Lectures on algebraic geometry: 2 Basic concepts, coherent cohomology, curves and their Jacobians
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Wiesbaden
Vieweg
2011
|
| Edition: | 1. ed. |
| Series: | Aspects of mathematics
E ; 39 |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 365 S. graph. Darst. |
| ISBN: | 9783834826862 9783834804327 |
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| 020 | |a 9783834826862 |9 978-3-8348-2686-2 | ||
| 020 | |a 9783834804327 |9 978-3-8348-0432-7 | ||
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| 035 | |a (DE-599)BVBBV023106504 | ||
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| 100 | 1 | |a Harder, Günter |d 1938-2025 |e Verfasser |0 (DE-588)1011622483 |4 aut | |
| 245 | 1 | 0 | |a Lectures on algebraic geometry |n 2 |p Basic concepts, coherent cohomology, curves and their Jacobians |c Günter Harder |
| 250 | |a 1. ed. | ||
| 264 | 1 | |c 2011 | |
| 264 | 1 | |a Wiesbaden |b Vieweg | |
| 300 | |a XIII, 365 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Aspects of mathematics : E |v 39 | |
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| 830 | 0 | |a Aspects of mathematics |v E ; 39 |w (DE-604)BV000018737 |9 39 | |
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| _version_ | 1835207664210018304 |
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| adam_text | |
| adam_txt |
Contents
Preface
v
Contents
vii
Introduction
xii
б
Basic Concepts
of the Theory of Schemes
1
6.1
Affine
Schemes
. 1
6.1.1
Localization
. 1
6.1.2
The Spectrum of a Ring
. 2
6.1.3
The Zariski Topology on Spec(A)
. 6
6.1.4
The Structure Sheaf on
Ѕрес(Л)
. 8
6.1.5
Quasicoherent Sheaves
. 11
6.1.6
Schemes as Locally Ringed Spaces
. 12
Closed Subschemes
. 14
Sections
. 15
A remark
. 15
6.2
Schemes
. 16
6.2.1
The Definition of a Scheme
. 16
The gluing
. 16
Closed subschemes again
. 17
Annihilators, supports and intersections
. 18
6.2.2
Functorial properties
. 18
Affine morphisms
. 19
Sections again
. 19
6.2.3
Construction of Quasi-coherent Sheaves
. 19
Vector bundles
. 20
Vector Bundles Attached to Locally Free Modules
. 20
6.2.4
Vector bundles and GLn-torsors
. 21
6.2.5
Schemes over a base scheme
S
. 22
Some notions of finiteness
. 22
Fibered products
. 23
Base change
. 28
6.2.6
Points,
Т
-valued Points and Geometric Points
. 28
Closed Points and Geometric Points on varieties
. 32
6.2.7
Flat Morphisms
. 34
The Concept of Flatness
. 35
Representability of functors
. 38
6.2.8
Theory of descend
. 40
Effectiveness for
affine
descend data
. 43
6.2.9
Galois descend
. 44
A geometric interpretation
. 47
Descend for general schemes of finite type
. 48
6.2.10
Forms of schemes
. 48
6.2.11
An outlook to more general concepts
. 51
Some Commutative Algebra
55
7.1
Finite A-Algebras
. 55
7.1.1
Rings With Finiteness Conditions
. 58
7.1.2
Dimension theory for finitely generated fc-algebras
. 59
7.2
Minimal prime ideals and decomposition into
irreducibles
. 61
Associated prime ideals
. 63
The restriction to the components
. 63
Decomposition into
irreducibles
for noetherian schemes
. 64
Local dimension
. 65
7.2.1
Affine
schemes over
к
and change of scalars
. 65
What is dim^
Π Ζ2)?
. 70
7.2.2
Local Irreducibility
. 71
The connected component of the identity of an
affine
group scheme
G/k
. 72
7.3
Low Dimensional Rings
. 73
Finite /c-Algebras
. 73
One Dimensional Rings and Basic Results from Algebraic Number
Theory
. 74
7.4
Flat morphisms
. 80
7.4.1
Finiteness Properties of Tor
. 80
7.4.2
Construction of flat families
. 82
7.4.3
Dominant morphisms
. 84
Birational
morphisms
. 88
The Artin-Rees Theorem
. 89
7.4.4
Formal Schemes and Infinitesimal Schemes
. 90
7.5
Smooth Points
. 91
The Jacobi Criterion
. 95
7.5.1
Generic Smoothness
. 97
The singular locus
. 97
7.5.2
Relative Differentials
. 99
7.5.3
Examples
. 102
7.5.4
Normal schemes and smoothness in codimension one
. 109
Regular local rings
. 110
7.5.5
Vector fields, derivations and infinitesimal automorphisms
.
Ill
Automorphisms
. 114
7.5.6
Group schemes
. 114
7.5.7
The groups schemes
(Ga,(Gřm
and
μη
. 116
7.5.8
Actions of group schemes
. 117
Projective
Schemes
121
8.1
Geometric Constructions
. 121
8.1.1
The Projective Space
Ψ\
. 121
Homogenous coordinates
. 123
8.1.2
Closed subschemes
. 125
8.1.3
Projective Morphisms and Projective Schemes
. 126
Locally Free Sheaves on P"
. 129
Opn(ď)
as Sheaf of Meromorphic Functions
. 131
The Relative Differentials and the Tangent Bundle of Pg
. 132
8.1.4
Seperated and Proper Morphisms
. 134
8.1.5
The Valuative Criteria
. 136
The Valuative Criterion for the Protective Space
. 136
8.1.6
The Construction
Proj(ñ)
. 137
A special case of a finiteness result
. 139
8.1.7
Ample and Very Ample Sheaves
. 140
8.2
Cohomology of Quasicoherent Sheaves
. 146
8.2.1
Cech
cohomology
. 148
8.2.2
The Kimneth-formulae
. 150
8.2.3
The cohomology of the sheaves Op» (r)
. 151
8.3
Cohomology of Coherent Sheaves
. 153
The Hubert polynomial
. 157
8.3.1
The coherence theorem for proper morphisms
. 158
Digression: Blowing up and contracting
. 159
8.4
Base Change
. 164
8.4.1
Flat families and intersection numbers
. 171
The Theorem of
Bertini
. 179
8.4.2
The
hyperplane
section and intersection numbers of line bundles
. 180
9
Curves and the Theorem of Riemann-Roch
183
9.1
Some basic notions
. 183
9.2
The local rings at closed points
. 185
9.2.1
The structure of
ÔC,P
. 186
9.2.2
Base change
.'. 186
9.3
Curves and their function fields
. 188
9.3.1
Ramification and the different ideal
. 190
9.4
Line bundles and Divisors
. 193
9.4.1
Divisors on curves
. 195
9.4.2
Properties of the degree
. 197
Line bundles on
non
smooth curves have a degree
. 197
Base change for divisors and line bundles
. 198
9.4.3
Vector bundles over a curve
. 198
Vector bundles on P1
. 199
9.5
The Theorem of Riemann-Roch
. 201
9.5.1
Differentials and Residues
. 203
9.5.2
The special case
С
=
P
l/k
. 207
9.5.3
Back to the general case
. 211
9.5.4
Riemann-Roch for vector bundles and for coherent sheaves
. 218
The structure of K'(C)
. 220
9.6
Applications of the Riemann-Roch Theorem
. 221
9.6.1
Curves of low genus
. 221
9.6.2
The moduli space
. 223
9.6.3
Curves of higher genus
. 234
The ''moduli space" of curves of genus
g
. 238
9.7
The Grothendieck-Riemann-Roch Theorem
. 239
9.7.1
A special case of the Grothendieck -Riemann-Roch theorem
. 240
9.7.2
Some geometric considerations
. 241
9.7.3
The Chow ring
. 244
Base extension of the Chow ring
. 247
9.7.4
The formulation of the Grothendieck-Riemann-Roch Theorem
. . . 249
9.7.5
Some special cases of the Grothendieck-Riemann-Roch-Theorem
. 252
9.7.6
Back to the case p2
:
X
=
C x C
—>
С
. 253
9.7.7
Curves over finite fields
. 257
Elementary properties of the
ζ
-function
.
258
The Riemann hypothesis
. 261
10
The
Picard
functor for curves and their Jacobians
265
Introduction:
. 265
10.1
The construction of the Jacobian
. 265
10.1.1
Generalities and heuristics
: . 265
Rigidification of VIC
. 267
10.1.2
General properties of the functor VTC
. 269
The locus of triviality
. 269
10.1.3
Infinitesimal properties
. 272
Differentiating a line bundle along a vector field
. 274
The theorem of the cube
. 274
10.1.4
The basic principles of the construction of the
Picard
scheme of a
curve
. 278
10.1.5
Symmetric powers
. 279
10.1.6
The actual construction of the
Picard
scheme of a curve
. 284
The gluing
. 291
10.1.7
The local representability of VlC9c/k
. 294
10.2
The
Picard
functor on X and on
J
. 297
Some heuristic remarks
. 297
10.2.1
Construction of line bundles on X and on J
. 297
The homomorphisms
фм
. 298
10.2.2
The projectivity of X and J
. 301
The morphisms
фм
are homomorphisms of functors
. 302
10.2.3
Maps from the curve
С
to X, local representability oiVlCx/k
,
VlCj/k
and the self duality of the Jacobian
. 303
10.2.4
The self duality of the Jacobian
. 310
10.2.5
General abelian varieties
. 311
10.3
The ring of endomorphisms End( J) and the f-adic modules Ti( J)
. . 314
Some heuristics and outlooks
. 314
The study of End( J)
. 315
The degree and the trace
. 318
The Weil Pairing
. 326
The Neron-Severi groups NS{J),NS{J
x J)
and End(J)
. 328
The ring of correspondences
. 331
10.4
Étale Cohomology
. 334
The cyclotomic character
. 334
10.4.1
Étale
cohomology groups
. 335
Galois cohomology
. 336
The geometric
étale
cohomology groups
. 338
10.4.2
Schemes over finite fields
.344
The global case
.346
The degenerating family of elliptic curves
.350
Bibliography
357
Index
362 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T19:46:50Z |
| indexdate | 2025-06-17T20:00:14Z |
| institution | BVB |
| isbn | 9783834826862 9783834804327 |
| language | English |
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| physical | XIII, 365 S. graph. Darst. |
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| publisher | Vieweg |
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| series | Aspects of mathematics |
| series2 | Aspects of mathematics : E |
| spelling | Harder, Günter 1938-2025 Verfasser (DE-588)1011622483 aut Lectures on algebraic geometry 2 Basic concepts, coherent cohomology, curves and their Jacobians Günter Harder 1. ed. 2011 Wiesbaden Vieweg XIII, 365 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Aspects of mathematics : E 39 (DE-604)BV023059274 2 Aspects of mathematics E ; 39 (DE-604)BV000018737 39 Digitalisierung UB Passau https://bvbr.bib-bvb.de:443/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016309168&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Harder, Günter 1938-2025 Lectures on algebraic geometry Aspects of mathematics |
| title | Lectures on algebraic geometry |
| title_auth | Lectures on algebraic geometry |
| title_exact_search | Lectures on algebraic geometry |
| title_exact_search_txtP | Lectures on algebraic geometry |
| title_full | Lectures on algebraic geometry 2 Basic concepts, coherent cohomology, curves and their Jacobians Günter Harder |
| title_fullStr | Lectures on algebraic geometry 2 Basic concepts, coherent cohomology, curves and their Jacobians Günter Harder |
| title_full_unstemmed | Lectures on algebraic geometry 2 Basic concepts, coherent cohomology, curves and their Jacobians Günter Harder |
| title_short | Lectures on algebraic geometry |
| title_sort | lectures on algebraic geometry basic concepts coherent cohomology curves and their jacobians |
| url | https://bvbr.bib-bvb.de:443/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016309168&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV023059274 (DE-604)BV000018737 |
| work_keys_str_mv | AT hardergunter lecturesonalgebraicgeometry2 |