Brief introduction to tropical geometry (1502.05950v1)

Abstract: The paper consists of lecture notes for a mini-course given by the authors at the G\"okova 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.

• The paper outlines tropical geometry fundamentals using max-plus arithmetic to connect classical and tropical algebraic structures.
• It demonstrates how tropical techniques, like combinatorial patchworking, effectively construct real algebraic curves for enumerative applications.
• The study extends concepts to higher dimensions, integrating tropical homology and modifications to deepen insights into geometric structures.

Overview of "Brief Introduction to Tropical Geometry"

The paper "Brief Introduction to Tropical Geometry" serves as a foundational exposition aimed at introducing tropical geometry, emphasizing its visual and algebraic simplicity. The discourse is structured as an incremental build-up from basic concepts in tropical arithmetic to more intricate topics like tropical manifolds and tropical homology.

Tropical Arithmetic and Tropical Curves

The authors begin with the fundamentals of tropical arithmetic, where operations are defined using max-plus algebra. This section also introduces the tropical semi-field and explores its properties, including idempotency. Understanding these operations is critical as they underpin much of tropical geometry, enabling the transition from classical to tropical varieties.

Tropical curves, initially explored in the tropical plane $\R^2$, are defined using tropical polynomials. These curves are visualized as piecewise linear graphs, emphasizing the graphical representation of algebraic curves. The paper explores balancing conditions for tropical curves, drawing parallels to classical algebraic geometry via dual subdivisions and the Newton polygon.

Tropical Geometry's Enumerative Applications

A significant portion of the paper is devoted to demonstrating the applicability of tropical geometry in enumerative geometry. It introduces combinatorial patchworking, illustrating how tropical techniques can construct real algebraic curves. This method requires transforming tropical curves into real algebraic curves within toric varieties, linking tropical and classical domains.

Extensions to Higher Dimensions

The exposition extends to tropical subvarieties of $\R^n$, generalizing the concepts from curves to hypersurfaces and beyond. The paper’s methodical approach introduces tropical hypersurfaces via algebraic, combinatorial, and geometric perspectives, ensuring a robust understanding. The balancing condition is reiterated and generalized for higher-dimensional tropical structures.

Tropical Modifications and Homology

Tropical modifications are presented as a significant tool in adjusting tropical varieties, equivalent to subdivisions or refinements in polyhedral complexes. The inclusion of tropical homology and cohomology gives an additional layer of theoretical depth, drawing parallels to familiar algebraic topology concepts. This is crucial for understanding the equivalences and distinctions between tropical and classical settings, particularly in how they handle intersection theory.

Implications and Future Directions

The theoretical foundations laid in this paper have broad implications for mathematical research. By substituting classical methods with tropical approaches, there are potential advancements in solving complex enumerative problems and exploring connections between algebraic varieties. The interplay between tropical mathematics and its classical counterpart could spur novel insights into both fields, especially in areas like mirror symmetry and moduli spaces.

From a practical standpoint, methodologies like patchworking have potential applications in computational geometry and optimization, where tropical methods simplify complex problems. As tropical geometry evolves, these foundations will be instrumental in forging new paths in mathematical research, fostering a deeper understanding of geometric structures.

In conclusion, the paper is a meticulously crafted introduction that prepares researchers to engage deeply with tropical geometry, equipping them with the necessary tools and insights to advance the field further.