Göttsche L. — Hilbert Schemes Of Zero-Dimensional Subschemes Of Smooth Varieties :: Электронная библиотека попечительского совета мехмата МГУ

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Göttsche L. — Hilbert Schemes Of Zero-Dimensional Subschemes Of Smooth Varieties

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Название: Hilbert Schemes Of Zero-Dimensional Subschemes Of Smooth Varieties

Автор: Göttsche L.

Аннотация:

In this book we study Hilbert schemes of zero-dimensional subschemes of smooth varieties and several related parameter varieties of interest in enumerative geometry. The main aim here is to describe their cohomology and Chow rings. Some enumerative applications are also given. The Weil conjectures are used to compute the Betti numbers of many of the varieties considered, thus also illustrating how this powerful tool can be applied. The book is essentially self-contained, assuming only a basic knowledge of algebraic geometry; it is intended both for graduate students and research mathematicians interested in Hilbert schemes, enumertive geometry and moduli spaces.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1994

Количество страниц: 196

Добавлена в каталог: 20.11.2009

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Предметный указатель

$(E)^{n}_{m}$ , contact bundle      108
$(e_{i})_{i}>0$ , jumping index      10
$(n_{1},..., n_{r}) = (l^{\alpha_{1}}, 2^{\alpha_{2}},...)$ , partitions      29
$Al^{n}(P(E)), Al^{n}(P_{d})$ , variety of aligned subschemes      151
$Al^{n}_{n}(P_{N})$ , aligned n-tuples      135
$A_{*}(X), A^{*}(X)$ , Chow ring      19
$b_{i}(X)$ , Betti number      5
$Cop^{3}(\textbf{P}_{d})$ , Hilbert scheme of subschemes in a plane in $\textbf{P}_{d}$       180
$D_{l, x}(Y)$ , $I^{th}$ -order datum of Y at x      97
$F_{n}(X), G_{n}(X)$ , Semple bundles      128 196
$F_{n}(X/T), G_{n}(X/T)$ , relative Semple bundles      144
$Gal(\widetilde{k}/k)$ , Galois group      29
$G_{m}$ , multiplicative group      19
, Hodge polynomial      37
$Hilb^{n}(P(E)/X)$ , relative Hilbert scheme of projective bundle      149
$Hilb^{n}(R/m^{n}), Hilb^{n}(R)_{red}$ , punctual Hilbert scheme      10
$Hilb^{n}(X/T)$ relative Hilbert scheme of subschemes of length n      2
$H^{i}(\overline{X}, \textbf{Q}_{l})$ , l-adic cohomology      5
$h^{p,g}(X)$ , Hodge number      37
$h_{u, v}(\alpha)$ , hook difference      23
$J_{n}(X), J_{n}_{i}(X)$ , jet bundles      14
$K(X, Y_{T})$ , class of second order contact      126
$KS_{n-1}, KA_{n-1}$ , higher order Kummer variaties      40
$K_{n}(X), K_{n, l}(X)$ , class of contact with lines      140
$len(Z_{1})$ , length of subscheme      29
$N_{H/H'}$ , Shintani descent      42
$P(X, \mathds{F}_{q})$ , primitive 0-cycles      31
$p(\overline{X}, \textit{z})$ , Poincare polynomial      5
$P_{r}(X, \mathds{F}_{q})$ , set of primitive cycles      8
$S[\nu]$ , set used for counting      42
$T_{3}, Q_{3}$ , tautological subbundle and quotient of $W^{3}_{m}(X)$       107
$t_{i}(\alpha), T(\alpha)$ , diagonal lengths of partition      23
$T_{n}(X, \mathds{F}_{q})$ , set of admissible functions      31
$V_{n}$ , punctual Hilbert scheme      29
$X^{(n)}$ , symmetric power      3
$X^{(n)}_{\nu}$ , stratum of $X^{(n)}$       3
$X^{[n]}$ , stratum of $X^{[n]}$       4
$X^{[n]}, Hilb^{n}(X)$ , Hilbert scheme of subschemes of length n      2
$X^{[n]}_{(n), c}$ , curvilinear subschemes concentrated in a point      18
$X_{y}(X)$ , $x_{y}(X)$ -genus      37
$Z^{al}_{n}(P(E)), Z^{al}_{n}(P_{d})$ , universal subscheme of aligned subschemes      151
$Z_{n}(X/T), Z_{n}(X)$ , universal family      2
$Z_{q}(X, t)$ , zeta function      5
$Z_{T}(X), G_{T}(X)$ , relative Hilbert function strata      17
$Z_{T}, G_{T}$ , Hilbert function strata      10
$\Delta(\tau), \eta(\tau)$ , Delta function, eta function      35
$\Delta, \Delta^{n}$ , (thickened) diagonal      14
$\Delta_{\Phi}, \Delta_{\Phi}^{n}$ , (thickened) diagonals of morphism      88
$\epsilon, \delta$ , modular forms      38
$\Gamma(\alpha)$ , graph of partition      23
$\mathcal{D}_{k}(\sigma)$ , degeneracy locus      98
$\omega_{n}$ , Hilbert-Chow morphism      4
$\overline{H}, \overline{A}, \overline{h}, \overline{p}, \overline{\alpha}, \overline{\beta}$ , classes in the Chow ring of $Hilb^{3}(\textbf{P}(E)/X)$       173
$\Phi_{m, n}, \Phi_{n}$ , morphisms of Hilbert schemes of projective bundle      149
$\Phi_{\mathfrak{L}, n}$ , morphism of Hilbert scheme induced by $\mathfrak{L}$       146
$\sigma_{1}(n)$ , sum of numbers dividing n      51
$\textit{ev}_{E}$ , evaluation map      115
$\textit{Hilb}(X/T)$ Hilbert functor      1
$\textit{res}$ , residual morphism      61
$\widehat{H}^{3}(X)$ , variety of complete triangles      63
$\widehat{\alpha}$ , dual partition      23
$\widetilde{Cop}^{3}(\textbf{P}_{d})$ , variety of triangles in a plane in $P_{d}$       170
$\widetilde{D}^{2}_{m}(Y/T)$ , relative data variety      126
$\widetilde{Hilb}^{3} (\textbf{P}(E)/X), W(\textbf{P}(E)), \widehat{H}^{3}(\textbf{P}(E)/X)$ , relative triangle varieties      160
$\widetilde{Hilb}^{n} (X)$ , incidence variety of subschemes of lengths n — 1 and n      60
axe, axial morphism      152
Axial morphism      152 154 161 163 165 167 174 177
Borel — Moore homology      19
Cell decomposition      12 19—28 34 79 166 175
Chern classes of symmetric powers      93 114 150
cl, cycle map      19
Constant over T      148 149
Contact      81 99
Contact of families of curves      85 143

Contact with linear subspaces      119—125
Contact with lines      124—125 128—142
Contact, bundle      88 108 115—118 119 121
Cycle map      19
Degeneracy cycle      98 119
Degeneracy locus      98 119 123—124
Division point      43
Dual partition      23
e(X, G), orbifold Euler number      54
Evaluation morphism      115 119 122—124
Formula of Macdonald      35 49 50 79
G(n), symmetric group on n letters      3
General one-parameter subgroup      21 25 80
Geometric Frobenius      5 7 31 43
Geometric point      4 5 10 15 29
Good reduction      5 6 35 49 63 78
Graph of partition      23
H, A, $H_{2}$ , P, $P_{2}$ , classes in the Chow ring of $\widetilde{Hilb}^{3}(P_{2})$       154
Higher-order Kummer varieties      12 40—59
Hilb(X/T), relative Hilbert scheme      2
Hilbert function      9
Hilbert function strata      9—11 16—18 23—28 64 67 74 91 97 110 131
Hilbert polynomial      1
Hilbert scheme      1—4
Hilbert scheme for coplanar subschemes      145 170 180
Hilbert scheme of aligned subschemes      133 138 145 151—153 155 156 163 168 171 174 177 180
Hilbert scheme of subschemes of length n      2
Hilbert — Chow morphism      4 32 40 42 54 61
Hook difference of partition      23
Incidence variety      133—137 140 152 165 170 180
Initial degree      9
Initial form      9 18
Jet-bundle      14 85 90 104 110 111 119
Jumping index      10 28
K-valued point      2
l-adic cohomology      5
len(f), “length” of function      30
Leray — Hirsch for Chow groups      166 175
Modular forms      35 52
Multiplicative group      19m21
N-very ample      146
One-parameter subgroup      21
Orbifold Euler number      12 54—56
p(n), p(n, l), number of partitions of n      20
P(n), set of partitions of n      29
Partition      3 20 22 26 29 42 44
Porteous formula      98 99 120 123—124
Primitive zero-cycle      8 31—34 45 75
Punctual Hilbert scheme      9—11 19 29 30 33
Relative Hilbert scheme      2 147—149
Relative Hilbert scheme of projective bundles      145—184
Relative n-very ample      147—149
Relative Semple-bundle varieties      142—144
Relative varieties of second order data      126—127
Representable functor      1 60 63 83 101 102 109
Residual morphism      61 63 154 155 160 164 171 176 181
Schubert cycle      135 138 139
Second order contact      122 126
Segre class      98 124
Semple-bundle varieties      128—144
Shintani-descent      41
sign(X), signature      37
Stratification of - by partitions      3 14 30 60 67
Symmetric power of a variety      3 7 32 34 54 150
T-valued point      1
Universal subscheme      2 62 65 83 88 110 116 147 149 151 152 154 156 164
Varieties of higher order data      101—127
Varieties of higher order data of curves      85 91 94 113 114 118
Varieties of second order data      81—100
Varieties of triangles      12 60—80
Varieties of triangles with a chosen side      60—62
Varieties of triangles, complete triangles      63—73 79 85
Weil Conjectures      5—8 29 35 41 49 74
Zero-cycle      3 40
Zeta-function      5

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