Brief introduction to tropical geometry

Abstract

The paper consists of lecture notes for a mini-course given by the authors at the Gökova Geometry \& Topology conference in May 2014. We start the exposition with tropical curves in the plane and their applications to problems in classical enumerative geometry, and continue with a look at more general tropical varieties and their homology theories.

75 pages, 37 figures, many examples and exercises

References in corpus (9)

• Tropical Geometry and its applications
• Moduli spaces of rational tropical curves
• Refined curve counting with tropical geometry
• Floor decompositions of tropical curves : the planar case
• Logarithmic equivalence of Welschinger and Gromov-Witten invariants
• Correspondence theorems for tropical curves I
• Lifting representations of finite reductive groups I: Semisimple conjugacy classes
• The Orlik-Solomon Algebra and the Bergman Fan of a Matroid
• Severi degrees on toric surfaces

Cited by in corpus (18)

• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Tropical curves in sandpiles
• Lefschetz section theorems for tropical hypersurfaces
• Tropicalized quartics and canonical embeddings for tropical curves of genus 3
• Poincaré duality for tropical Dolbeault cohomology of non-archimedean Mumford curves
• Tropical approach to Nagata's conjecture in positive characteristic
• Rational quintics in the real plane
• Tropical geometry of genus two curves
• On the Severi problem in arbitrary characteristic
• Euler characteristic and signature of real semi-stable degenerations
• The Newton polygon of a planar singular curve and its subdivision
• Haas' theorem revisited
• A guide to tropical modifications
• Some statistics about Tropical Sandpile Model
• Tropical $\psi$ classes
• The tropical discriminant of a polynomial map on a plane
• Tropical Weil's reciprocity law and Weil's pairing
• Mirror symmetry for tropical hypersurfaces and patchworking