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Debarre O. — Higher-Dimensional Algebraic Geometry
Debarre O. — Higher-Dimensional Algebraic Geometry

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Название: Higher-Dimensional Algebraic Geometry

Автор: Debarre O.

Аннотация:

Higher-Dimensional Algebraic Geometry studies the classification theory of algebraic varieties. This very active area of research is still developing, but an amazing quantity of knowledge has accumulated over the past twenty years. The author's goal is to provide an easily accessible introduction to the subject. Based on lectures given at Harvard University, the book begins with preparatory and standard definitions and results, moves on to discuss various aspects of the geometry of smooth projective varieties with many rational curves, and finishes in taking the first steps toward Mori's minimal model program of classification of algebraic varieties by proving the cone and contraction theorems.


Язык: en

Рубрика: Математика/Алгебра/Алгебраическая геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 233

Добавлена в каталог: 06.05.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$C_R$, contraction of the extremal ray R      154 205
$Exc(\Pi)$, exceptional locus of $\Pi$      28
$f^{\shark}T_{X/Y}$      46
$i_X$, Fano index of the Fano variety X      133
$j_X$, index of the normal variety X      178
$j_{x, \Delta}$, index of the pair $(X, \Delta)$      185
$K_X$, canonical divisor of X      27 70 174
$K_{X/Y}$ relative canonical divisor      71 73
$Mor_d(P^1, X)$      39
$Mor_S(Y,X;g)$      45
$NE(\Pi)$, relative cone of curves for a morphism $\Pi$      11
$N^1(X)$, group of Cartier divisors on X modulo numerical equivalence      10
$N_1(X)$, group of 1-cycles on X modulo numerical equivalence      10
$P_{r, s}$      25
$T_X$, tangent bundle to the smooth variety X      27
$X^d_N$, Fermat hypersurface of degree d in $P^N$      50
$X_{can}$, the canonical model of X      174
$X_{min}$, a minimal model of X      173
$X_{r\cdot s$}$      25 34 75 159
$Y_{r,s}$      25 34 75 159
$\equiv$, linear equivalence      3
$\hat{Y}_{r\cdot s}$      26 34 159
$\mathscr{C}$-equivalence      111 117
$\mathscr{I}_D$, multiplier ideal of the effective Q-divisor D      187
$\mathscr{R}$-equivalence      111 128 130 131
$\overline{NE}(X)$, closure of NE(X)      19
$\Pi$-ample divisor      see divisor
$\Pi$-big divisor      see divisor
$\Pi$-nef divisor      see divisor
$\rho_X$, Picard number of X      10 140 205 210
1-Cycle      4
1-cycle, numerically equivalent      10
Abelian surface      147
Abelian variety      76 78
Abundance conjecture      173 195
Albanese morphism      77
Albanese variety      76
Ample, $\Pi$-ample divisor      see divisor
Base-point-free theorem      195
Bend-and-break      56 58 64 70 79
Big, $\Pi$-big divisor      see divisor
Canonical algebra      170
Canonical divisor      see divisor
Canonical model      see model
Canonical singularities      177
Cartier divisor      2
Cone      149
Cone of curves      11
Cone theorem (singular case)      202
Cone theorem (smooth case)      144
Cone, extremal      see extremal subcone
Cone, relative      11 154
Conic bundle      76
Constructible set      63 113 114 118
Contraction      13 27 154 156 157 195 205
Contraction Theorem      205
Contraction, divisorial      156 158 160 207
Contraction, fiber      207
Contraction, small      156 157 159 161 207
CURVES      4
Curves, cone of      11
Curves, rational      see rational curve
Curves, relative cone of      11
Divisor with simple normal crossings      183
Divisor, $\Pi$-ample      207
Divisor, $\Pi$-big      216
Divisor, $\Pi$-nef      181 182 216
Divisor, ample      13 16 19
Divisor, big      20 21 36
Divisor, canonical      27 70 174
Divisor, Cartier      2
Divisor, effective      3
Divisor, fractional part of a      183
Divisor, generically ample      69 74
Divisor, generically nef      69 74 77
Divisor, linearly equivalent      3
Divisor, nef      16—18
Divisor, nef R-      20
Divisor, numerically equivalent      10
Divisor, principal      3
Divisor, Q      4
Divisor, Q-Cartier      4
Divisor, R-      4
Divisor, ramification      28
Divisor, relative canonical      71 73
Divisor, round-up of a      183
Divisor, support of a      3
Divisor, supporting      153
Divisor, theta      188 219
Divisor, very ample      14 29
Divisor, Weil      4
Effective 1-cycle      4
Effective divisor      3
Equivalence, class      3
Equivalence, linear      3
Equivalence, numerical      10
Evaluation map      40
Exceptional locus      27
Extremal ray      23 26 27 144 147—149 157 160 203 205 214 217
Extremal ray, contraction of      2 35 153 154 157 158 195
Extremal ray, length of      156 164 210
Extremal ray, locus of      154
Extremal subcone      2 12 13 23 26 149 159
Fano variety      see also variety
Fano variety have finitely many deformation types      134
Fano variety in positive characteristic      131
Fano variety is covered by rational curves      61
Fano variety is rationally chain-connected      130
Fano variety is rationally connected      107
Fano variety is simply connected      108
Fano variety with a positive Kaehler — Einstein metric      133
Fano variety with Picard number 1 has a very free curve      121
Fano variety, degree of a      132
Fano variety, Fano index of a      133
Fano variety, singular      135 136 207
Fano variety, transverse rational curves on a      80 82
Fermat hypersurface      38 41 49 52 93 99
Fibration      128
Fibration, rationally chain-connected      128
Flip      157 159 162 208 214
FLOP      166 173
Free rational curve      see rational curve
Frobenius morphism      11 55 61 62 67 72 81 82 119
Gauss map      77
Generic flatness      115
Generically ample divisor      see divisor
Generically nef divisor      see divisor
Hilb(X), Hilbert scheme of X      116
Hilbert polynomial      4 40 116
Hilbert scheme      116 120 135
Hironaka theorem      see theorem
Hodge index theorem      36 60 66 69 134 181
Hodge theory      10 76 97 183
Index of a pair      185 195 197
Index of a variety      178 180
Intersection number      7
Kaehler — Einstein metric      134
Kleiman's criterion      19
Kodaira dimension      94
Length of an extremal ray      see extremal ray
Leray spectral sequence      175 212 218
lg(B), length of a finite scheme B      45
Linearly equivalent divisors      3
Lines on a subvariety of a projective space      27 47 99
Locally factorial variety      see variety
Locus(ft), locus of the extremal ray R      154
Log terminal pair      185 187
Minimal model      see model
Minimal Model Program      2 13 35 94 135 154 173 182 208 210 216
Model, canonical      170 174
Model, minimal      165 173
Model, minimal, isomorphism in codimension 1      181
Mor(Y, X), space of morphisms from Y to X      39
Mor(Y,X;g)      45
Mori's program      see minimal model program
Morphism space      39
Morphism space with extra structures      44
Morphism space, local structure of      42
Morphism space, tangent space to      41
Multiplier ideal      186 219
Nakai — Moishezon criterion      14
NE(X), cone of curves of X      11
Nef, $\Pi$-nef divisor      see divisor
Nonvanishing theorem      see theorem
Numerically equivalent 1-cycles      10
Numerically equivalent divisors      10
P(V), projectivization of the vector space V      171
Picard number      10 205 208
Pluricanonical map      171
Principal divisor      3
Projection formula      9
R(X), rational quotient of X      112 128
r-free      see rational curve
Ramification divisor      28
Rat(X), curves with rational components on X      116
Rational comb      103
Rational curve      30—34 38 41 47 55 56 58 60 61 64 66—68 70 80 82 85 144 203 213 214 217 219
Rational curve, free      85 93 94 103 120
Rational curve, minimal      93 109 130
Rational curve, r-free      90—93 102
Rational curve, very free      86 96 98 107 110 121
Rational quotient      111 116 127—129 131
Rational tree      100
Rationality theorem      197
Rationally chain-connected variety      see variety
Rationally connected variety      see variety
Riemann — Roch theorem      21
Rigidity lemma      12
Round-up      183
Ruled variety      see also variety
Simple connectedness of smooth rationally chain-connected varieties      108
Simple connectedness of smooth rationally connected varieties      97 108
Singularities, canonical      177
Singularities, terminal      178
Stein factorization      11
Support of a divisor      3
Supporting divisor      153
surface      145
Surface, abelian      147
Surface, cubic      27
Surface, Del Pezzo      148 165
Surface, ruled      23 69 75
Terminal singularities      178
Theorem, base-point-free      195
Theorem, cone, logarithmic case      214
Theorem, cone, relative logarithmic case      217
Theorem, cone, singular case      202
Theorem, cone, smooth case      144
Theorem, contraction      205
Theorem, Hironaka desingularization      181 184 187
Theorem, rationality      197
Theorem, Shokurov nonvanishing      190
Theorem, vanishing, Kawamata — Viehweg      182 183 217 218
Theorem, vanishing, Kodaira      79 133 134 183
Theorem, vanishing, logarithmic, Kawamata — Viehweg      186
Theorem, vanishing, Nadel      188
Theorem, Zariski      28 29 62 149 172 207
Theta divisor      188 219
TREE      100
Uniruled variety      see variety
Universal property      40
Vanishing theorem      see theorem
Variety of general type      75 165 167 173 219
Variety, abelian      76 78
Variety, Fano      60 61 76 80 82 121 134 148
Variety, index of a      178 180
Variety, locally factorial      4 160
Variety, locally Q-factorial      28 29 208
Variety, rationally chain-connected      86 99 107
Variety, rationally connected      86 88 96 107
Variety, ruled      31
Variety, singular Fano      210
Variety, uniruled      85 86 94 121 154 207
Variety, with Picard number 1      73 120 121 131 132 134 135 145
Very ample divisor      see divisor
Very free rational curve      see rational curve
Very general      94
Weil divisor      4
Zariski main theorem      see theorem
[D], round-up of the divisor D      183
~, numerical equivalence      10
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