A tropical version of Martens' theorem for metric graphs

We study the conjecture stated by Jensen and Len on a tropical version on Martens’ theorem via the Brill–Noether rank of a tropical curve. We recall Coppens’ counterexample of Martens-special chain of cycles, and we generalize the construction defining another class of graphs, Martens-special trees of cycles, for which the conjecture does not hold in a similar setting. These are not the only counterexamples. However, we prove that the conjecture holds for all metric graphs with a stricter assumption on the degree in the Brill–Noether rank.

💡 Research Summary

This paper conducts a thorough investigation into a tropical analogue of the classical Martens’ theorem within the framework of Brill–Noether theory for metric graphs.

The classical Martens’ theorem states that for a smooth algebraic curve C of genus g and integers d, r with 0 < 2r ≤ d < g, the dimension of the Brill–Noether locus W^r_d(C) is at most d - 2r, with equality holding if and only if C is hyperelliptic. In tropical geometry, the direct analogue using the dimension of the tropical Brill–Noether locus dim W^r_d(Γ) fails, as non-hyperelliptic metric graphs can achieve equality. To address this, Jensen and Len proposed using the Brill–Noether rank w^r_d(Γ), which exhibits upper semi-continuity on the moduli space, making it a more suitable candidate. They conjectured that for a metric graph Γ, w^r_d(Γ) ≤ d - 2r, with equality precisely when Γ is hyperelliptic.

The paper first revisits the necessary background on divisor theory on metric graphs, including definitions, linear equivalence, the Baker–Norine rank, Riemann–Roch theorem, and Dhar’s burning algorithm.

The core of the paper then explores the validity of this conjecture. It begins by recalling Coppens’ counterexamples: so-called Martens-special chains of cycles. These are specific chains of cycles of genus g ≥ 2r+3 where only certain designated cycles are non-hyperelliptic (having edge length ratios m_i ≠ 2), with at least r hyperelliptic cycles separating any two non-hyperelliptic ones. Coppens showed that for such graphs, at the specific degree d = g - 2 + r, the equality w^r_d(Γ) = d - 2r holds despite Γ being non-hyperelliptic, thus disproving the conjecture in its full generality.

The authors significantly contribute by generalizing this construction. They define a broader class of graphs termed Martens-special trees of cycles, which are tree-like gluings of cycles rather than simple chains, and prove that they similarly serve as counterexamples to the Jensen–Len conjecture. They further note that other counterexamples exist, such as graphs obtained by slightly altering edge lengths in a hyperelliptic graph.

However, the paper’s main positive result (Theorem A) establishes that the conjecture does hold under a stricter bound on the degree. Specifically, it is proven that for any metric graph Γ of genus g and integers d, r satisfying 0 < 2r ≤ d ≤ g - 3 + r, the inequality w^r_d(Γ) ≤ d - 2r holds, and equality w^r_d(Γ) = d - 2r occurs if and only if Γ is hyperelliptic. This proof extends Coppens’ earlier result for chains of cycles to all metric graphs and provides an alternative proof for the tropical Clifford’s theorem under the given degree constraints.

In conclusion, the paper provides a nuanced analysis of the tropical Martens’ theorem. It delineates the limits of the Jensen–Len conjecture by presenting and extending families of counterexamples, while simultaneously salvaging it by proving its truth under a more restrictive, yet natural, condition on the degree (d ≤ g - 3 + r). This work clarifies the precise conditions under which the tropical analogue of a fundamental algebraic geometry theorem holds.