🌴Tropical Geometry Unit 8 – Tropical moduli spaces

Key Concepts and Definitions

• Tropical geometry studies geometric objects defined by piecewise linear functions and polyhedral complexes
• Tropical arithmetic operations replace addition with minimum and multiplication with addition
• Tropical polynomials are piecewise linear functions obtained by replacing arithmetic operations in classical polynomials with their tropical counterparts
• Tropical varieties are the solution sets of systems of tropical polynomial equations
  • Consist of polyhedral complexes and exhibit a rich combinatorial structure
• Moduli spaces parameterize geometric objects up to equivalence and capture their essential properties
• Tropical moduli spaces parameterize tropical curves and encode their combinatorial and geometric data
• Berkovich analytification provides a connection between classical and tropical geometry by associating a tropical variety to an algebraic variety

Foundations of Tropical Geometry

• Tropical semiring $(\mathbb{T}, \oplus, \odot)$ is the basis for tropical arithmetic
  • $\mathbb{T} = \mathbb{R} \cup \{\infty\}$
  • $a \oplus b = \min(a, b)$
  • $a \odot b = a + b$
• Tropical hypersurfaces are the tropical analogs of classical algebraic hypersurfaces
  • Defined as the corner locus of a tropical polynomial
  • Dual to a regular subdivision of the Newton polytope of the polynomial
• Tropical convexity plays a crucial role in the study of tropical varieties
  • Tropical line segments are concatenations of ordinary line segments with slopes in $\{0, \infty\}$
• Tropical intersection theory allows for the computation of intersection multiplicities of tropical varieties
• Tropical Bézout's theorem relates the degrees of intersecting tropical hypersurfaces to the degree of their intersection

Introduction to Moduli Spaces

• Moduli spaces are geometric objects that parameterize classes of geometric objects up to equivalence
• Moduli spaces of curves $\mathcal{M}_g$ parameterize smooth algebraic curves of genus $g$
  • Deligne-Mumford compactification $\overline{\mathcal{M}}_g$ includes stable nodal curves
• Moduli spaces of stable maps $\overline{\mathcal{M}}_{g,n}(X, \beta)$ parameterize stable maps from curves of genus $g$ with $n$ marked points to a target variety $X$ in the class $\beta$
• Tropical moduli spaces are combinatorial analogs of classical moduli spaces
  • Parameterize tropical curves and encode their combinatorial and geometric properties
• Moduli spaces have applications in enumerative geometry, string theory, and mathematical physics

Tropical Curves and Their Properties

• Tropical curves are metric graphs with additional data (genus function, marking)
  • Edges have lengths in $\mathbb{R}_{>0} \cup \{\infty\}$
  • Vertices have a genus assigned by the genus function
• Tropical curves can be obtained as limits of amoebas of algebraic curves under a valuation
• Stable tropical curves are tropical curves satisfying stability conditions
  • Admit a finite automorphism group
  • Contracted edges have lengths in $\mathbb{R}_{>0}$
• Combinatorial types of tropical curves are determined by the underlying graph, genus function, and marking
• Tropical curves exhibit a rich combinatorial structure
  • Can be studied using techniques from graph theory and combinatorics
• Tropical intersection theory on tropical curves allows for the computation of intersection numbers

Construction of Tropical Moduli Spaces

• Tropical moduli spaces $\mathcal{M}_{g,n}^{\text{trop}}$ parameterize stable tropical curves of genus $g$ with $n$ marked points
• Constructed as a geometric realization of the moduli space of weighted metric graphs
• Tropical moduli spaces are polyhedral complexes
  • Cells correspond to combinatorial types of tropical curves
  • Gluing of cells is determined by edge contractions and expansions
• Tropical forgetful maps $\pi_i: \mathcal{M}_{g,n+1}^{\text{trop}} \to \mathcal{M}_{g,n}^{\text{trop}}$ forget the $i$-th marked point
• Tropical clutching maps glue two tropical curves to obtain a curve with higher genus or more marked points
• Tropical moduli spaces admit a natural stratification by the combinatorial types of tropical curves
• Tropical tautological classes on tropical moduli spaces are analogs of classical tautological classes

Applications in Algebraic Geometry

• Tropical geometry provides new insights and techniques for studying classical algebraic geometry
• Tropical methods can be used to compute Gromov-Witten invariants and Hurwitz numbers
  • Tropical Gromov-Witten invariants count tropical curves in tropical target spaces
  • Tropical Hurwitz numbers count covers of tropical curves with prescribed ramification data
• Tropical moduli spaces can be used to study the geometry of classical moduli spaces
  • Provide a combinatorial framework for understanding the boundary structure and intersection theory
• Tropical compactifications of moduli spaces offer new perspectives on compactifications in algebraic geometry
• Tropical geometry has applications in mirror symmetry and the study of Calabi-Yau manifolds
• Connections between tropical and non-Archimedean geometry provide a bridge between different areas of mathematics

Computational Techniques and Tools

• Tropical computation relies heavily on techniques from polyhedral geometry and combinatorics
• Gröbner bases can be used to compute tropical varieties and study their structure
  • Tropical basis is a generating set for a tropical ideal
• Newton polytopes play a central role in tropical geometry
  • Used to study the combinatorial structure of tropical hypersurfaces
• Polymake is a software system for polyhedral and tropical geometry
  • Provides algorithms for computing tropical varieties, moduli spaces, and other objects
• Sage and Macaulay2 have packages for tropical geometry and tropical computations
• Tropical linear algebra studies matrices and linear equations over tropical semirings
• Tropical convex hull algorithms are used to compute tropical convex hulls and study tropical polytopes

Advanced Topics and Current Research

• Tropical mirror symmetry aims to establish a tropical analog of the mirror symmetry conjecture
  • Relates tropical counts of curves to Gromov-Witten invariants of mirror manifolds
• Tropical Donaldson-Thomas theory studies the enumeration of tropical curves in tropical threefolds
• Tropical Hodge theory investigates the Hodge structures associated with tropical varieties
• Tropical Fock-Goncharov coordinates provide a tropical analog of cluster algebras and cluster varieties
• Tropical integrable systems and tropical soliton equations have been developed
• Tropical geometric representation theory studies representations of algebraic groups and Lie algebras using tropical geometry
• Connections between tropical geometry and mathematical physics, including string theory and quantum field theory, are an active area of research