Tropical algebraic geometry:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2009
|
| Edition: | 2. ed. |
| Series: | Oberwolfach seminars
35 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VIII, 104 S. Ill., graph. Darst. |
| ISBN: | 9783034600477 |
Staff View
MARC
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| 100 | 1 | |a Itenberg, Il'ja V. |d 1966- |e Verfasser |0 (DE-588)122288599 |4 aut | |
| 245 | 1 | 0 | |a Tropical algebraic geometry |c Ilia Itenberg ; Grigory Mikhalkin ; Eugenii Shustin |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2009 | |
| 300 | |a VIII, 104 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Oberwolfach seminars |v 35 | |
| 650 | 4 | |a Geometry, Algebraic | |
| 650 | 4 | |a Tropical geometry | |
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| 689 | 0 | 0 | |a Tropische Geometrie |0 (DE-588)7566380-6 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Mikhalkin, Grigory |e Verfasser |4 aut | |
| 700 | 1 | |a Shustin, Eugenii |e Verfasser |4 aut | |
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Record in the Search Index
| _version_ | 1813502148030234624 |
|---|---|
| adam_text |
Contents
Preface
vii
Preface
to the second edition
ix
1
Introduction to tropical geometry
1
1.1
Images under the logarithm
. 1
1.2
Families of amoebas
. 4
1.3
Non-
Archimedean amoebas
. 5
1.4
Non-standard complex numbers
. 7
1.5
The tropical semifield
Τ
. 9
1.6
Tropical curves and integer
affine
structure
. 11
1.7
Exercises
. 14
2
Patchworking of algebraic varieties
17
2.1
Introduction: A general idea of the patchworking construction
. . . 17
2.2
Elements of toric geometry
. 19
2.2.1
Construction of toric varieties
. 19
2.2.2
A toric variety associated with a fan
. 20
2.2.3
A toric variety associated with a convex lattice polyhedron
21
2.2.4
Embedding of Tor(A) into
a projective
space
. 22
2.2.5
The real part of a toric variety and the moment map
. 22
2.2.6
Hypersurfaces in toric varieties
. 24
2.3
Viro's patchworking method
. 25
2.3.1
Chart of a real polynomial
. 25
2.3.2
Patchworking of real nonsingular hypersurfaces
. 27
2.3.3
Combinatorial patchworking
. 30
2.3.4
A tropical point of view on the combinatorial
Viro
patch-
working
. 34
2.3.5
Patchworking of pseudo-holomorphic curves on ruled surfaces
37
2.4
Patchworking of singular algebraic hypersurfaces
. 45
2.4.1
Initial data
. 46
2.4.2
Transversality conditions
. 46
2.4.3
The patchworking theorem
. 47
2.4.4
Some ¿>-transversality criteria
. 50
2.5
Tropicalization and patchworking in the enumeration of nodal curves
51
2.5.1
Plane tropical curves
. 52
2.5.2
Algebraic enumerative problem and its tropical analogue
. . 54
2.5.3
Tropical formulas for the Gromov-
Witten
and Welschinger
invariants
. 56
2.5.4
Tropical limit
. 56
2.5.5
Tropicalization of nodal curves
. 57
2.5.6
Reconstruction of a simple tropical curve
. 61
2.5.7
Reconstruction of the limit curve C^
. 63
2.5.8
Refinement of a tropical limit
. 64
2.5.9
Refinement of the condition to pass through a fixed point
. 67
2.5.10
Refined patchworking theorem
. 69
2.5.11
The real case: Welschinger invariants
. 73
2.6
Exercises
. 74
3
Applications of tropical geometry to enumerative geometry
77
3.1
Introduction
. 77
3.2
Tropical hypersurfaces in Mn
. 78
3.3
Geometric description of plane tropical curves
. 81
3.4
Count of complex nodal curves
. 83
3.5
Correspondence theorem
. 84
3.6
Mikhalkin's algorithm
. 85
3.7
Welschinger invariants
. 86
3.8
Welschinger invariants Wm for small
m
. 88
3.9
Tropical calculation of Welschinger invariants
. 89
3.10
Asymptotic enumeration of real rational curves
. 90
3.11
Recurrence formula for Welschinger invariants
. 92
3.12
Welschinger invariants Wm>i
. 93
3.13
Exercises
.'. 95
List of Figures
97
Bibliography
99 |
| any_adam_object | 1 |
| author | Itenberg, Il'ja V. 1966- Mikhalkin, Grigory Shustin, Eugenii |
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| building | Verbundindex |
| bvnumber | BV035409070 |
| callnumber-first | Q - Science |
| callnumber-label | QA564 |
| callnumber-raw | QA564 |
| callnumber-search | QA564 |
| callnumber-sort | QA 3564 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 240 |
| ctrlnum | (OCoLC)310400815 (DE-599)DNB992751756 |
| dewey-full | 516.35 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.35 |
| dewey-search | 516.35 |
| dewey-sort | 3516.35 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV035409070 |
| illustrated | Illustrated |
| indexdate | 2024-10-21T06:00:21Z |
| institution | BVB |
| isbn | 9783034600477 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017329605 |
| oclc_num | 310400815 |
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| owner_facet | DE-20 DE-19 DE-BY-UBM DE-384 DE-739 DE-11 DE-188 DE-83 |
| physical | VIII, 104 S. Ill., graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Oberwolfach seminars |
| series2 | Oberwolfach seminars |
| spelling | Itenberg, Il'ja V. 1966- Verfasser (DE-588)122288599 aut Tropical algebraic geometry Ilia Itenberg ; Grigory Mikhalkin ; Eugenii Shustin 2. ed. Basel [u.a.] Birkhäuser 2009 VIII, 104 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oberwolfach seminars 35 Geometry, Algebraic Tropical geometry Tropische Geometrie (DE-588)7566380-6 gnd rswk-swf Tropische Geometrie (DE-588)7566380-6 s DE-604 Mikhalkin, Grigory Verfasser aut Shustin, Eugenii Verfasser aut Oberwolfach seminars 35 (DE-604)BV019806550 35 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017329605&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Itenberg, Il'ja V. 1966- Mikhalkin, Grigory Shustin, Eugenii Tropical algebraic geometry Oberwolfach seminars Geometry, Algebraic Tropical geometry Tropische Geometrie (DE-588)7566380-6 gnd |
| subject_GND | (DE-588)7566380-6 |
| title | Tropical algebraic geometry |
| title_auth | Tropical algebraic geometry |
| title_exact_search | Tropical algebraic geometry |
| title_full | Tropical algebraic geometry Ilia Itenberg ; Grigory Mikhalkin ; Eugenii Shustin |
| title_fullStr | Tropical algebraic geometry Ilia Itenberg ; Grigory Mikhalkin ; Eugenii Shustin |
| title_full_unstemmed | Tropical algebraic geometry Ilia Itenberg ; Grigory Mikhalkin ; Eugenii Shustin |
| title_short | Tropical algebraic geometry |
| title_sort | tropical algebraic geometry |
| topic | Geometry, Algebraic Tropical geometry Tropische Geometrie (DE-588)7566380-6 gnd |
| topic_facet | Geometry, Algebraic Tropical geometry Tropische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017329605&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV019806550 |
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