Tropical Algebraic Geometry

By: Ilia Itenberg, Grigory Mikhalkin, Eugenii I. Shustin

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Paperback | 16 April 2009 | Edition Number 2

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116 Pages

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Revised

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24.41 x 16.99 x 0.61

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This book is based on the lectures given at the Oberwolfach Seminar on Tropical Algebraic Geometry in October 2004. Tropical Geometry ?rst appeared as a subject of its own in 2002, while its roots can be traced back at least to Bergman's work [1] on logarithmic limit sets. Tropical Geometry is now a rapidly developing area of mathematics. It is int- twined with algebraic and symplectic geometry, geometric combinatorics, in- grablesystems, and statistical physics. Tropical Geometry can be viewed as a sort of algebraic geometry with the underlying algebra based on the so-called tropical numbers. The tropicalnumbers (the term "tropical" comesfrom computer science and commemorates Brazil, in particular a contribution of the Brazilian school to the language recognition problem) are the real numbers enhanced with negative in?nity and equipped with two arithmetic operations called tropical addition and tropical multiplication. The tropical addition is the operation of taking the m- imum. The tropical multiplication is the conventional addition. These operations are commutative, associative and satisfy the distribution law. It turns out that such tropical algebra describes some meaningful geometric objects, namely, the Tropical Varieties. From the topological point of view the tropical varieties are piecewise-linearpolyhedral complexes equipped with a particular geometric str- ture coming from tropical algebra. From the point of view of complex geometry this geometric structure is the worst possible degeneration of complex structure on a manifold.

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Non-Fiction Mathematics Geometry Algebraic Geometry

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Non-Fiction Mathematics Geometry Algebraic Geometry

Preface	p. vii
Preface to the second edition	p. ix
Introduction to tropical geometry	p. 1
Images under the logarithm	p. 1
Families of amoebas	p. 4
Non-Archimedean amoebas	p. 5
Non-standard complex numbers	p. 7
The tropical semifield T	p. 9
Tropical curves and integer affine structure	p. 11
Exercises	p. 14
Patchworking of algebraic varieties	p. 17
Introduction: A general idea of the patchworking construction	p. 17
Elements of toric geometry	p. 19
Construction of toric varieties	p. 19
A toric variety associated with a fan	p. 20
A toric variety associated with a convex lattice polyhedron	p. 21
Embedding of Tor(¿) into a projective space	p. 22
The real part of a toric variety and the moment map	p. 22
Hypersurfaces in toric varieties	p. 24
Viro's patchworking method	p. 25
Chart of a real polynomial	p. 25
Patchworking of real nonsingular hypersurfaces	p. 27
Combinatorial patchworking	p. 30
A tropical point of view on the combinatorial Viro patch-working	p. 34
Patchworking of pseudo-homomorphic curves on ruled surfaces	p. 37
Patchworking of singular algebraic hypersurfaces	p. 45
Initial data	p. 46
Transversality conditions	p. 46
The patchworking theorem	p. 47
Some S-transversality criteria	p. 50
Tropicalization and patchworking in the enumeration of nodal curves	p. 51
Plane tropical curves	p. 52
Algebraic enumerative problem and its tropical analogue	p. 54
Tropical formulas for the Gromov-Witten and Welschinger invariants	p. 56
Tropical limit	p. 56
Tropicalization of nodal curves	p. 57
Reconstruction of a simple tropical curve	p. 61
Reconstruction of the limit curve C(0)	p. 63
Refinement of a tropical limit	p. 64
Refinement of the condition to pass through a fixed point	p. 67
Refined patchworking theorem	p. 69
The real case: Welschinger invariants	p. 73
Exercises	p. 74
Applications of tropical geometry to enumerative geometry	p. 77
Introduction	p. 77
Tropical hypersurfaces in Rn	p. 78
Geometric description of plane tropical curves	p. 81
Count of complex nodal curves	p. 83
Correspondence theorem	p. 84
Mikhalkin's algorithm	p. 85
Welschinger invariants	p. 86
Welschinger invariants Wm for small m	p. 88
Tropical calculation of Welschinger invariants	p. 89
Asymptotic enumeration of real rational curves	p. 90
Recurrence formula for Welschinger invariants	p. 92
Welschinger invariants Wm,i	p. 93
Exercises	p. 95
List of Figures	p. 97
Bibliography	p. 99
Table of Contents provided by Ingram. All Rights Reserved.

Tropical Algebraic Geometry

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