Tropical Trigonal Construction Insights

• Tropical Trigonal Construction is a framework in tropical geometry that establishes a degree-3 harmonic morphism from tropical curves to trees using divisors of degree 3 and Baker–Norine rank arguments.
• The method employs chip-firing techniques, admissible representatives, and the concept of hyperelliptic halves to equate divisorial trigonality with the existence of a non-degenerate morphism.
• Recillas-type constructions extend the approach by linking free double covers of tropical trigonal curves to generic tetragonal maps, shedding light on tropical Prym varieties and moduli space structures.

Searching arXiv for the specified paper and closely related work on tropical trigonal constructions, low-genus embeddings, and the tropical $n$-gonal/Recillas framework. arXiv search results located for:

• (Melo et al., 11 Sep 2025) — "Tropical trigonal curves: the general case"
• (Melo et al., 7 Jan 2025) — "Tropical trigonal curves"
• (Markwig et al., 2 Feb 2026) — "Trigonal and embedded tropical curves of low genus"
• (Zakharov, 8 Jul 2025) — "The trigonal construction and the second moment of the tropical Prym variety"
• (Röhrle et al., 2022) — "The tropical $n$-gonal construction"

Proceeding with an overview grounded in these sources. Tropical trigonal construction denotes a set of closely related constructions in tropical geometry centered on degree-$3$ harmonic morphisms from metric graphs to trees. In one direction, it is the explicit procedure that starts from a divisor $D$ of degree $3$ and Baker–Norine rank at least $1$ on a tropical curve $\Gamma$, and constructs a non-degenerate harmonic morphism of degree $3$ from a tropical modification of $\Gamma$ to a tropical rational curve. In another, Recillas-type direction, it starts from a free double cover of a tropical trigonal curve and produces a tetragonal tropical curve whose Jacobian is identified with the corresponding tropical Prym. These constructions form tropical analogues both of the classical equivalence between a $g^1_3$ and a degree-$n$0 map to $n$1, and of Recillas’ trigonal construction (Melo et al., 11 Sep 2025, Röhrle et al., 2022).

1. Foundational objects and terminology

A tropical curve is a finite connected metric graph $n$2, where $n$3 is a finite graph and $n$4 assigns a positive real length to each edge. In the setting under discussion the curves are unweighted. The canonical loopless model $n$5 of $n$6 is obtained by contracting all loops of $n$7. A tropical rational curve is a metric tree $n$8, i.e. a finite connected metric graph without cycles (Melo et al., 11 Sep 2025).

A divisor on $n$9 is a formal integer combination

$3$0

with degree $3$1. Two divisors $3$2 and $3$3 are linearly equivalent if $3$4 is principal, namely the divisor of a continuous piecewise-linear function with integral slopes; chip-firing and Dhar’s burning algorithm provide the operative combinatorial model. The Baker–Norine rank is

$3$5

In the trigonal setting one considers divisors of degree $3$6; for genus $3$7, one has $3$8 because there are no rank-$3$9 degree-$D$0 divisors by tropical Riemann–Roch and Clifford (Melo et al., 11 Sep 2025).

A harmonic morphism of metric graphs is specified by the image of each vertex and each edge, together with integer expansion factors. For an edge $D$1,

$D$2

The morphism is non-degenerate if at each vertex $D$3 at least one incident edge has positive index. Harmonicity at a vertex $D$4 means that for each tangent direction at $D$5,

$D$6

is independent of the chosen target edge $D$7. The quantity $D$8 is the local degree. The total degree is

$D$9

which is independent of $3$0 when the target is a tree (Melo et al., 11 Sep 2025).

A tropical modification of $3$1 is obtained by attaching metric trees at points and/or subdividing edges. Edge connectivity is central in degree $3$2: a graph is $3$3-edge-connected if removing fewer than $3$4 edges never disconnects it. A necklace is a metric graph whose underlying graph contains a cycle with at least three separating vertices. Necklaces are exceptional because chip-firing around the cycle introduces degeneracies that obstruct the tree-target construction in general (Melo et al., 11 Sep 2025).

2. Divisorial trigonality and harmonic trigonality

The central equivalence theorem in the general case states that if $3$5 is not a necklace and its canonical loopless model has at least four vertices, then the following are equivalent: $3$6 is divisorially trigonal, meaning $3$7, and $3$8 is trigonal, meaning that there exists a non-degenerate harmonic morphism of degree $3$9 from a tropical modification $1$0 of $1$1 to a metric tree $1$2. In the broader formulation of Melo–Zheng, if $1$3 is not a necklace with canonical loopless model $1$4, then

• $1$5 or $1$6 is trigonal, and
• $1$7 is divisorially trigonal, are equivalent; in particular, when $1$8, divisorial trigonality is equivalent to trigonality (Melo et al., 11 Sep 2025).

The antecedent 3-edge-connected case was established earlier. For a $1$9-edge connected tropical curve $\Gamma$0, the existence of a divisor of degree $\Gamma$1 and Baker–Norine rank at least $\Gamma$2 is equivalent to the existence of a non-degenerate harmonic morphism of degree $\Gamma$3 from a tropical modification of $\Gamma$4 to a tropical rational curve; in the loopless $\Gamma$5-edge-connected case, no tropical modification is needed (Melo et al., 7 Jan 2025).

The reverse implication, from morphism to divisor, is structurally simple. Given a non-degenerate harmonic morphism $\Gamma$6 of degree $\Gamma$7 to a tree, one pulls back a generic point $\Gamma$8: $\Gamma$9 This is an effective divisor of degree $3$0, and its rank is at least $3$1 because moving $3$2 along $3$3 produces chip-firing motions compatible with linear equivalence. Hence $3$4 after identifying $3$5 with a tropical modification of $3$6 (Melo et al., 11 Sep 2025).

Several refined equivalences sit around the main theorem. If $3$7 has no separating vertices and no multiple edges, then “divisorially trigonal,” “trigonal,” and “admits a tropical admissible cover of degree $3$8” are equivalent. Conversely, if a degree-$3$9 harmonic morphism to a tree is not an admissible cover, then $\Gamma$0 has multiple edges or separating vertices. At low genus, one has the corollary that for $\Gamma$1, divisorial trigonality is equivalent to trigonality (Melo et al., 11 Sep 2025).

3. Explicit divisor-to-morphism construction

The constructive direction starts from a metric graph $\Gamma$2 together with a divisor $\Gamma$3 of degree $\Gamma$4 and rank at least $\Gamma$5. Rank certification is carried out by Dhar’s burning algorithm: for any point $\Gamma$6, one checks that $\Gamma$7 is linearly equivalent to an effective divisor, and similarly for every effective divisor of degree $\Gamma$8. This is the chip-firing certificate that $\Gamma$9 (Melo et al., 11 Sep 2025).

The first structural split isolates hyperelliptic subgraphs. A $g^1_3$0-hyperelliptic half $g^1_3$1 is a connected subcurve for which there exists $g^1_3$2 such that $g^1_3$3 with $g^1_3$4, and such that the supports of effective divisors linearly equivalent to $g^1_3$5 sweep out precisely $g^1_3$6. These halves are subject to two overlap conditions: $g^1_3$7 $g^1_3$8 meets its complement only at two points $g^1_3$9, with $n$00; and $n$01 distinct $n$02-hyperelliptic halves meet in at most one point. Under these hypotheses one uses the degree-$n$03 hyperelliptic morphism $n$04, then raises degree from $n$05 to $n$06 by attaching at $n$07 a copy of $n$08 mapped identically to $n$09. Gluing all such local constructions gives a non-degenerate degree-$n$10 harmonic morphism on the union of the halves (Melo et al., 11 Sep 2025).

On the complementary part, where no $n$11-hyperelliptic halves remain, the construction uses maximal admissible representatives. For each vertex $n$12, an admissible representative of $n$13 has the form $n$14, with $n$15 and $n$16 not both in the interior of the same edge. The $n$17-maximal representative $n$18 is the admissible representative maximizing the coefficient at $n$19. If $n$20, then $n$21; if $n$22, then $n$23. Consecutive maximal representatives determine the local combinatorics of the target tree: their supports lie on a $n$24-edge cut with $n$25, or on a bridge. If $n$26, the three edges have equal lengths; if $n$27, with edges $n$28, then $n$29 in the refined metric (Melo et al., 11 Sep 2025).

The target tree $n$30 is built by subdividing $n$31 at all points appearing in supports of the $n$32, forming a refined graph $n$33, and then introducing one vertex $n$34 in $n$35 for each maximal representative $n$36. Whenever $n$37 and $n$38 are consecutive, one adds an edge $n$39 to $n$40. The map $n$41 sends the support points of $n$42 to $n$43, and each source edge whose endpoints lie in supports of consecutive maximal divisors maps to $n$44 with an index dictated by the cut type.

Source configuration	Edge indices	Target-edge length
$n$45-edge cut	$n$46	preserved
$n$47-edge cut $n$48	$n$49 on $n$50, $n$51 on $n$52	$n$53
bridge $n$54	$n$55 on $n$56	$n$57

These assignments make

$n$58

independent of the chosen tangent direction, so vertex-wise harmonicity holds. The total degree is $n$59, and since there are no contractions in the refined model, non-degeneracy holds. It is further checked that $n$60 is a tree because each edge of $n$61 corresponds to a cut in the source that disconnects the target when removed (Melo et al., 11 Sep 2025).

The final step glues the morphism on the complement to the morphisms on the $n$62-hyperelliptic halves. Because intersections of distinct halves are single points by $n$63, and because the local degrees at glue points agree, the glued target remains a tree and the global map remains harmonic and non-degenerate of degree $n$64 (Melo et al., 11 Sep 2025).

In the earlier $n$65-edge-connected loopless case, the same procedure takes a simpler form. For each vertex $n$66, there is a unique admissible representative $n$67; consecutive representatives are separated by a $n$68-edge cut of equal-length edges; all non-contracted edges carry index $n$69; and no tropical modification is needed. When loops are present, one adds a leaf at a point $n$70 determined by a relation $n$71, and maps the two half-edges of the loop together with the added leaf to a leaf of the target, restoring the degree-$n$72 count (Melo et al., 7 Jan 2025).

4. Necklaces, hyperelliptic blocks, and exceptional behavior

Lower edge connectivity introduces phenomena absent in the $n$73-edge-connected case. Bridges force edges of index $n$74, while separating vertices produce path-uniqueness and $n$75-edge cuts that constrain chip motion. These effects destroy the disjoint-support behavior of admissible representatives that underlies the rigid $n$76-edge-connected construction. The general construction resolves this by separating off $n$77-hyperelliptic halves and treating the complement with maximal admissible representatives and precise index-length relations (Melo et al., 11 Sep 2025).

The principal obstruction is the necklace. A necklace is a connected graph with a cycle $n$78 whose vertices are all separating, with at least three such vertices. On $n$79, any two edges form a $n$80-edge cut, and chips can slide freely around the cycle. This flexibility prevents the construction of a non-degenerate degree-$n$81 harmonic morphism to a tree in general; Luo’s example in ABBR is presented precisely as an instance where a divisor of degree $n$82 and rank $n$83 exists but no degree-$n$84 harmonic morphism to a tree exists (Melo et al., 11 Sep 2025).

For non-hyperelliptic necklaces the paper replaces the tree target by a metric “tree of triangles.” The corresponding theorem states that for a non-hyperelliptic necklace $n$85, divisorial trigonality is equivalent to the existence of a non-degenerate harmonic morphism $n$86 of degree $n$87, where $n$88 is a tropical modification of $n$89 and $n$90 is a graph whose minimal cycles are triangles with equal edge lengths and no edges in common, such that the preimage of any cycle in $n$91 is the corresponding cycle in $n$92 with edge lengths divided by $n$93 (Melo et al., 11 Sep 2025).

Hyperelliptic substructures occupy the opposite extreme. If $n$94 is hyperelliptic and $n$95, then $n$96 is divisorially trigonal and trigonal. The construction begins with the degree-$n$97 hyperelliptic morphism $n$98 and attaches, at a preimage of a leaf $n$99, a copy of $3$00 mapped identically to $3$01; this raises the total degree to $3$02 while preserving harmonicity and non-degeneracy (Melo et al., 11 Sep 2025).

Cycle geometry imposes additional restrictions. A cycle cannot admit a non-degenerate degree-$3$03 harmonic morphism to a tree that has three vertices mapping to distinct leaves with local degree $3$04. Even after allowing tropical modifications, there are precise distance and balancing constraints on the positions of points of local multiplicity $3$05. These cycle constraints explain why necklace obstructions are global rather than merely local (Melo et al., 11 Sep 2025).

5. Maps to a line and low-genus embedded models

A complementary formulation of tropical trigonality, developed for genera $3$06 and $3$07, replaces the target tree by a line. A connected metric graph $3$08 is called tropical trigonal if there exists a continuous, piecewise-linear map $3$09 with integer slopes, harmonic with non-degenerate fibers, of degree $3$10. Equivalently, $3$11 is a well-contracted tropical cover of degree $3$12: a non-degenerate tropical cover to a path such that no loops are contracted. For such maps, tropical Riemann–Hurwitz gives

$3$13

so $3$14 in genus $3$15 and $3$16 in genus $3$17 (Markwig et al., 2 Feb 2026).

The embedding-theoretic realization uses Hirzebruch polygons. A tropical plane realization of a trigonal curve embedded in a Hirzebruch surface $3$18 is a smooth tropical plane curve dual to a unimodular triangulation of

$3$19

The projection $3$20 tropicalizes to projection $3$21, and for a smooth tropical plane curve dual to $3$22, the restriction $3$23 is a realizable, well-contracted degree-$3$24 tropical cover. At each $3$25-valent vertex, harmonicity follows from balancing: either two edges map in the same horizontal direction and the third in the opposite direction with matching weight, or one edge is contracted and the other two map with equal and opposite slopes (Markwig et al., 2 Feb 2026).

In genus $3$26, the paper gives a complete criterion for maximal combinatorial types. An abstract tropical curve of genus $3$27 and maximal combinatorial type is realizable in $3$28 if and only if its combinatorial type is one of $3$29, $3$30, $3$31, $3$32. For type $3$33, with edge lengths labeled as in Figure 1 of the paper, the conditions are

$3$34

For types $3$35, $3$36, and $3$37, the constraints coincide with the plane degree-$3$38 case (Markwig et al., 2 Feb 2026).

In genus $3$39, the Appendix gives necessary edge-length conditions for types realizable in $3$40 or $3$41. For example, for type $3$42A one has in $3$43

$3$44

whereas in $3$45

$3$46

For type $3$47, the conditions are in $3$48

$3$49

and in $3$50,

$3$51

The structural theorem is that if $3$52 has genus $3$53 or $3$54 and planar maximal combinatorial type, then after tropical modification $3$55 admits a degree-$3$56 well-contracted cover if and only if $3$57 is realizable in some $3$58, up to contractions that do not change the combinatorial type (Markwig et al., 2 Feb 2026).

The same work isolates obstructions that are not visible from tree gonality alone. Graphs with sprawling nodes, crowded graphs, and TIE-fighter graphs do not admit degree-$3$59 well-contracted covers, even after tropical modification. This explains the non-realizability of several maximal types; for genus $3$60, type $3$61 has a degree-$3$62 tropical cover only in a non-well-contracted form. When the plane embedding is non-smooth, the obstruction may be resolved by unfolding: a linear tropical re-embedding replaces a crossing dual to a parallelogram by a tropical modification of the ambient plane, separates the crossing edges, produces a new edge, and preserves the degree-$3$63 projection morphism (Markwig et al., 2 Feb 2026).

6. Recillas-type tropical trigonal construction and Prym geometry

In the broader tropical $3$64-gonal framework, the trigonal construction is the case $3$65 of a fiberwise procedure applied to a tower

$3$66

where $3$67 is a connected free double cover and $3$68 is a degree-$3$69 harmonic morphism to a metric tree. The construction considers the set of fiber sections

$3$70

This set carries a natural metric graph structure, and the projection

$3$71

is harmonic of degree $3$72. The graph $3$73 has two isomorphic connected components exchanged by the covering involution; either component is denoted $3$74, and the induced map $3$75 has degree $3$76. The genus satisfies $3$77, and there is an isomorphism of principally polarized tropical abelian varieties

$3$78

(Zakharov, 8 Jul 2025).

The local fiber structure is classified in the original $3$79-gonal construction by the type of a point $3$80 under the degree-$3$81 map $3$82. Type I means $3$83 with local degree $3$84; Type II means $3$85 with local degrees $3$86 and $3$87; Type III means $3$88 with all local degrees $3$89. For the resulting tetragonal curve $3$90, the fiber dilation profiles are therefore $3$91, $3$92, or $3$93, and never $3$94 or $3$95. In this sense the output is a generic tetragonal map (Röhrle et al., 2022).

The construction is reversible. Starting from a generic tetragonal map $3$96, one defines a graph parametrizing unordered pairs in fibers of $3$97; under the genericity assumption that no fiber has profile $3$98 or $3$99, the induced involution is free, the quotient gives a free double cover $D$00, and one recovers a degree-$D$01 harmonic morphism $D$02. This tropical Recillas theorem establishes a bijection between free double covers of tropical trigonal curves and generic tetragonal maps (Röhrle et al., 2022).

The construction has direct arithmetic consequences for tropical Pryms. For $D$03, every metric graph is trigonal in the harmonic sense used in the paper, so the above passage to $D$04 allows one to compute the second moment of the tropical Prym via the Jacobian of $D$05. The resulting formula is

$D$06

where $D$07 is a polynomial term analogous to the Jacobian formula and $D$08 is a piecewise-polynomial term depending solely on the signed graphic matroid of the double cover through its $D$09-sets. The piecewise term vanishes when there are no $D$10-sets with $D$11, and its appearance is tied to Friedman–Smith degenerations and the failure of the Prym–Torelli map to extend over all boundary strata (Zakharov, 8 Jul 2025).

7. Moduli and combinatorial models

The divisor-to-morphism construction leads to a polyhedral moduli theory for tropical trigonal covers and curves. A trigonal type is a triple $D$12, where $D$13 is a stable weighted graph and $D$14 is a non-degenerate harmonic morphism of degree $D$15 from a graph whose stabilization is $D$16 to a tree $D$17, with the condition that for every $D$18, the fiber $D$19 meets $D$20. The morphism induces an equivalence relation $D$21 on edges of the source when they have the same image, and an order relation by refinement. For a fixed type, the admissible edge lengths form the cone

$D$22

cut out by the linear relations implied by $D$23 and the inequalities implied by $D$24 (Melo et al., 7 Jan 2025).

The moduli space of tropical trigonal covers is the colimit

$D$25

taken over trigonal types of genus $D$26, with face maps induced by $D$27-contractions. Forgetting the morphism yields the moduli of tropical trigonal curves as a locus inside $D$28; the $D$29-edge-connected subspaces are defined analogously by restricting to $D$30-edge-connected types (Melo et al., 7 Jan 2025).

Maximal cells in the $D$31-edge-connected locus are parameterized by $D$32-ladders. Given a tree $D$33 with $D$34 vertices, one forms $D$35 by taking three disjoint copies $D$36 and connecting the copies at each vertex by prescribed “vertical” edges depending on whether the original valence is $D$37 or $D$38. The natural map $D$39 is a non-degenerate harmonic morphism of degree $D$40, and after stabilization the resulting graph is $D$41-edge connected and trigonal. These are precisely the graphs parameterizing the maximal cells of the $D$42-edge-connected tropical trigonal locus (Melo et al., 7 Jan 2025).

The combinatorics yield the expected dimension. If $D$43, then $D$44 and $D$45. After stabilization the number of independent edge-length parameters remains $D$46 for $D$47, and equals $D$48 for $D$49. Consequently, the maximal cones have dimension $D$50 for $D$51 and $D$52 for $D$53. This matches the dimension of the algebraic trigonal locus in $D$54. Moreover, after passing to tropical modifications with no contractions, the local Riemann–Hurwitz equalities hold at every vertex, so the degree-$D$55 harmonic morphisms define tropical admissible covers in the sense of Cavalieri–Markwig–Ranganathan (Melo et al., 7 Jan 2025).