2.5 Tropical genus and Riemann-Roch theorem

Tropical genus and the Riemann-Roch theorem are key concepts in tropical geometry. They measure curve complexity and relate divisor ranks to degrees and curve genus, providing powerful tools for understanding tropical curve geometry.

These concepts bridge classical and tropical geometry, enabling the study of linear systems, specialization from algebraic curves, and tropical versions of important theories. They highlight the combinatorial nature of tropical geometry while maintaining connections to classical algebraic geometry.

Tropical genus

Definition of tropical genus

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• The tropical genus of a tropical curve is a non-negative integer that measures the complexity of the curve
• Defined as the first Betti number of the underlying graph of the tropical curve
• Can be calculated by counting the number of independent cycles in the graph
• Intuitively captures the number of "holes" or "loops" in the tropical curve

Relationship to first Betti number

• The tropical genus is equal to the first Betti number of the graph underlying the tropical curve
• First Betti number counts the number of independent cycles in a graph
• For a connected graph with $e$ edges and $v$ vertices, the first Betti number is given by $e - v + 1$
• Higher tropical genus indicates more complex structure and topology of the tropical curve

Examples of tropical curves and genus

• A tropical line has genus 0, as it has no cycles and is topologically equivalent to a tree
• A tropical elliptic curve has genus 1, with a single cycle forming a loop
• Tropical curves of higher genus can be constructed by gluing together edges to create multiple cycles
  • For example, a genus 2 tropical curve can be obtained by joining two cycles at a common vertex
• The genus of a disconnected tropical curve is the sum of the genera of its connected components

Riemann-Roch theorem for tropical curves

Statement of tropical Riemann-Roch theorem

• The tropical Riemann-Roch theorem relates the rank of a divisor on a tropical curve to its degree and the genus of the curve
• For a divisor $D$ on a tropical curve $\Gamma$ of genus $g$, the theorem states: $r(D) - r(K - D) = \deg(D) - g + 1$
  • $r(D)$ is the rank of the divisor $D$
  • $K$ is the canonical divisor on $\Gamma$
• Provides a powerful tool for understanding the geometry of tropical curves and divisors

Divisors on tropical curves

• A divisor on a tropical curve is a formal sum of points on the curve with integer coefficients
• Can be thought of as a way to assign integer "weights" to points on the tropical curve
• The degree of a divisor is the sum of its coefficients
• Divisors capture important geometric and combinatorial information about the tropical curve

Rank of divisors

• The rank of a divisor $D$ on a tropical curve $\Gamma$, denoted $r(D)$, is a non-negative integer
• Measures the dimension of the space of rational functions on $\Gamma$ that are bounded above by $D$
• Can be computed using the tropical Riemann-Roch theorem
• Higher rank divisors correspond to larger spaces of functions and more intricate geometry

Canonical divisor

• The canonical divisor $K$ on a tropical curve $\Gamma$ is a special divisor of degree $2g-2$, where $g$ is the genus of $\Gamma$
• Plays a crucial role in the tropical Riemann-Roch theorem
• Can be explicitly constructed using the vertex weights and edge lengths of the tropical curve
• Captures important intrinsic geometric properties of the tropical curve

Proof of tropical Riemann-Roch theorem

• The proof of the tropical Riemann-Roch theorem relies on a careful analysis of the combinatorics of the tropical curve
• Involves studying the behavior of rational functions and their divisors on the edges and vertices of the curve
• Key steps include:
  • Establishing a local Riemann-Roch formula for each edge of the tropical curve
  • Gluing the local contributions together using the topology of the curve
• Ultimately reduces to a combinatorial statement about the genus and divisor degrees

Applications of tropical Riemann-Roch

Computing dimensions of linear systems

• The tropical Riemann-Roch theorem allows for the computation of dimensions of linear systems on tropical curves
• A linear system is a space of divisors on the curve satisfying certain constraints
• The rank of divisors, determined by tropical Riemann-Roch, gives the dimension of the corresponding linear system
• Enables the study of maps between tropical curves and their geometry

Specialization from algebraic curves

• Tropical curves can be obtained as limits of algebraic curves over valued fields
• The tropical Riemann-Roch theorem is compatible with this specialization process
• Allows for the transfer of geometric information from algebraic curves to their tropical counterparts
• Provides a bridge between classical algebraic geometry and tropical geometry

Tropical Brill-Noether theory

• Brill-Noether theory studies the geometry of linear systems on algebraic curves
• The tropical Riemann-Roch theorem forms the foundation for a tropical analog of Brill-Noether theory
• Investigates the existence and properties of divisors of prescribed rank and degree on tropical curves
• Leads to interesting combinatorial and geometric questions in the tropical setting

Tropical Abel-Jacobi maps

• The Abel-Jacobi map is a fundamental object in the study of algebraic curves
• Tropical Riemann-Roch allows for the construction of tropical Abel-Jacobi maps
• These maps relate divisors on a tropical curve to points on its Jacobian variety
• Provides a way to study the geometry of tropical curves using their Jacobians

Tropical Jacobians and Jacobi inversion

• The Jacobian of a tropical curve is a tropical torus that parametrizes divisor classes on the curve
• Tropical Riemann-Roch plays a key role in understanding the structure of tropical Jacobians
• Jacobi inversion problem asks for the preimage of a point under the Abel-Jacobi map
• Can be approached using the tropical Riemann-Roch theorem and combinatorial techniques
• Relates to the study of integrable systems and soliton equations in the tropical setting

Comparisons to classical Riemann-Roch

Analogies between tropical and algebraic curves

• Tropical curves share many analogies with algebraic curves, despite their different nature
• Both have a notion of genus, divisors, and a Riemann-Roch theorem
• Tropical curves can be thought of as "skeletons" or "degenerations" of algebraic curves
• Many geometric properties and theorems have tropical counterparts

Key differences in tropical case

• Tropical curves are piecewise linear objects, while algebraic curves are defined by polynomial equations
• The tropical Riemann-Roch theorem has a more combinatorial flavor compared to the classical version
• Divisors on tropical curves are formal sums of points, rather than line bundles
• The proof of tropical Riemann-Roch relies on graph theory and discrete geometry

Limits of tropical Riemann-Roch

• While the tropical Riemann-Roch theorem captures many aspects of the classical theory, it has some limitations
• Not all properties of algebraic curves and their divisors can be directly translated to the tropical setting
• Some geometric phenomena, such as inflection points or higher-order contact, are not easily visible in the tropical world
• The tropical theory is most effective for studying certain aspects of algebraic curves, such as their degenerations and combinatorial properties
• Further research continues to explore the connections and differences between tropical and classical Riemann-Roch, and their implications for algebraic and tropical geometry