Lectures on algebraic geometry: 2 Basic concepts, coherent cohomology, curves and their Jacobians
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Wiesbaden
Vieweg
2011
|
| Ausgabe: | 1. ed. |
| Schriftenreihe: | Aspects of mathematics
E ; 39 |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016309168&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XIII, 365 S. graph. Darst. |
| ISBN: | 9783834826862 9783834804327 |
Internformat
MARC
| LEADER | 00000nam a2200000 cc4500 | ||
|---|---|---|---|
| 001 | BV023106504 | ||
| 003 | DE-604 | ||
| 005 | 20190114 | ||
| 007 | t| | ||
| 008 | 080130s2011 xx d||| |||| 00||| eng d | ||
| 020 | |a 9783834826862 |9 978-3-8348-2686-2 | ||
| 020 | |a 9783834804327 |9 978-3-8348-0432-7 | ||
| 035 | |a (OCoLC)227281835 | ||
| 035 | |a (DE-599)BVBBV023106504 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-739 |a DE-824 |a DE-20 |a DE-19 |a DE-29T |a DE-11 |a DE-355 |a DE-188 | ||
| 100 | 1 | |a Harder, Günter |e Verfasser |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 | |a Wiesbaden |b Vieweg |c 2011 | |
| 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 | |
| 490 | 0 | |a Aspects of mathematics : E |v ... | |
| 773 | 0 | 8 | |w (DE-604)BV023059274 |g 2 |
| 830 | 0 | |a Aspects of mathematics |v E ; 39 |w (DE-604)BV000018737 |9 39 | |
| 856 | 4 | 2 | |m Digitalisierung UB Passau |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016309168&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-016309168 | |
Datensatz im Suchindex
| _version_ | 1819315168554778624 |
|---|---|
| adam_text | 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
|
| any_adam_object | 1 |
| author | Harder, Günter |
| author_facet | Harder, Günter |
| author_role | aut |
| author_sort | Harder, Günter |
| author_variant | g h gh |
| building | Verbundindex |
| bvnumber | BV023106504 |
| ctrlnum | (OCoLC)227281835 (DE-599)BVBBV023106504 |
| edition | 1. ed. |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01355nam a2200337 cc4500</leader><controlfield tag="001">BV023106504</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20190114 </controlfield><controlfield tag="007">t|</controlfield><controlfield tag="008">080130s2011 xx d||| |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9783834826862</subfield><subfield code="9">978-3-8348-2686-2</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9783834804327</subfield><subfield code="9">978-3-8348-0432-7</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)227281835</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV023106504</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-739</subfield><subfield code="a">DE-824</subfield><subfield code="a">DE-20</subfield><subfield code="a">DE-19</subfield><subfield code="a">DE-29T</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-355</subfield><subfield code="a">DE-188</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Harder, Günter</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Lectures on algebraic geometry</subfield><subfield code="n">2</subfield><subfield code="p">Basic concepts, coherent cohomology, curves and their Jacobians</subfield><subfield code="c">Günter Harder</subfield></datafield><datafield tag="250" ind1=" " ind2=" "><subfield code="a">1. ed.</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Wiesbaden</subfield><subfield code="b">Vieweg</subfield><subfield code="c">2011</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XIII, 365 S.</subfield><subfield code="b">graph. Darst.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Aspects of mathematics : E</subfield><subfield code="v">39</subfield></datafield><datafield tag="490" ind1="0" ind2=" "><subfield code="a">Aspects of mathematics : E</subfield><subfield code="v">...</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="w">(DE-604)BV023059274</subfield><subfield code="g">2</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Aspects of mathematics</subfield><subfield code="v">E ; 39</subfield><subfield code="w">(DE-604)BV000018737</subfield><subfield code="9">39</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Passau</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016309168&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="943" ind1="1" ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-016309168</subfield></datafield></record></collection> |
| id | DE-604.BV023106504 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T13:08:47Z |
| institution | BVB |
| isbn | 9783834826862 9783834804327 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016309168 |
| oclc_num | 227281835 |
| open_access_boolean | |
| owner | DE-739 DE-824 DE-20 DE-19 DE-BY-UBM DE-29T DE-11 DE-355 DE-BY-UBR DE-188 |
| owner_facet | DE-739 DE-824 DE-20 DE-19 DE-BY-UBM DE-29T DE-11 DE-355 DE-BY-UBR DE-188 |
| physical | XIII, 365 S. graph. Darst. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Vieweg |
| record_format | marc |
| series | Aspects of mathematics |
| series2 | Aspects of mathematics : E |
| spellingShingle | Harder, Günter 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_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 | http://bvbr.bib-bvb.de:8991/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 |