[Paper Review] Jacobian varieties of reduced tropical curves

Shuhei Yoshitomi|ArXiv.org|Dec 28, 2006

Polynomial and algebraic computation5 references3 citations

TL;DR

This paper establishes that for a reduced tropical curve in ℝ² whose bunch (a topological quotient collapsing tentacle edges and rays) forms a bouquet of genus g, the Jacobian variety is isomorphic to the g-fold product of the bouquet's cycles. The key result is a complete proof of the bijectivity of the map φ: Λ₁×⋯×Λg → Jac(C) defined by (P₁,…,Pg) ↦ P₁+⋯+Pg−g𝒪, resolving an incomplete argument in earlier work and extending the group structure of tropical elliptic curves to higher-genus cases.

ABSTRACT

On tropical geometry in $ r^2$, the divisor and the Jacobian variety are defined in analogy to algebraic geometry. For study of these objects, it is important to think of the `bunch' of a tropical curve. In this paper, we will show that if the bunch is a bouquet, then the Jacobian is a higher-dimensional torus.

Motivation & Objective

• To complete the proof of Vigeland's conjecture that the map φ: Bunch(C) → Jac(C) is bijective for tropical elliptic curves.
• To generalize the group structure of tropical elliptic curves to higher-genus tropical curves.
• To establish a precise characterization of the Jacobian variety of a reduced tropical curve in ℝ² when its bunch is a bouquet.
• To clarify the relationship between the algebraic and geometric definitions of linear equivalence and Jacobian varieties in tropical geometry.

Proposed method

• Define the Jacobian variety Jac(C) as the quotient group Div⁰(C)/∼, where ∼ denotes linear equivalence via stable intersection with tropical curves.
• Introduce the 'bunch' of a tropical curve C as the quotient space obtained by collapsing all tentacle edges and rays to points.
• Define a bouquet as a wedge sum of g circles sharing a common basepoint, with genus g equal to the number of cycles.
• Construct a map φ: Λ₁×⋯×Λg → Jac(C) by sending (P₁,…,Pg) to P₁+⋯+Pg−g𝒪, and prove it is well-defined and bijective.
• Use local parameterization and continuity of stable intersection to show that the intersection number map σ(L) = λ(C·L) is locally constant across continuous families of tropical curves.
• Apply tropical Bézout’s theorem and moment conditions to prove injectivity by showing that equal intersection counts force point-wise equality of corresponding points on cycles.

Experimental results

Research questions

• RQ1 Under what topological conditions on the bunch of a tropical curve is its Jacobian variety isomorphic to a higher-dimensional torus?
• RQ2 Is the map φ: Λ₁×⋯×Λg → Jac(C) defined by (P₁,…,Pg) ↦ P₁+⋯+Pg−g𝒪 bijective for reduced tropical curves with bouquet bunches?
• RQ3 How does the algebraic Jacobian variety relate to the geometric Jacobian variety in the context of tropical curves?
• RQ4 Can the incomplete proof of Vigeland’s bijectivity claim for genus-1 tropical curves be completed and generalized to higher genus?
• RQ5 What topological and combinatorial constraints ensure that two divisors on a tropical curve are linearly equivalent?

Key findings

• The map φ: Λ₁×⋯×Λg → Jac(C) is bijective when the bunch of a reduced tropical curve C is a bouquet of genus g.
• The Jacobian variety Jac(C) of such a curve is isomorphic to the g-dimensional real torus ℝ^g/ℤ^g, i.e., a higher-dimensional torus.
• The map φ is well-defined: points on the same tentacle edge or ray are linearly equivalent, so the map respects the quotient structure of the bunch.
• The intersection number map σ(L) = λ(C·L) is locally constant over continuous families of tropical curves with fixed Newton complex.
• Injectivity of φ is proven using the tropical Bézout theorem and moment conditions, showing that equal intersection counts across cycles imply pointwise equality of divisor supports.
• The parameter space 𝒫(𝒩,ℝ²) of tropical curves with fixed Newton complex 𝒩 is a connected, relatively open convex cone in ℝ^{l+2}, implying that curves with the same Newton polygon lie in the same connected component.