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Liu Q., Erne R. — Algebraic Geometry and Arithmetic Curves
Liu Q., Erne R. — Algebraic Geometry and Arithmetic Curves

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Название: Algebraic Geometry and Arithmetic Curves

Авторы: Liu Q., Erne R.


This book is a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularization (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the fundamental theorem of stable reduction of Deligne-Mumford. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 463

Добавлена в каталог: 21.05.2007

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Предметный указатель
$(R_{k})$, property      338—345
$(S_{k})$, property      338—345
$Ann\mathcal{F}$, annihilator of an $\mathcal{O}_X$-module $\mathcal{F}$      173
$A_K$, algebra obtained by extension of scalars      89
$D \cdot E$, intersection of a divisor D with a vertical divisor E      383
$degtr_k K$, transcendence degree of K over k      73
$deg_{k} D$, degree of a Cartier divisor D      275
$deg_{k} \mathcal{L}$, degree of an invertible sheaf      282
$Der_{A} (B,M)$, derivations of B into M      210
$dim_{x} X$, dimension of X at $x \in X$      68
$Div_s (X)$, group of divisors with support in $X_s$      381
$Div_s (X)_{\mathbb{R}}$, real vector space $Div_{s} (X) \oplus_{\mathbb{Z}} \mathbb{R}$      385
$Div_{+}(X)$, effective Cartier divisors      256
$D_{+}(f)$, open subset associated to a homogeneous element f      51
$D|_{E}$, restriction of a Cartier divisor to a closed subscheme E      377
$E^2$, self-intersection of a vertical divisor E      383
$f \times g$, product of two morphisms      80
$f^{*} \mathcal{G}$, pull-back of a sheaf of modules      163
$F_X$, absolute Frobenius      94
$f_{*} Z$, direct image of a cycle      271
$f_{S'}$, morphism obtained by base change $S'\rightarrowS$      81
$F_{X/S}$, relative Frobenius      94
$gr_{\mathfrak{m}} (A)$, graded ring associated to an ideal m      135
$G^0$, identity component of an algebraic group G      496
$H^{p} (X,\mathcal{F})$, Cech cohomology group of $\mathcal{F}$      182
$i_{x} (D,E)$, intersection number of D and E at x      377
$k(\nu)$, residue field of a valuation $\nu$      355
$K_{X/S}$, canonical divisor on a fibered surface $X \rightarrow S$      389
$length_{A} (M)$, length of an A-module M      258
$M \oplus_{A} N$, tensor product over A      2
$Mor_{S} (X,Y)$, set of morphisms of S-schemes from X to Y      81
$mult_{x} (D)$, multiplicity of a Cartier divisor at a point x      260
$mult_{x} (Z)$, multiplicity of a cycle at a point x      267
$M[\alpha]$, the $\alpha$-torsion elements of M      198
$M_f$, localization of M at f      10
$M_{\mathfrak{p}}$, localization of M at a prime ideal $\mathfrak{p}$      10
$n_{G}$, multiplication by n in a commutative group G      306
$Pic^{0} (X)$, group of divisors of degree 0      299 306 430
$p_{a}(X)$, arithmetic genus of a curve X      279
$p_{a}(Z)$, arithmetic genus of a vertical divisor Z      431
$p_{g}(X)$, geometric genus      279
$R^{p} f_{*} \mathcal{F}$, higher direct image of a sheafs      189
$s_x$, germ of a section s      35
$s| _{V}$, restriction of a section s to an open subset V      34
$T_{f, x}$, tangent map      126
$V(\mathcal{J})$, closed subscheme associated to a quasi-coherent sheaf of ideals      164
$V_{+}(I)$, closed subset defined by a homogeneous ideal I      50
$X \times_{s} Y$, fibered product of the S-schemes X, Y      80
$X^{(p)}$ twist by the Frobenius      94
$X_f$, open subset of X associated to a function $f \in \mathcal{O} _{X} (X)$      44
$X_s$, open subset defined by a section s of an invertible sheaf on X      166
$X_{red}$, reduced scheme associated to X      60
$X_{S'}$, S'-scheme obtained by base change $S' \rightarrow S$      81
$Z(P_{1}... P_{m})$, set of common zeros of the polynomials $P_{1},..., P{m}$      30
$\chi_{k}(\mathcal{F})$, Euler — Poincare characteristic of a coherent sheaf $\mathcal{F}$      205
$\Delta_{W}$, discriminant of a Weierstrass model      446
$\Delta_{X/Y}$, diagonal morphism      100
$\hat{A}$, formal completion of A for the I-adic topology      18
$\langle \cdot , \cdot \rangle$, symmetric bilinear form on $Div_{s} (X)_{\mathbb{R}}$      385
$\mathbb{P}(V)$, projective space associated to a vector space V      53
$\mathbb{P}^{n} _{A}$, projective space over a ring A      50
$\mathbb{P}^{n} _{S}$, projective space over a scheme S      82
$\mathcal{C}_{X/Y}$, conormal sheaf of X in Y      229
$\mathcal{F|} _{U}$, restriction of a sheaf to an open subset U      34
$\mathcal{F} \otimes _{\mathcal{O} _{X}} \mathcal{G}$, tensor product of two $\mathcal{O} _{X}$-modules      158
$\mathcal{F} ^{\vee}$, dual of an ${\mathcal{O} _{X}}$-module      173
$\mathcal{F}(n)$, twist of $\mathcal{F}$      166
$\mathcal{F}_s$, pull-back of $\mathcal{F}$ to a fiber      201
$\mathcal{H}om_{\mathcal{O}_{X}(\mathcal{F},\mathcal{G})}$, sheaf of homomorphisms from $\mathcal{F}$ to $\mathcal{G}$      172
$\mathcal{L}^{n}$, nth tensor power of an invertible sheaf      169
$\mathcal{N}_{X/Y}$ normal sheaf of X in Y      229
$\mathcal{O} _{X} (n)$, twist of $\mathcal{O} _{X}$      165
$\mathcal{O} _{X} ^{I}$, direct sum indexed by I      158
$\mathcal{O} _{X}$, structure sheaf      37
$\mathcal{O}_{K}$, valuation ring of K      106
$\mu_{x} (D)$, multiplicity of a hypersurface at a point x      402
$\nu_{\Gamma}$, valuation associated to a closed irreducible subset Г of codimension 1      354
$\nu_{\xi}$, valuation associated to a point of codimension 1      354
$\Omega^{1} _{X/Y}$ or $\Omega^{1} _{X}$ sheaf of relative differential forms      216
$\Omega^{r} _{X/Y}$ differential forms of order r      238
$\omega_{X/Y}$ dualizing (or canonical) sheaf      239
$\Phi_{E}$, group of components of the Neron model of E      497
$\pi_{0} (X)$, scheme of connected components of X      496
$\sigma$-Process      395
$\sqrt{I}$, radical of an ideal I      27
${\bar{x}}$, Zariski closure of {x}      63
a(X), Abelian rank of a curve      314
A*, set of invertible elements of a ring A      44
Abelian rank of a curve      314 316 521 531 555
Abelian variety      298
Abelian variety, good reduction      502
Abelian variety, models      456
Abelian variety, potential good reduction      502
Abelian variety, serni-Abelian reduction      see “Reduction”
Abelian variety, torsion points      299
Abhyankar's theorem on the desingularization of surfaces      362
Abhyankar's theorem on the exceptional locus of birational morphisms      411
Absolute value      467
Adjunction formula      239 390
Affine line      27 43
Affine morphism      172 191
Affine morphism, and base change      193
Affine open subset      44
Affine open subset, complement has codimension      1 124
Affine open subset, of a projective space      76
Affine open subset, of an affine space      76
Affine scheme      43
Affine scheme, and quasi-coherent sheaves      160
Affine scheme, cohomology of sheaves      186
Affine scheme, is separated      100
Affine scheme, Serre's criterion      187
Affine space      47
Affine space, open subset      192
Affine variety      55
Algebra      5
Algebra, finite      29
Algebra, finitely generated      20 29
Algebra, flat      6
Algebra, graded      20
Algebra, homogeneous      53
Algebra, simple      227
Algebraic group      307 314
Algebraic set      30
Algebraic set, and rational points      49
Algebraic variety      55
Algebraic variety, complete      107
Algebraic variety, Jacobian criterion of smoothness      130
Algebraic variety, proper      105 109 112
Algebraic variety, smooth      141—142 219 220
Algebraization of coherent sheaves      553
Alteration      361 374 407
Ample divisor      266 (see also “Ample sheaf”)
Ample divisor, on a curve      315
Ample divisor, the support is connected      266
Ample sheaf      169—178 194 197 198
Ample sheaf, on a curve      305
Ample sheaf, on a projective scheme over a Dedekind scheme      205
Ample sheaf, on a proper scheme      196
Ann(M), annihilator of a module M      13
Annihilator      13 173
Approximation theorem      379 531
Arcwise connectedness      331
Arithmetic surface      353
Arithmetic surface, minimal      418 423—424
Arithmetic surface, relatively minimal      418
Arithmetic surface, smooth      368—370 370 388 418 526
Arithmetic surface, with normal crossings      404 406—408 425 426 530
Artin — Rees lemma      21
Ass(M), set of associated prime ideals of M      253
Associated point      254
Associated prime ideal      253
Automorphisms, of a curve      300
Automorphisms, of a minimal arithmetic surface      418
Automorphisms, of a projective space      176
Automorphisms, of a stable curve      520 529
Automorphisms, of models      460
A[[T]], ring of formal power series with coefficients in A      18
Base change      81
Base point      176
Betti number      472
Betti number, of the dual graph associated to a curve      511
Bezout identity      384
Birational map      112
Birational map, and base change      146 194
Birational map, of normal fibered surfaces      354—355
Birational map, of smooth fibered surfaces      370
Birational morphism      96 304
Birational morphism, and blowing-up      328 332
Birational morphism, and dimension      334
Birational morphism, and multiplicities      410
Birational morphism, and Picard groups      408
Birational morphism, decomposition into a sequence of blowing-ups      396
Birational morphism, elimination of indeterminacy      396
Birational morphism, exceptional locus      see “Exceptional locus”
Birational morphism, of normal fibered surfaces      369—371 417
Birational morphism, of regular fibered surfaces      395
Birational morphism, quasi-finite      152
Birational morphism, to a Dedekind scheme      124
Birational morphism, to a normal scheme      201
Birational morphism, to a regular scheme      272
Blowing-up      318—332
Blowing-up, along a closed point      402
Blowing-up, along a regular closed subscheme      325
Blowing-up, and finite morphisms      344
Blowing-up, universal property      324
Branch locus      290 347
CaCl(X), group of Cartier divisors modulo linear equivalence      257
Canonical divisor      282 389
Canonical divisor, on a curve      282 288
Canonical divisor, on a fibered surface      389 429
Canonical map      288 296
Canonical model      440 457
Canonical sheaf      239 (see also “Dualizing sheaf”)
Cartier divisor      256
Cartier divisor, effective      256
Castelnuovo's criterion      416
Catenary ring      332—333
Catenary scheme      333
Cayley — Hamilton identity      122
Chevalley's Theorem      154
Chow's Lemma      109 196
Class group      58 268
Closed fiber      84 347
Closed immersion      38 155 164
Closed point      27
Closed point, a scheme without closed point      114
Closed point, in an algebraic variety      60 76
Closed subscheme      46
Closed subscheme, of a projective scheme      53
Closed subscheme, of an affine scheme      47
Cochain      180
Cochain, alternating      180
Codifferent      250 289
codim(Z,X), codimension of Z in X      69
Codimension      69 75 333
Cohen — Macaulay      see “Module” “Ring” “Scheme”
Coherent sheaf      161
Coherent sheaf, direct image      163
Coherent sheaf, finiteness of cohomology      195
Coherent sheaf, higher direct image      195
Coherent sheaf, on a projective scheme      167
Cohomological flatness      209 385 393
Cohomology (Cech)      182
Cohomology and flat base change      190
Complete intersection      206—207 287
Complete intersection, and duality      250
Complete intersection, is cohomologically flat      209
Complete intersection, is geometrically connected      207
Completion      see “Formal completion”
Completion of a curve      125 546
COMPLEX      4 35
Complex, exact      4
Component group      see “Group of components”
Conductor      250
conic      147—148 285 418
Conic, good reduction      479
Conic, reduction      529
Conic, regular model      426
Connected component      66 97
Constructive subset      98
Contraction      357—360 371—372 430—437 450
Contraction, of (–2)-curves      440 447 516
Contraction, of exceptional divisors      416—417
Coordinates, system of      129
Covering      544
Cup product      194
Curve      75 275
Curve, (-2)-curve      434
Curve, (-l)-curve      412
Curve, a proper curve over a field is projective      315
Curve, a regular proper flat curve over a Dedekind domain is projective      353
Curve, affine      315
Curve, affine plane      302
Curve, completion      125 546
Curve, flat curve over a scheme      347
Curve, normal but not smooth      278
Curve, of arithmetic genus $\leq 0$      see “Conic”
Curve, projective plane      110 280 300
CYCLE      252 267
Cycle, direct image      271 397
Cycle, of codimension 1      267
Cycle, positive      267
Cycle, prime      267
Cycle, restriction to an open subset      269
Cycle, support      267
D(f), open subset associated to a function f      27
d-uple embedding      176 209
de Jong's theorem on alteration      407
Decomposition group      146 532
Dedekind domain      11 115 344
Dedekind scheme      115—116
Dedekind scheme, is regular      128
Dedekind scheme, open subscheme      123
Degeneration      84
Degree, of a Cartier divisor      275
Degree, of a divisor in a projective space      287
Degree, of a finite morphism      176
Degree, of an invertible sheaf      282
Degree, transcendence      73
Deligne — Mumford's theorem on stable reduction      533
Depth      335—345
depth M, depth of a module      335
Derivation      210 218
Derived functor      185
Desingularization      361
Desingularization, a scheme without desingularization      373
Desingularization, embedded resolution      404
Desingularization, in the strong sense      361
Desingularization, minimal      424
Determinant      236 241
Differential      178
Differential forms      127 216
Differential forms, of higher order      238
Dilatation      395
dim A, dimension of a ring A      69
dim X, dimension of a topological space X      68
DIMENSION      68—77 96 144 333—335
Dimension, and surjective morphism      76
Dimension, and surjective morphisms      110
Dimension, formula      333
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