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Fulton W. — Introduction to toric varieties
Fulton W. — Introduction to toric varieties

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Название: Introduction to toric varieties

Автор: Fulton W.

Аннотация:

Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories.The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.


Язык: en

Рубрика: Математика/Алгебра/Алгебраическая геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1993

Количество страниц: 157

Добавлена в каталог: 12.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\mathbb{Q}$-Cartier divisor      62 65 89
Action of torus      19 23 31
Adjunction formula      91
Affine toric variety      4 16
Alexandrov — Fenchel inequality      119
Ample divisor      70—72 74 99
An singularity      47
Arithmetic genus      75 91
Betti numbers      91—95
Bezout theorem      121—124
Blowing up      6 40—43 50 71
Borel — Moore homology      103 108
Boundary of a cone      10—11
Boundary of a polytope      25 90 111
Brunn — Minkowski inequality      120
Canonical divisor      85—86 88
Cartier divisor      60
CHARACTER      37
Chern class      59—60 108—109 113
Chow group      63 96—101
Chow's Lemma      72
Cohen — Macaulay      30 31 73 89
Cohomology betti      58 93 101—108 128
Cohomology of line bundle      67 74 110
Compatible fan      73
Complete fan      39
Complete toric variety      39
Convex function      67
Convex polyhedral cone      9
Convex polytope, polyhedron      23 25
Cotangent bundle      86 108—109
Cotangent space      28
cube      24 27 50 65 69—70 105—106 114
Cyclic polytope      129
Cyclic quotient singularity      32 35
Dehn — Somerville equations      126
Differentials      86
Dimension of a cone      9
Distinguished point      28 37—38 42 51 55—56 61
Divisor      60—63
Dual face      12
Dual of a cone      4 9
Duality theorem for cones      9
Dualizing sheaf      89
Equivariant      48 60 80
Euler's formula      124—126
Euler-characteristic      59 80
Exceptional divisor      47
Face      9 23
Facet      10 24
Fan (in a lattice)      20
Farkas' theorem      11
Fiber bundle      29 41 70 93
Fundamental group      56—57
Generated by sections      67—68
Generators of a cone      9
Gordon's lemma      12 30
Grothendieck duality      89
Gubeladze's theorem      31
Half-space      11
Hard Lefschetz theorem      105
Hirzebruch surface      7—8 43 70
Hirzebruch — Jung continued fraction      46
Hirzebruch — Riemann — Roch      109
Hodge index theorem      119—120
Interior of a cone      12
Intersection homology      94 105 128
Intersection multiplicity      99 122
Intersection product      97—101
Intersection ring      106—108
Inversion formula      91
Isoperimetric inequalities      121
Lattice      4
Laurent polynomial      16 121
Line bundle      8 44 59—60 63—77
Local cohomology group      74
Logarithmic poles      87
Lower bound conjecture      129
Macaulay vector      127—128
Manifold with corners      78—80
McMullen conjecture      126
Minkowski sum      114
Mixed volume      114—121
Moment map      81—85
Monomials      17 19
Mori's program      50
Moving lemma (algebraic)      106—107
Multiplicity of cone      48
Newton polytope, polyhedron      121
Noether's formula      86
Non-projective variety      71—72 102
Nonsingular      28—29
normal      29 73
octahedron      24 27 69—70 114
One-parameter subgroup      36—39
Orbifold      34
Orbit      51—56
Pick's formula      113
Piecewise linear function      66 68
Poincare duality      104
polar      24
Polyhedron      66
Polytope toric variety of      23—27
Principal divisor      60
Projective bundle      8 42
Projective space      6—7 22 70 76 113—114 123
Proper intersection      97—98 100
Proper morphism      39—40
Quadric cone      5 17 27 49 72
Quotient singularity      31—36 100
Rational cone      20
Rational normal curve      32
Rational polytope      24
Rational ruled surface      8
Rational singularities      47 76
Refinement of fan      45
Relative interior      12
Residue      87 91
Resolution of singularities      45—50
Riemann — Roch      108—114
Saturated      18—19 30
Self-intersection number      8 47 111
Separation lemma      13
Serre duality      87—91
Simple polytope      129—130
Simplex, simplicial cone      15
Simplicial fan      34 65
Stanley — Reisner ring      108
Stanley's theorem      124—130
Star of cone      52
Steiner decomposition      117
Strictly convex function      68
Strongly convex cone      4 12 14
Support of Cartier divisor      96—97
Support of fan      38
Surface (nonsingular)      42—44
T-Cartier divisor      61—64 66
T-Weil divisor      60 63
Todd class      108—114
Toric variety      4 20
Torus      36 79
Torus action      23 51—56
Twisted projective space      35—36
Upper bound conjecture      129
V-manifold      34
Veronese embedding      35 73
Very ample divisor      69
Virtual Poincare polynomial      92—93
Volume      111
Weighted projective space      35—36
Weil Conjectures      94
Weil divisor, T-Weil divisor      60
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