Kollar J. — Rational curves on algebraic varieties :: Электронная библиотека попечительского совета мехмата МГУ

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Kollar J. — Rational curves on algebraic varieties

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Название: Rational curves on algebraic varieties

Автор: Kollar J.

Аннотация:

The aim of this book is to provide an introduction to the structure theory of higher dimensional algebraic varieties by studying the geometry of curves, especially rational curves, on varieties. The main applications are in the study of Fano varieties and of related varieties with lots of rational curves on them. This Ergebnisse volume provides the first systematic introduction to this field of study. The book contains a large number of examples and exercises which serve to illustrate the range of the methods and also lead to many open questions of current research.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2001

Количество страниц: 338

Добавлена в каталог: 01.12.2013

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Предметный указатель

$(*\cdot...\cdot*)$       294
$(*^{*})$       294
$*\leq_{*}$ , $*<_{*}$       160
$*^{-1}_{*}(\ )$       3
$*^{sn}$ , $*^{wn}$       85
$*^{[-1]}(\ )$       46
$*^{\cap}$       124
$Aut(\ )$ , $Aut^{\tau}(\ )$       17
$A_{*}(\ )$ , $AE_{*}(\ )$       122
$a_{*}(\ )$ , $a_{*}(\ ,\ )$       119
$B_{*}(\ )$ , $BE_{*}(\ )$       122
$Chow(\ )$ , $Chow_{*}(\ )$       51 74
$Chow^{big}(\ )$ , $Chow^{big}_{*}(\ )$       71
$Chow^{small}(\ )$ , $Chow^{small}_{*}(\ )$       71
$Ch^{-1}( )$       56 58
$C_{r}$ -field      230
$E(\ )$ , $E^{*}(\ )$       23 28
      78
$Grass(\ ,\ )$       11
, $Hilb_{*}(\ )$       9
$Hom(\ ,\ )$       16
$Hom{\ ,\ ,\ )$       94
$k^{ch}(\ )$       56
$N(\ )$ , $NE(\ )$ , $\overline{NE}(\ )$       126
$N^{*}(\ )$       123
$N_{*}(\ )$ , $NE_{*}(\ )$       122
$R(\ ,\ )$       173
$RatCurves^{n}(\ )$ , $RatCurves^{n}(\ ,\ )$       108
$RC_{*}$       214
, $\bar{u}(\ )$       186
$Z_{*}(\ )$ , $ZE_{*}(\ )$       121
$\ast$ -negative extremal ray      127
$\bar{*}$       210
$\equiv$       122
$\mathbb{P}(*,...,*)$       240
$\mathbb{P}^{1}$ -bundle      105
$\mathbb{Q}$ -Fano      240
$\mathrm{Aut}( )$ , $\mathrm{Aut}^{\tau}(\ )$       17
$\mathrm{CDiv}(\ )$       18 123
$\mathrm{Chain}_{*}(\ )$       212
$\mathrm{cont}_{*}$       166
$\mathrm{deg}_{(-K)}(\ )$       119
$\mathrm{deg}_{*}(\ )$       41
$\mathrm{Grass}(\ ,\ )$       11
$\mathrm{Hom}^{free}(\ ,\ )$       115
$\mathrm{Hom}^{n}_{bir}(\ ,\ )$       111
$\mathrm{Hom}_{bir}(\ ,\ )$       105
$\mathrm{Univ}^{rc}(\ )$       109
$\rho(\ )$ , $\rho_{*}(\ )$       126
$\rightarrow$ , $--\rightarrow$       3
${a\atop{\approx}}}$ , ${e\atop{\approx}}}$ , ${r\atop{\approx}}}$       122
${e\atop{\prec}}}$ , ${a\atop{\prec}}}$       123
Algebraic cycle      41
Algebraic equivalence      122
Algebraic realization      210
Algebraic relation      210
Ample (vector bundle)      116 265
Anticanonical degree      119
Attaching trees      155
Auticanonical degree      119
Auticanonical ring      173
Bend-and-break      134
Big      3
Birational transform      3
Canonical ring      173
Cayley form      56
CDiv( )      17
ch( )      56
Chain (of smooth rational curves)      155
Chow field      56
Chow field condition      70
Chow form      56
Chow functor      51 74
Chow pull-back      50 51
Chow( )      52
Closed under $\ast$       192
Comb      156
Cone of curves      126
Cone of effective cycles      122
Connected by a $\ast$ -chain      212
Connected by a $\ast$ -chain (of smooth rational curves)      212
Contractible      134
Cycle theoretic fiber      45 46
Cycle, defined over a subfield      42
Cycle, theoretic fiber      45 46
Cyclic cover      149
Degree $\ast$ uniruling      181
Degree (of a cycle)      41
Degree of a cycle      41
Del Pezzo surface      171
Del Pezzo, surface      171
DVR      3
Effective Cartier divisor      17
Effective cycle      41
Equivalence (algebraic, effective algebraic, effective rational, numerical, rational)      121—123
Equivalence relation      210
Equivalent (algebraically, rationally, numerically)      121—123
Essentially independent (family of cycles)      45
Essentially the same cycle      45
Exceptional set      286
Extremal ray      127
Extremal subcone      127
Fam( )      119 123
Family of algebraic cycles      46
Family of rational curves      108
Family of rational curves through $\ast$       109
Fano variety      240
Fano variety of lines      266
Field condition      70
Field of definition      19
Finite type (property)      144
Flat pull back      41
Flat pull-back      41
Flat section      135
Free morphism      113
Free over $\ast$       113

Fundamental cycle      41
General deformation      115
General point      3
Generically unobstructed      33
Geometrically irreducible      108
Geometrically normal      79 108
Geometrically rational      103 199
Geometrically rational components      103
Geometrically reduced      79 108
Geometrically ruled      181
Geometrically smooth      103
Grassmann functor      11
Group scheme      17
Handle      156
Hilb( ), $\mathrm{Hilb}_{*}(\ )$       10 74
Hilbert functor      9 74
Hilbert polynomial      9
Hom functor      16
Hom( , )      16
Hom( , , )      94
Incidence correspondence      53
Index (of a Fano variety)      245
Inseparably unirational      206
Inseparably uniruled      206
Intersection number      294 295
Irreducible algebraic relation      210
Irreducible relation      210
Line      248
Locally unobstructed      33
Locus      104 164
Locus( )      104 164
MAP      3
Maximal rationally chain connected fibration      222
Maximal rationally connected fibration      223
Minimal (free morphism)      195
Minimal free      195
Mod $\ast$ -reduction      144
Morphism      3
MRC-fibration      223
MRCC-fibration      222
Nearby smoothable      154
Nef vector bundle      265
nef, $\ast$ -nef      124
Nonnegative cycle      41
Normal form      211
Normal point      79
Normic form      230
Numerical equivalence      123
Obs( )      29
Obstruction      23 28
Obstruction space      29
Obtained from $\ast$ by attaching trees      155
Open algebraic relation      210
Open relation      210
Picard number      126
Prerelation      212
Prime divisor (of a field)      286
Proalgebraic relation      210
Product (of algebraic relations)      210
Product (of relations)      209
Proper algebraic relation      210
Proper relation      210
Push forward      41 81
Quot scheme      77
Quot( , ), $Quot(\ ,\ )$       77
rational      103 198
Rational equivalence      122
Rationally chain (of smooth rational curves) connected      199
Rationally chain connected      199
Rationally chain connected fibration      222
Rationally connected      199
RatLocus( )      104
Reduced point      79
Relation      209
Relation class      210 212
Relative Cartier divisor      17
Represent (a functor)      8
Root, $*^{th}$ root of $\ast$       149
Ruled      181
Ruled modification      289
S*( )      10
Semi normal      84
Semi normalization      84
Semi positive (vector bundle)      116
Separably rationally connected      199
Separably ruled      181
Separably unirational      199
Separably uniruled      181
Seshadri constant      305
Set theoretically equivalent      210
Smooth point      79
Smoothable      98 154
Smoothable fixing $\ast$       154
Subcomb      156
Supporting function      127
Tooth      156
TREE      155
Two general points can be connected by a -chain      213
Unirational      199
Uniruled      181
Uniruled modification      289
Uniruled with curves of $\ast$ -degree $\ast$       183
Univ( ), $\mathrm{Univ}_{*}(\ )$       10 52
Universal element      9
Universal family      9
Unobstructed      30
Unsplit (family of morphisms or rational curves )      192
Unsplit family of rational curves      192
Very general point      3
WDiv( )      123
Weak normalization      84
Weakly normal      84
Weighted projective space      240
Well defined family of algebraic cycles      46 47
Well formed (weighted projective space)      240
[]      41

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