🌴Tropical Geometry Unit 6 – Tropical intersection theory

What's Tropical Intersection Theory?

• Subfield of tropical geometry focusing on intersections of tropical curves and hypersurfaces
• Studies properties and structures arising from intersections in the tropical setting
• Extends classical algebraic geometry concepts to the tropical world
• Utilizes the min-plus algebra ($a \oplus b = min(a, b), a \odot b = a + b$) as the underlying algebraic structure
• Provides a combinatorial approach to studying intersections of geometric objects
• Offers new insights and techniques for solving problems in algebraic geometry and combinatorics
• Connects to other areas of mathematics such as toric geometry, combinatorics, and phylogenetics

Key Concepts and Definitions

• Tropical semiring consists of the real numbers $\mathbb{R}$ with the min-plus algebra operations
• Tropical polynomial $f(x) = \bigoplus_{i=1}^n a_i \odot x^{\odot i} = min(a_1 + i_1x_1, \ldots, a_n + i_nx_n)$ where $a_i \in \mathbb{R}$ and $i = (i_1, \ldots, i_n) \in \mathbb{N}^n$
• Tropical hypersurface defined by a tropical polynomial $f$ is the set of points $x$ where $f$ is not differentiable
• Tropical variety is the intersection of finitely many tropical hypersurfaces
• Dual subdivision of a tropical hypersurface is a polyhedral complex induced by the polynomial coefficients
  • Each cell corresponds to a monomial attaining the minimum value
• Stable intersection occurs when the intersection is transversal and remains unchanged under small perturbations
• Tropical cycle is a weighted sum of tropical varieties satisfying the balancing condition
• Tropical intersection product of two tropical cycles $A$ and $B$ is a new tropical cycle $A \cdot B$ representing their stable intersection

Tropical Varieties and Their Properties

• Tropical varieties are piecewise linear objects defined by tropical polynomials
• Can be represented as the intersection of finitely many tropical hypersurfaces
• Have a polyhedral structure determined by the dual subdivisions of the defining polynomials
• Satisfy the balancing condition each facet is weighted, and the sum of weights around each codimension-1 face is zero
• Exhibit a rich combinatorial structure captured by the associated tropical cycles
• Admit a fan structure, which encodes the asymptotic behavior of the variety
• Can be studied using techniques from polyhedral geometry and combinatorics
• Arise naturally in the study of amoebas and non-Archimedean analytic spaces

Intersection Multiplicity in the Tropical Setting

• Tropical intersection multiplicity measures the complexity of the intersection of two tropical varieties
• Defined as the sum of the weights of the facets in the stable intersection
• Captures the number of ways the varieties intersect while accounting for multiplicities
• Can be computed using the dual subdivisions of the defining polynomials
  • Each cell in the common refinement contributes to the intersection multiplicity
• Satisfies properties analogous to the classical intersection multiplicity
  • Invariant under tropical rational equivalence
  • Respects the balancing condition
• Provides a tool for studying the enumerative geometry of tropical curves and hypersurfaces
• Connects to the classical intersection theory through the process of tropicalization

Tropical Bezout's Theorem

• Generalizes the classical Bezout's theorem to the tropical setting
• States that the degree of the intersection of two tropical hypersurfaces is bounded by the product of their degrees
  • Degree of a tropical hypersurface is the maximum of the sum of coordinates of its vertex
• Provides an upper bound for the number of intersection points counted with multiplicities
• Equality holds when the hypersurfaces intersect transversally and the intersection is stable
• Can be proved using the dual subdivisions and the balancing condition
• Has applications in the study of enumerative problems and the complexity of tropical varieties
• Extends to higher-dimensional tropical varieties with appropriate modifications

Applications in Enumerative Geometry

• Tropical intersection theory provides a powerful tool for solving enumerative problems
• Allows counting the number of geometric objects satisfying certain conditions
• Enumerative problems can be translated into the tropical setting using tropicalization
  • Solutions correspond to the intersection points of tropical varieties
• Tropical Bezout's theorem gives bounds on the number of solutions
• Techniques such as lifting and projecting can be used to extract solutions
• Has been applied to various problems in algebraic geometry and combinatorics
  • Counting curves on toric surfaces (Mikhalkin's correspondence theorem)
  • Enumerating rational curves in projective spaces
  • Studying Gromov-Witten invariants and Hurwitz numbers
• Offers a combinatorial approach to classical enumerative questions

Computational Techniques and Tools

• Tropical intersection theory can be studied using computational methods
• Gröbner bases techniques can be adapted to the tropical setting
  • Compute tropical varieties and their intersections
  • Determine the dual subdivisions and intersection multiplicities
• Software packages and libraries are available for tropical computations
  • Gfan for computing Gröbner fans and tropical varieties
  • Polymake for polyhedral geometry and tropical hypersurfaces
  • Singular for symbolic computations in algebraic geometry
• Combinatorial algorithms can be used to study tropical varieties and their properties
  • Computing the Newton polygon and the dual subdivision
  • Determining the weights and balancing condition
• Visualization tools help in understanding the geometric structure of tropical varieties
• Computational methods are essential for exploring large-scale examples and applications

Challenges and Open Problems

• Developing a comprehensive intersection theory for higher codimension tropical cycles
• Extending tropical intersection theory to non-constant coefficient fields
• Investigating the relationship between tropical and non-Archimedean intersection theory
• Studying the intersection theory of tropical varieties over fields with valuations
• Exploring connections between tropical intersection theory and other areas of mathematics
  • Mirror symmetry and Hodge theory
  • Berkovich spaces and non-Archimedean geometry
  • Matroid theory and combinatorial optimization
• Applying tropical intersection theory to problems in physics and other sciences
• Developing efficient algorithms and software for computing tropical intersections in high dimensions
• Investigating the role of tropical intersection theory in the study of limit linear series and moduli spaces