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Bibliographic Details
Main Authors:	Itenberg, Il'ja V. 1966- (Author), Mikhalkin, Grigory (Author), Shustin, Eugenii (Author)
Format:	Book
Language:	English
Published:	Basel [u.a.] Birkhäuser 2009
Edition:	2. ed.
Series:	Oberwolfach seminars 35
Subjects:	Geometry, Algebraic Tropical geometry Tropische Geometrie
Links:	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
Physical Description:	VIII, 104 S. Ill., graph. Darst.
ISBN:	9783034600477

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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
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author	Itenberg, Il'ja V. 1966- Mikhalkin, Grigory Shustin, Eugenii
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illustrated	Illustrated
indexdate	2024-12-20T13:31:50Z
institution	BVB
isbn	9783034600477
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-017329605
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physical	VIII, 104 S. Ill., graph. Darst.
publishDate	2009
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publisher	Birkhäuser
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series	Oberwolfach seminars
series2	Oberwolfach seminars
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
work_keys_str_mv	AT itenbergiljav tropicalalgebraicgeometry AT mikhalkingrigory tropicalalgebraicgeometry AT shustineugenii tropicalalgebraicgeometry

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