8.3 Tropical Deligne-Mumford compactification

Tropical Deligne-Mumford compactification is a powerful tool in tropical geometry. It provides a combinatorial approach to studying moduli spaces of curves, offering a simpler structure compared to classical counterparts while retaining key geometric properties.

This compactification, denoted as $\overline{M}_{g,n}^{trop}$, is a polyhedral complex that parametrizes stable tropical curves. It has applications in intersection theory, Gromov-Witten invariants, and mirror symmetry, bridging tropical and classical algebraic geometry in meaningful ways.

Tropical moduli spaces

• Tropical moduli spaces are parameter spaces that classify tropical curves up to certain equivalence relations
• They provide a combinatorial approach to studying moduli problems in algebraic geometry
• Tropical moduli spaces often have a simpler structure compared to their classical counterparts, making them more amenable to explicit computations

Moduli of tropical curves

Top images from around the web for Moduli of tropical curves

• 

  genus 1 graphs View original

  Is this image relevant?

• 

  Frontiers | Pattern Formation and Tropical Geometry View original

  Is this image relevant?

• 

  Moduli of algebraic curves - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  genus 1 graphs View original

  Is this image relevant?

• 

  Frontiers | Pattern Formation and Tropical Geometry View original

  Is this image relevant?

1 of 3

Top images from around the web for Moduli of tropical curves

• 

  genus 1 graphs View original

  Is this image relevant?

• 

  Frontiers | Pattern Formation and Tropical Geometry View original

  Is this image relevant?

• 

  Moduli of algebraic curves - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  genus 1 graphs View original

  Is this image relevant?

• 

  Frontiers | Pattern Formation and Tropical Geometry View original

  Is this image relevant?

1 of 3

• The moduli space of tropical curves $M_{g,n}^{trop}$ parametrizes metric graphs of genus $g$ with $n$ marked points
• Each point in $M_{g,n}^{trop}$ represents a tropical curve, which is a metric graph with possibly unbounded edges (leaves)
• The space $M_{g,n}^{trop}$ has a natural stratification based on the combinatorial types of the underlying graphs
• The top-dimensional stratum corresponds to trivalent graphs, while lower-dimensional strata represent graphs with higher valence vertices

Tropical Deligne-Mumford compactification vs classical

• The tropical Deligne-Mumford compactification $\overline{M}_{g,n}^{trop}$ is an analogue of the classical Deligne-Mumford compactification $\overline{M}_{g,n}$ in algebraic geometry
• While $\overline{M}_{g,n}$ is a complex projective variety that compactifies the moduli space of smooth algebraic curves, $\overline{M}_{g,n}^{trop}$ is a polyhedral complex that compactifies $M_{g,n}^{trop}$
• The boundary strata in $\overline{M}_{g,n}$ correspond to stable nodal curves, while the boundary strata in $\overline{M}_{g,n}^{trop}$ correspond to stable tropical curves with vertices of higher valence
• There are many similarities between the combinatorial structure of $\overline{M}_{g,n}^{trop}$ and the stratification of $\overline{M}_{g,n}$, which can be exploited to study the geometry of $\overline{M}_{g,n}$ using tropical techniques

Cones in tropical moduli spaces

• The tropical moduli spaces $M_{g,n}^{trop}$ and $\overline{M}_{g,n}^{trop}$ have a natural decomposition into cones
• Each cone corresponds to a combinatorial type of tropical curve, and its dimension is determined by the number of bounded edges in the underlying graph
• The cones are glued together along their faces, which represent tropical curves obtained by contracting edges
• The cone complex structure of tropical moduli spaces is essential for studying their geometry and intersection theory

Constructions of tropical Deligne-Mumford compactification

• There are several equivalent constructions of the tropical Deligne-Mumford compactification $\overline{M}_{g,n}^{trop}$, each highlighting different aspects of its structure
• These constructions provide a combinatorial framework for studying the geometry of $\overline{M}_{g,n}^{trop}$ and its relationship to classical moduli spaces

Moduli of stable tropical curves

• One construction of $\overline{M}_{g,n}^{trop}$ is as the moduli space of stable tropical curves
• A stable tropical curve is a metric graph with marked points that satisfies certain stability conditions (e.g., each vertex of valence 1 must have a marked point)
• The space of stable tropical curves is naturally compactified by allowing edge lengths to become infinite, leading to the tropical Deligne-Mumford compactification

Tropical Hassett spaces

• Another construction of $\overline{M}_{g,n}^{trop}$ is via tropical Hassett spaces, which are moduli spaces of weighted tropical curves
• In this setting, each marked point is assigned a weight, and the stability condition is modified accordingly (e.g., the sum of weights at each vertex must exceed a certain threshold)
• By varying the weight parameters, one obtains a family of tropical moduli spaces that interpolate between $M_{g,n}^{trop}$ and $\overline{M}_{g,n}^{trop}$

Tropical forgetful maps

• The tropical Deligne-Mumford compactification admits natural forgetful maps $\pi_i: \overline{M}_{g,n+1}^{trop} \rightarrow \overline{M}_{g,n}^{trop}$ that forget the i-th marked point
• These forgetful maps are morphisms of polyhedral complexes and are compatible with the cone complex structure
• The fibers of the forgetful maps are tropical analogues of the universal curve over $\overline{M}_{g,n}$

Tropical gluing maps

• Another key feature of $\overline{M}_{g,n}^{trop}$ is the existence of gluing maps, which allow one to construct new tropical curves by gluing together existing ones
• The gluing maps are defined by identifying marked points on different tropical curves and adjusting the edge lengths accordingly
• These maps play a crucial role in the study of the boundary strata of $\overline{M}_{g,n}^{trop}$ and the intersection theory on tropical moduli spaces

Properties of tropical Deligne-Mumford compactification

• The tropical Deligne-Mumford compactification $\overline{M}_{g,n}^{trop}$ has several remarkable properties that make it a powerful tool for studying the geometry of moduli spaces
• These properties are often analogous to those of the classical Deligne-Mumford compactification, but with a more combinatorial flavor

Tropical stability conditions

• The stability conditions for tropical curves are combinatorial analogues of the stability conditions for algebraic curves
• A tropical curve is stable if it has no automorphisms (i.e., no nontrivial isometries that fix the marked points)
• This condition translates into combinatorial restrictions on the underlying graph (e.g., each vertex must have valence at least 3 or contain a marked point)
• The tropical stability conditions ensure that $\overline{M}_{g,n}^{trop}$ is a well-behaved compactification of $M_{g,n}^{trop}$

Boundary stratification

• The boundary $\overline{M}_{g,n}^{trop} \setminus M_{g,n}^{trop}$ has a natural stratification based on the combinatorial types of the stable tropical curves
• Each stratum corresponds to a cone in the cone complex structure of $\overline{M}_{g,n}^{trop}$, and its codimension is determined by the number of edges that need to be contracted to obtain the corresponding combinatorial type
• The boundary strata can be described in terms of products of lower-dimensional moduli spaces, reflecting the decomposition of stable tropical curves into simpler components

Tropical Psi-classes

• Tropical Psi-classes are analogues of the Psi-classes in the cohomology of the classical moduli space $\overline{M}_{g,n}$
• They are defined as the pullbacks of certain divisors on $\overline{M}_{g,n}^{trop}$ under the forgetful maps $\pi_i$
• Tropical Psi-classes have a simple combinatorial description in terms of the valences of the vertices adjacent to the marked points
• They play a key role in the intersection theory on tropical moduli spaces and the computation of tropical Gromov-Witten invariants

Tropical Picard group

• The tropical Picard group $Pic(\overline{M}_{g,n}^{trop})$ is the group of divisors on $\overline{M}_{g,n}^{trop}$ modulo linear equivalence
• It is a finitely generated abelian group that captures important information about the geometry of $\overline{M}_{g,n}^{trop}$
• The generators of $Pic(\overline{M}_{g,n}^{trop})$ can be described in terms of tropical Psi-classes and boundary divisors
• The structure of the tropical Picard group is closely related to the intersection theory on $\overline{M}_{g,n}^{trop}$ and the study of tropical moduli spaces as toric varieties

Applications of tropical Deligne-Mumford compactification

• The tropical Deligne-Mumford compactification $\overline{M}_{g,n}^{trop}$ has numerous applications in tropical geometry, algebraic geometry, and mathematical physics
• Its combinatorial structure and close relationship with classical moduli spaces make it a powerful tool for solving problems and gaining new insights

Intersection theory on tropical moduli spaces

• The intersection theory on $\overline{M}_{g,n}^{trop}$ is a tropical analogue of the intersection theory on the classical moduli space $\overline{M}_{g,n}$
• It involves the study of intersections of tropical cycles, which are weighted polyhedral complexes that can be thought of as tropical analogues of algebraic cycles
• Many key results from the classical intersection theory, such as the string equation and the dilaton equation, have tropical counterparts that can be proven using the combinatorial structure of $\overline{M}_{g,n}^{trop}$

Tropical Gromov-Witten invariants

• Tropical Gromov-Witten invariants are tropical analogues of the classical Gromov-Witten invariants, which count algebraic curves satisfying certain incidence conditions
• They are defined as intersection numbers of tropical Psi-classes and pull-backs of evaluation morphisms on $\overline{M}_{g,n}^{trop}$
• Tropical Gromov-Witten invariants can be computed using combinatorial techniques, such as the tropical correspondence theorem and the tropical WDVV equations
• They provide a new perspective on the enumerative geometry of algebraic curves and have applications in mirror symmetry and mathematical physics

Correspondence theorems

• Correspondence theorems relate tropical and classical enumerative invariants, such as Gromov-Witten invariants and Hurwitz numbers
• They state that, under certain conditions, the tropical invariants coincide with the classical ones, providing a powerful tool for computing enumerative invariants using tropical geometry
• The proofs of correspondence theorems often rely on the combinatorial structure of $\overline{M}_{g,n}^{trop}$ and its relationship with the classical moduli space $\overline{M}_{g,n}$
• Examples include the Mikhalkin correspondence theorem for plane curves and the Gross-Siebert correspondence theorem for toric degenerations

Tropical mirror symmetry

• Mirror symmetry is a profound duality between symplectic geometry and complex geometry that has far-reaching implications in mathematics and physics
• Tropical mirror symmetry is a version of mirror symmetry that relates tropical geometry to complex geometry
• The tropical Deligne-Mumford compactification $\overline{M}_{g,n}^{trop}$ plays a key role in the study of tropical mirror symmetry, as it provides a natural setting for defining tropical enumerative invariants
• Tropical mirror symmetry has led to new insights into the structure of Gromov-Witten invariants, the geometry of Calabi-Yau manifolds, and the physics of string theory