Kollar J., Mori S. — Birational geometry of algebraic varieties :: Электронная библиотека попечительского совета мехмата МГУ

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Kollar J., Mori S. — Birational geometry of algebraic varieties

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Название: Birational geometry of algebraic varieties

Авторы: Kollar J., Mori S.

Аннотация:

One of the major discoveries of the past two decades in algebraic geometry is the realization that the theory of minimal models of surfaces can be generalized to higher dimensional varieties. This generalization, called the minimal model program, or Mori's program, has developed into a powerful tool with applications to diverse questions in algebraic geometry and beyond. This book provides the first comprehensive introduction to the circle of ideas developed around the program, the prerequisites being only a basic knowledge of algebraic geometry. It will be of great interest to graduate students and researchers working in algebraic geometry and related fields.

Язык:

Рубрика: Математика/Алгебра/Алгебраическая геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 254

Добавлена в каталог: 16.03.2005

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Предметный указатель

$\cong_{et}$ etale equivalence      149
$\equiv$ , numerical equivalence      4
$\equiv_f$ , numerical f-equivalence      5
$(L_1…L_d Z), intersection number      29
, d-fold intersection number      29
$A \leq B$ , A - B is effective      4
, a type of a Du Val point      122
$discrep(X, \Delta)$       52
$D^{ \perp}={Z|(DZ) = 0}$       19
, a type of a Du Val point      122
$D_{ \geq 0} = {Z | (D Z) \geq 0}$       19
$f_{\ast}(Z)$ , the birational transform of Z by f      5
$g_{\ast}^{-1}(Z)=(g^{-1})_{\ast}(Z)$       5
, a $\mu_r$ -action      171
$L^{[i]}$ , the double dual of $L^{ \otimes i}$       64
$L^{[i]}$ , the double dual of $L^{\otimes i}$       64
      19
$Nm_{Y/X}: g_{\ast}\mathcal{O}_Y \rightarrow \mathcal{O}_X$       154
, the space of 1-cycles of X modulo numerical equivalence      18
      45
, Serre’s condition      153
for       180
$Trace_{X/Y}: f_{\ast}\omega_X\rightarrow\omega_Y$       185
$v_E{ )$ , the valuation at E      218
$\bar{NE}(X)$       46
$\bar{NE}(X/Y)$ , the closure of NE(X)      19
$\bar{NE}_{D\geq 0}(X) = \bar{NE}(X) \cap D_{\geq 0}$       19
$\bar{NE}_{D\geq 0}(X/Y$ )      46 95
$\lceil D\rceil$ , the round up of divisor D      5
$\lceil d\rceil$ |, the round up of $d\in\mathbb{R}$       5
$\lfloor d\rfloor$ , the round down of $d\in\mathbb{R}$       5
$\lfloor D\rfloor$ , the round down of divisor D      5
$\mathbb{Q}$ -Cartier, divisor      4
$\mathbb{Q}$ -Divisor      4
$\mathbb{Q}$ -factorial      4 47 158
$\mathbb{Q}$ -factorial, globally analytically      47
$\mathbb{Q}$ -factorial, locally analytically      47
$\mathbb{Q}$ -Factorialization      195
$\mathbb{Q}$ -Fano fibration      46
$\mathbb{R}_{\geq 0}$ , the set of real numbers $\geq 0$       6
$\mathbb{Z}_{>0}$ , the set of positive integers      6
$\mathcal{O}_{E,Y} = \mathcal{O}_{\xi,Y} for generic$ \xi\in E$      50
$\omega^{|q|}_{X/S}$ , the double dual of $\ omega^{\otimes |q|}_{X/S}$       164
$\phi_{|D|/Y}: X \rightarrow \mathbb{P}_Y(f_{\ast}\mathcal{O}_X(D))$       94
$\phi_{|D|}: X \rightarrow \mathbb{P}(H^0(X,\mathcal{O}(D)))$       68
$\rho(X)$ , the Picard number of X      19
$\rho(X/Y)$ , relative Picard number      45
$\sim$ , linear equivalence      4
$\sim_f$ , linear f-equivalence      5
$\tau(X)$ , Tyurina number      129
(-l)-Curve      8
(K + D)-Flip      99
(K + D)-Flipping contraction      99 189
1-Complement      221
1-Cycle      11 18
1-Cycle, effective      18
1-Cycle, relative      45
:=, is defined by      6
Abelian surface      22
Abundance conjecture      81
Adjunction      173
Adjunction, formula      21 182
Adjunction, inversion of      173 174
Algebra, symbolic power      189
Algebraic equivalence      10 11
Algebraic function      148
Algebraic space      148
Algebraic valuation      50
Almost all      14
Ample, /-      34 35
Analytic case      47
Analytic singularity type      193
Analytic surface      25
Aut, the automorphism group      165 217
Automorphism      22 165
Basepoint-free theorem      75 78
Basepoint-free theorem, relative      94
Bend and break      9 11
Big      67
Big, /-      94
Bimeromorphic map      6
Birational map      6
Birational transform      5
Blow up, symbolic      199
Blow up, weighted      142 143 167 170
Branch      114
Bundle, canonical      47
Bundle, conic      28
Bundle, lattice      10
C, a type of an extremal ray      28
Calabi — Yau threefold      239
Canonical      42 56 165 238
Canonical bundle      47
Canonical bundle formula      52
Canonical bundle formula, Kodaira’s      49
Canonical divisor      47
Canonical flop      191
Canonical model      107
Canonical model, weak      107
Canonical ring      80 237
Canonical singularity      42
Cartier, $\mathbb{Q}-$       4
Castelnuovo’s Theorem      8
cDV, compound Du Val      165 199
Center      51
Chain      114
Chow’s Lemma      32
CM, Cohen — Macaulay      153
Codimension in      159
Codimension of pure      8 70
Cohen — Macaulay, CM      153
Compact complex manifold      13
Compound Du Val, cDV      165
Cone      18
Cone theorem      22 27 38 43 76 81
Cone theorem, relative      95
Cone, example of      21-22
Conic bundle      28
cont, a contraction      25
Contraction      25 46 76
Contraction, (K + D)-flipping      99
Contraction, divisorial      38 43 98
Contraction, extremal      38 43 100
Contraction, Fano      99
Contraction, fiber type      38
Contraction, flipping      43 98
Contraction, flopping      191
Contraction, morphism      27
Contraction, small      38
Contraction, theorem      76
Cover, cyclic      63
Cover, ramified cyclic      64
Cover, unbranched      13
Cover, unramified cyclic      63
Crepant      195
Criterion, Kleiman’s      19 33 35
Criterion, Nakai — Moishezon      31 34
Cubic surface      22
Curve, (-l)-      8
Curve, existence of rational      12 16 22 76 95
Curve, flipped      100
Curve, flipping      100
Curve, multiple      11
Curve, stable      230
Cusp      116
CYCLE      114 115
Cycle, 1-      11 18
Cycle, relative 1-      45
Cyclic cover      63
Cyclic cover, ramified      64

Cyclic cover, unramified      63
D, a type of an extremal ray      28
Deformation      144
Deformation, miniversal      145
Deformation, space      12 232
Deformation, theory      12
Deformation, versal      145
Degeneration      11
Del Pezzo fibration      28
Del Pezzo surface      22 39
Dimension at      149 150
Dimension, embedding      149 150
Dimension, fiber      9
Dimension, Kodaira      237
discrep(center $\cap S\subset Z, X, \Delta$ )      172
discrep(center $\subset Z, X, \Delta$ )      172
Discrepancy      50
Discrepancy of a divisor      51 112
Discrepancy of a pair      52
Discrepancy, total      52
Divisor      4
Divisor, $\mathbb{Q}$ -Cartier      4
Divisor, $\mathbb{Q}-$       4
Divisor, canonical      47
Divisor, effective      4
Divisor, over X      51
Divisor, prime      4
Divisor, simple normal crossing, snc      5
Divisor, support of      4
Divisorial contraction      38 43 98
Divisorial log terminal, dlt      58
Divisorial log terminal, dlt, morphism      208
Dlt, divisorial log terminal      58
Dlt, divisorial log terminal, morphism      208
Dlt, divisorial log terminal, numerically      112
Double dual      64
Du Val, compound, cDV      165
Du Val, singularity      113 143 165
Dual graph      113
Dual graph, extended      114
Dualizing sheaf, $\omega_X$       47 180
e(X) =#{crepant divisor}      198
Easy termination theorem      203
Effective 1-cycle      18
Effective divisor      4
El-5, types of an extremal ray      28 37
Elimination theory      15
Elliptic singularity      138 165
Elliptic threefold      239
Embedding, dimension      149 150
Embedding, minimal      150
En, a type of a Du Val point      122
Endomorphism      13
Endomorphism, Frobenius      14
Equivalence, algebraic      10 11
Equivalence, etale      149
Equivalence, linear      4
Equivalence, linear /-      5
Equivalence, numerical      4 18
Equivalence, numerical f-      5
Etale      148 150
Etale, equivalence      149
Ex(f), the exceptional set of f      5
Example of $\nexists$ simultaneous resolution      128
Example of cones      21- 22
Example of degenerations to cones      231
Example of extremal contractions      44
Example of extremal rays      25
Example of f-nef divisors      35
Example of flips      39
Example of Kummer threefolds      37
Example of minimal models      107
Example of non-algebraic varieties      10
Example of non-CM lc singularities      162
Example of simultaneous resolution      132
Example of Tyurina number      129
Exceptional set      5
Existence of $\mathbb{Q}$ -factorialization      195
Existence of flips      189- 191
Existence of flops      192 204
Existence of rational curves      12 16 22 76 95
Existence of semi-stable flips      210 224
Existence of special semi-stable flips      220
Existence of terminalization      195
Extended dual graph      114
Extremal contraction      38 43 100
Extremal contraction, face      18
Extremal contraction, ray      18
Extremal contraction, subcone      18
F, a type of an extremal ray      28
f-equivalence, linear      5
f-equivalence, numerical      5
f-Minimal      106
Face, extremal      18
Factorial, $\mathbb{Q}-$       4 47 158
Factorialization,       195
Family, one-parameter      13
Fano contraction      99
Fano fiber space      44
Fano threefold      239
Fano variety      28
Fiber dimension      9
Fiber space, Fano      44
Fiber type contraction      38
Fibration, -Fano      46
Fibration, Del Pezzo      28
Finite kernel      217
Finite quasi-      149 150
Finite sequence is      193 194 203 205 211
Fixed point of an action      40
Flatness, generic      14
Flip      39 41 190 237
Flip, existence of semi-stable      210 224
Flip, existence of special semi-stable      220
Flip, infinite sequence of      193
Flip, semi-stable      viii
Flip, sequence of      193
Flip, termination of      193 210 211
Flipped curve      100
Flipping contraction      43 98
Flipping curve      100
FLOP      191
Flop, canonical      191
Flop, existence of      192
Flop, sequence of      194
Flop, terminal      191 192
Flop, termination of      194 203 205
Flopping contraction      191
Fork      114
Formula, adjunction      21
Formula, canonical bundle      52
Formula, Hurwitz      210
Formula, Kodaira’s canonical bundle      49
Fractional part, { }      5
Free, f-      94
Frobenius endomorphism      14
Frobenius morphism      13 14 16
Fujita’s conjecture      238
Function, algebraic      148
Function, regular      148
G-Minimal model      48
GAGA principle      62
General Kodaira vanishing      73
Generic flatness      14
Geometric invariant theory      237
Globally analytically Q-factorial      47
Group, action      47 217
Hensel’s Lemma      125
Hironaka theorem      3 4
Hironaka theorem, weak      3 4

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