rational functions      260
Function, support      18
Gale diagram      58
Gale transform      52
Gale’s evenness condition      44
Generated ideal      200
Generators of a monoid      154
Generators, minimal system of      154
Global section      263
Gluing map      226
Gordan’s Lemma      154
Group, Cech cohomology      319
Group, chain      308
Group, cochain      319
Group, combinatorial Picard      170
Group, homology      310
Group, Picard      271
Group, relative homology      310
h-vector      83
Half-space      6
Half-space, supporting      12
Hausdorff, distance      107
Hausdorff, metric      108
Height      211
Hilbert’s Nullstellensatz      204
Hirzebruch surface      228
Hodge, toric-index theorem      296
Hodge, toric-inequality      295
Homologous chains      309
Homologous relative to a subcomplex      310
Homology group      310
Homology group with respect to a subcomplex      310
Homology group, relative      310
Homomorphism, module      260
Hull, affine      6
Hull, convex      6
Hull, inear      5
Hull, positive      6
Hyperplane, arrangement      139
Hyperplane, supporting      12
Ideal      200
Ideal, generated      200
Ideal, maximal      203
Ideal, prime      203
Ideal, principal      200
Ideal, vanishing      201
Improper faces      14
Independent zonotope      138
Index of a critical point      301
Inner, normal      12
Interior, relative      6
Intersection complex      75
Intersection number      291
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