Tropical algebraic geometry / Ilia Itenberg, Grigory Mikhalkin, Eugenii Shustin.

Format:

Book

Author/Creator:

Itenberg, I. V. (Ilʹi︠a︡ Vladimirovich)

Contributor:

Mikhalkin, Grigory.

Shustin, Eugenii.

Series:

Oberwolfach seminars ; 35.

Oberwolfach seminars ; 35

Language:

English

Subjects (All):

Geometry, Algebraic.

Physical Description:

viii, 103 pages : illustrations ; 24 cm.

Place of Publication:

Basel ; Boston : Birkhäuser, [2007]

Summary:

Tropical geometry is algebraic geometry over the semifield of tropical numbers, i.e., the real numbers and negative infinity enhanced with the (max, +)-arithmetics. Geometrically, tropical varieties are much simpler than their classical counterparts. Yet they carry information about complex and real varieties.

These notes present an introduction to tropical geometry and contain some applications of this rapidly developing and attractive subject. It consists of three chapters which complete each other and give a possibility for non-specialists to make the first steps in the subject which is not yet well represented in the literature. The intended audience is graduate, post-graduate, and Ph.D. students as well as established researchers in mathematics.

Contents:

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 T 9

1.6 Tropical curves and integer affine structure 11

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([Delta]) 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 S-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([superscript 0]) 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

3 Applications of tropical geometry to enumerative geometry 77

3.2 Tropical hypersurfaces in R[superscript n] 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 W[subscript m] 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 W[subscript m,i] 93.

Notes:

Bibliographical references (pages 99-103).

ISBN:

9783764383091

3764383097

OCLC:

85243485