11.4 Tropical compactifications

Tropical compactifications extend tropical varieties by adding points at infinity, providing a framework to study their global geometry. This process involves embedding tropical varieties into tropical projective space, preserving their structure while allowing for analysis of asymptotic behavior and degenerations.

Compactifications are crucial for understanding the closure of tropical varieties and their intersections. They enable the application of algebraic geometry techniques to tropical settings, offering insights into the geometry of tropical varieties and their moduli spaces.

Tropical projective space

• Tropical projective space is a key concept in tropical geometry that extends the notion of classical projective space to the tropical setting
• It provides a framework for studying tropical varieties and their compactifications
• Understanding tropical projective space is crucial for constructing and analyzing tropical compactifications

Tropical projective torus

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• The tropical projective torus is the tropicalization of the classical projective torus
• It is obtained by taking the quotient of the tropical affine space by the diagonal action of the tropical semiring
• The tropical projective torus serves as the building block for tropical projective space

Tropical homogeneous coordinates

• Tropical homogeneous coordinates are used to represent points in tropical projective space
• They are defined as equivalence classes of tropical affine coordinates under the diagonal action of the tropical semiring
• Tropical homogeneous coordinates allow for the study of tropical varieties in a coordinate-free manner

Tropical projective space vs affine space

• Tropical projective space is a compactification of tropical affine space
• Points at infinity are added to tropical affine space to obtain tropical projective space
• Tropical projective space provides a more complete and geometrically meaningful setting for studying tropical varieties compared to tropical affine space

Construction of tropical compactifications

• Tropical compactifications are obtained by embedding tropical varieties into tropical projective space
• The process of constructing tropical compactifications involves extending the tropical variety to include points at infinity
• Tropical compactifications allow for the study of the asymptotic behavior and degenerations of tropical varieties

Compactifying tropical varieties

• To compactify a tropical variety, it is embedded into a suitable tropical projective space
• The embedding is chosen in such a way that the structure of the tropical variety is preserved
• Compactifying tropical varieties is important for understanding their global geometry and for applying techniques from algebraic geometry

Closure of tropical varieties

• The closure of a tropical variety in tropical projective space is obtained by taking the topological closure of the embedded variety
• The closure operation adds limit points to the tropical variety, making it compact
• Understanding the closure of tropical varieties is crucial for studying their compactifications and intersections

Tropical hypersurfaces in projective space

• Tropical hypersurfaces are a special class of tropical varieties defined by a single tropical polynomial
• In tropical projective space, tropical hypersurfaces are defined by homogeneous tropical polynomials
• Studying tropical hypersurfaces in projective space provides insights into the geometry of tropical varieties and their compactifications

Divisors in tropical geometry

• Divisors play a central role in the study of algebraic curves and their generalizations to higher dimensions
• In tropical geometry, divisors are used to study the geometry of tropical curves and surfaces
• Tropical divisors provide a way to measure the singularities and intersections of tropical varieties

Tropical Cartier divisors

• Tropical Cartier divisors are a tropical analog of classical Cartier divisors
• They are defined as piecewise linear functions on a tropical variety satisfying certain compatibility conditions
• Tropical Cartier divisors capture the local geometry of tropical varieties and are used to define intersection theory on tropical varieties

Tropical Weil divisors

• Tropical Weil divisors are a tropical analog of classical Weil divisors
• They are defined as formal sums of codimension-one subvarieties of a tropical variety
• Tropical Weil divisors provide a global perspective on the geometry of tropical varieties and are related to the topology of the variety

Picard group of tropical varieties

• The Picard group of a tropical variety is the group of isomorphism classes of line bundles on the variety
• In the tropical setting, the Picard group is closely related to the group of tropical Cartier divisors modulo linear equivalence
• Studying the Picard group of tropical varieties provides insights into their geometry and helps classify them

Tropical toric varieties

• Tropical toric varieties are a class of tropical varieties that arise from classical toric varieties
• They are defined using combinatorial data encoded in fans and polytopes
• Tropical toric varieties provide a rich source of examples and serve as a testing ground for general theories in tropical geometry

Toric varieties vs tropical toric varieties

• Classical toric varieties are algebraic varieties defined by combinatorial data (fans and polytopes)
• Tropical toric varieties are obtained by tropicalizing classical toric varieties
• While classical toric varieties are defined over fields, tropical toric varieties are defined over the tropical semiring

Tropical fans and polytopes

• Tropical fans are combinatorial objects that generalize classical fans used in toric geometry
• They encode the combinatorial data needed to define tropical toric varieties
• Tropical polytopes are convex hulls of finite sets of points in tropical affine space and are dual to tropical fans

Compactifications of tropical toric varieties

• Tropical toric varieties can be compactified using techniques similar to those used for general tropical varieties
• The compactifications of tropical toric varieties are often easier to study due to their combinatorial nature
• Understanding the compactifications of tropical toric varieties provides insights into the compactifications of more general tropical varieties

Tropical stable intersection theory

• Tropical stable intersection theory is a framework for defining and computing intersections of tropical varieties
• It extends classical intersection theory to the tropical setting while taking into account the non-transverse nature of tropical intersections
• Tropical stable intersection theory is a powerful tool for studying the geometry of tropical varieties and their moduli spaces

Tropical stable intersections

• Tropical stable intersections are a generalization of classical transverse intersections to the tropical setting
• They are defined using a stability condition that ensures the intersections are well-behaved
• Computing tropical stable intersections involves techniques from polyhedral geometry and combinatorics

Tropical intersection products

• Tropical intersection products are operations that take tropical varieties and produce new tropical varieties representing their intersections
• They are defined using tropical stable intersections and satisfy properties analogous to classical intersection products
• Tropical intersection products allow for the computation of intersection numbers and the study of enumerative problems in tropical geometry

Applications of tropical intersection theory

• Tropical intersection theory has applications in various areas of mathematics, including enumerative geometry, mirror symmetry, and mathematical physics
• It provides a framework for solving problems related to counting curves and their intersections (Gromov-Witten invariants)
• Tropical intersection theory also plays a role in the study of moduli spaces of curves and their compactifications

Tropical moduli spaces

• Tropical moduli spaces are parameter spaces that classify tropical varieties of a given type
• They are tropical analogs of classical moduli spaces and provide a way to study the geometry of families of tropical varieties
• Tropical moduli spaces often have a rich combinatorial structure and are related to important objects in algebraic geometry

Moduli spaces of tropical curves

• The moduli space of tropical curves is a parameter space that classifies tropical curves of a given genus and number of marked points
• It is a tropical analog of the classical moduli space of algebraic curves
• The moduli space of tropical curves has a natural structure as a tropical variety and is closely related to the moduli space of graphs

Compactifications of tropical moduli spaces

• Tropical moduli spaces can be compactified to include degenerate tropical varieties
• The compactifications of tropical moduli spaces are often easier to construct and study than their classical counterparts
• Compactifications of tropical moduli spaces provide a way to study the asymptotic behavior of families of tropical varieties

Tropical Deligne-Mumford compactification

• The tropical Deligne-Mumford compactification is a specific compactification of the moduli space of tropical curves
• It is a tropical analog of the classical Deligne-Mumford compactification of the moduli space of stable curves
• The tropical Deligne-Mumford compactification has a rich combinatorial structure and is related to the theory of tropical intersection products