Tropical Algebraic Geometry

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.

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
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