Parshin A.N., Shafarevich I.R. — Algebraic Geometry V : Fano Varieties :: Электронная библиотека попечительского совета мехмата МГУ

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Parshin A.N., Shafarevich I.R. — Algebraic Geometry V : Fano Varieties

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Название: Algebraic Geometry V : Fano Varieties

Авторы: Parshin A.N., Shafarevich I.R.

Аннотация:

The aim of this survey, written by V.A. Iskovskikh and Yu.G. Prokhorov, is to provide an exposition of the structure theory of Fano varieties, i.e. algebraic vareties with an ample anticanonical divisor. Such varieties naturally appear in the birational classification of varieties of negative Kodaira dimension, and they are very close to rational ones. This EMS volume covers different approaches to the classification of Fano varieties such as the classical Fano-Iskovskikh "double projection" method and its modifications, the vector bundles method due to S. Mukai, and the method of extremal rays. The authors discuss uniruledness and rational connectedness as well as recent progress in rationality problems of Fano varieties. The appendix contains tables of some classes of Fano varieties. This book will be very useful as a reference and research guide for researchers and graduate students in algebraic geometry.

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Рубрика: Математика/Алгебра/Алгебраическая геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1999

Количество страниц: 247

Добавлена в каталог: 12.03.2005

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Предметный указатель

-crepant blow-up      182
$\Delta$ -genus      50
$\mathbb{Q}$ -Cartier divisor      7
$\mathbb{Q}$ -Divisor      7
$\mathbb{Q}$ -factorial variety      11
$\mathbb{Q}$ -Fano fibration      178
$\mathbb{Q}$ -Fano variety      23
$\mathbb{Q}$ -Gorenstein variety      8
Abel — Jacobi map      162
Abundance conjecture      13 16
Anti-flips      180
Binary icosahedral group      115
Binary octahedral group      60
Boundary      10
Canonical class (divisor)      8
Canonical ring      13
Canonical threshold      178
cDV-point      9
Chain of rational curves      120
Coindex      50 108
Conic bundle      17 131
Conic bundle, standard      17 131
Cremona group      210
D-dimension      12
D-flop      22
Degree of $(H,\varphi: X \rightarrow S)$       182
Degree of a polarized variety      50
Degree of an invertible sheaf      7
Degree of Fano variety      24
Del Pezzo fibration      17
Del Pezzo surface      25
Del Pezzo variety      53
Discrepancy      8
Divisor, big      7
Divisor, ncf      7
Divisor, numerically effective      7
Effective cycle      11
Elementary rational map      66
Extremal ray      11
Fano $\mathbb{P}^1$ -bundle      133
Fano conic bundle      133
Fano variety      23
Fano variety, hyperelliptic      33 70
Fano variety, imprimitive      129
Fano variety, primitive      129
Fano variety, quasi-Goreistein      209
Fano variety, singular      23
Fano variety, trigonal      70
Fibration, maximally rationally connected      122
Fibration, rationally connected      122
Flip      20
FLOP      22
Fractional part of a divisor      14
Frobenius morphism      119
Fundamental divisor      24
Fundamental system      24
Genus of a Fano variety      32
Gorenstein index      8
Group of $\mathbb{Q}$ -Cartier divisors      7
Group of Cartier divisors      7
Group of invertible sheaves      7
Group of r-dimensional cycles      7
Group of Weil divisors      7
Hodge conjecture      212
Homological $\mathbb{P}^n$       205
Iitaka D-dimension      12
Index of Fano variety      24
Integral part of a divisor      14
Intermediate Jacobian      154
Intersection number      7
Kleiman's criterion for ampleness      12

Kodaira D-dimension      12
Kodaira dimension      12
Ladder of a polarized variety      51
Length of extremal ray      11
Log canonical divisor      10
Log canonical threshold      178
Log del Pezzo surface      24
Log Fano variety      23
Log flip      21
Log resolution      10
Log variety      10
Maximal blow-up      179
Maximal singularity      171 172
Minimal model      16
Mori cone      11
Mori Minimal Model Program      13
MRC-fibration      122
Net of quadrics      165
Non-negative vector bundle      117
Null-correlation vector bundle      145
Numerical dimension      12
Numerically equivalent cycles      7
Picard group      7
Pieard number      11
Positive part of the Mori cone      12
Projective space bundle      144
Quasi-effective $\mathbb{Q}$ -divisor      201
Quasi-effective threshold      201
Rational quotient      122
RC-fibratiou      122
Relative Fano model      16
Rung of a polarized variety      50
Rung of a polarized variety, regular      51
Sarkisov's degree      182
SCROLL      144
Scroll of vector bundle      38
Sectional genus      51
Semistable curve      121
Singularity, canonical      8
Singularity, Du Val      9
Singularity, log canonical      8
Singularity, log terminal      8
Singularity, terminal      8
Small contraction      66
Small extremal contraction      15
Small resolution      65
Standard form of a conic bundle      17 131
Terminal modification      209
Theorem, base point free      14
Theorem, cone      15
Theorem, contraction      15
Theorem, existence of minimal models      17
Theorem, non-vanishing      14
Theorem, rationality      15
Theorem, vanishing      14
Variety of general type      13
Variety polarized      50
Variety, $\mathbb{Q}$ -Gorenstein      8
Variety, Moishezon      204
Variety, non-degenerate      37
Variety, Q-factorial      11
Variety, rationally connected      120
Variety, separably uniruled      118
Variety, toric      30
Variety, uniruled      118
Vector bundle      104 106 135
Vector bundle, ample      203
Vector bundle, flat      204
Vector bundle, nef      203
Vector bundle, numerically effective      203
Veronese surface      8
Weighted projective space      26

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