--- title: Tropical Trigonal Construction Insights url: https://www.emergentmind.com/topics/tropical-trigonal-construction type: topic --- # Tropical Trigonal Construction Insights 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: - 2509.09502 — "Tropical trigonal curves: the general case" - 2501.03903 — "Tropical trigonal curves" - 2602.02257 — "Trigonal and embedded tropical curves of low genus" - 2507.06401 — "The trigonal construction and the second moment of the tropical Prym variety" - 2210.02267 — "The tropical \(n\)-gonal construction" Proceeding with a synthesis 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-\(3\) map to \(\mathbb P^1\), and of Recillas’ trigonal construction [2509.09502] [2210.02267]. ## 1. Foundational objects and terminology A tropical curve is a finite connected metric graph \(\Gamma=(G,l)\), where \(G\) is a finite graph and \(l\) assigns a positive real length to each edge. In the setting under discussion the curves are unweighted. The canonical loopless model \((G^{-},l^{-})\) of \(\Gamma\) is obtained by contracting all loops of \(G\). A tropical rational curve is a metric tree \(T\), i.e. a finite connected metric graph without cycles [2509.09502]. A divisor on \(\Gamma\) is a formal integer combination \[ D=\sum a_i x_i,\qquad x_i\in \Gamma,\ a_i\in \mathbb Z, \] with degree \(\deg(D)=\sum a_i\). Two divisors \(D\) and \(D'\) are linearly equivalent if \(D-D'\) 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 \[ r(D)=\max\Big\{ r\in\mathbb Z_{\ge -1}\ \Big|\ \forall\,E\ge 0,\ \deg(E)=r,\ \exists\,E'\ge 0\text{ with }D-E\sim E'\Big\}. \] In the trigonal setting one considers divisors of degree \(3\); for genus \(g\ge 3\), one has \(r(D)=1\) because there are no rank-\(2\) degree-\(3\) divisors by tropical Riemann–Roch and Clifford [2509.09502]. 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 \(e\), \[ \mu_{\varphi}(e)= \begin{cases} \dfrac{l_T(\varphi(e))}{l(e)}\in\mathbb Z_{\ge 0}, & \text{if }\varphi(e)\in E(T),\\[4pt] 0, & \text{if }\varphi(e)\in V(T). \end{cases} \] The morphism is non-degenerate if at each vertex \(x\in V(\Gamma')\) at least one incident edge has positive index. Harmonicity at a vertex \(v\) means that for each tangent direction at \(\varphi(v)\), \[ m_{\varphi}(v):=\sum_{\substack{e\in E_v(\Gamma')\\ \varphi(e)=e'}} \mu_{\varphi}(e) \] is independent of the chosen target edge \(e'\). The quantity \(m_{\varphi}(v)\) is the local degree. The total degree is \[ \deg(f)=\sum_{x\in f^{-1}(y)}\deg_x(f), \] which is independent of \(y\) when the target is a tree [2509.09502]. A tropical modification of \(\Gamma\) is obtained by attaching metric trees at points and/or subdividing edges. Edge connectivity is central in degree \(3\): a graph is \(k\)-edge-connected if removing fewer than \(k\) 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 [2509.09502]. ## 2. Divisorial trigonality and harmonic trigonality The central equivalence theorem in the general case states that if \(\Gamma\) is not a necklace and its canonical loopless model has at least four vertices, then the following are equivalent: \(\Gamma\) is divisorially trigonal, meaning \(W^1_3(\Gamma)\neq\emptyset\), and \(\Gamma\) is trigonal, meaning that there exists a non-degenerate harmonic morphism of degree \(3\) from a tropical modification \(\Gamma'\) of \(\Gamma\) to a metric tree \(T\). In the broader formulation of Melo–Zheng, if \(\Gamma\) is not a necklace with canonical loopless model \((G^{-},l^{-})\), then - \(|V(G^{-})|=2,3\) or \(\Gamma\) is trigonal, and - \(\Gamma\) is divisorially trigonal, are equivalent; in particular, when \(|V(G^{-})|\ge 4\), divisorial trigonality is equivalent to trigonality [2509.09502]. The antecedent 3-edge-connected case was established earlier. For a \(3\)-edge connected tropical curve \(\Gamma\), the existence of a divisor of degree \(3\) and Baker–Norine rank at least \(1\) is equivalent to the existence of a non-degenerate harmonic morphism of degree \(3\) from a tropical modification of \(\Gamma\) to a tropical rational curve; in the loopless \(3\)-edge-connected case, no tropical modification is needed [2501.03903]. The reverse implication, from morphism to divisor, is structurally simple. Given a non-degenerate harmonic morphism \(f:\Gamma'\to T\) of degree \(3\) to a tree, one pulls back a generic point \(y\in T\): \[ D:=f^*(y)=\sum_{x\in f^{-1}(y)} \deg_x(f)\cdot x. \] This is an effective divisor of degree \(3\), and its rank is at least \(1\) because moving \(y\) along \(T\) produces chip-firing motions compatible with linear equivalence. Hence \(D\in W^1_3(\Gamma)\) after identifying \(\Gamma'\) with a tropical modification of \(\Gamma\) [2509.09502]. Several refined equivalences sit around the main theorem. If \(\Gamma\) has no separating vertices and no multiple edges, then “divisorially trigonal,” “trigonal,” and “admits a tropical admissible cover of degree \(3\)” are equivalent. Conversely, if a degree-\(3\) harmonic morphism to a tree is not an admissible cover, then \(\Gamma\) has multiple edges or separating vertices. At low genus, one has the corollary that for \(g\le 5\), divisorial trigonality is equivalent to trigonality [2509.09502]. ## 3. Explicit divisor-to-morphism construction The constructive direction starts from a metric graph \(\Gamma\) together with a divisor \(D\) of degree \(3\) and rank at least \(1\). Rank certification is carried out by Dhar’s burning algorithm: for any point \(w\in\Gamma\), one checks that \(D-w\) is linearly equivalent to an effective divisor, and similarly for every effective divisor of degree \(1\). This is the chip-firing certificate that \(r(D)\ge 1\) [2509.09502]. The first structural split isolates hyperelliptic subgraphs. A \(D\)-hyperelliptic half \(\Gamma_1\subset \Gamma\) is a connected subcurve for which there exists \(p\in \Gamma\setminus \Gamma_1\) such that \(D-p\sim H\) with \(H\in W^1_2(\Gamma_1)\), and such that the supports of effective divisors linearly equivalent to \(H\) sweep out precisely \(\Gamma_1\). These halves are subject to two overlap conditions: \((\mathrm{H}1)\) \(\Gamma_1\) meets its complement only at two points \(\{x_0,y_0\}\), with \(D-p\sim x_0+y_0\); and \((\mathrm{H}2)\) distinct \(D\)-hyperelliptic halves meet in at most one point. Under these hypotheses one uses the degree-\(2\) hyperelliptic morphism \(\psi:\Gamma_1\to T_1\), then raises degree from \(2\) to \(3\) by attaching at \(p\) a copy of \(T_1\) mapped identically to \(T_1\). Gluing all such local constructions gives a non-degenerate degree-\(3\) harmonic morphism on the union of the halves [2509.09502]. On the complementary part, where no \(D\)-hyperelliptic halves remain, the construction uses maximal admissible representatives. For each vertex \(x\in V(G_0)\), an admissible representative of \(D\) has the form \(x+x_1+x_2\), with \(x_1\) and \(x_2\) not both in the interior of the same edge. The \(x\)-maximal representative \(D_x^{\max}\) is the admissible representative maximizing the coefficient at \(x\). If \(D_x^{\max}(x)=1\), then \(D_x^{\max}=x+x_1+x_2\); if \(D_x^{\max}(x)=2\), then \(D_x^{\max}=2x+x_1\). Consecutive maximal representatives determine the local combinatorics of the target tree: their supports lie on a \(k\)-edge cut with \(k\in\{2,3\}\), or on a bridge. If \(k=3\), the three edges have equal lengths; if \(k=2\), with edges \(e_1,e_2\), then \(2l(e_1)=l(e_2)\) in the refined metric [2509.09502]. The target tree \(T_D\) is built by subdividing \(\Gamma\) at all points appearing in supports of the \(D_x^{\max}\), forming a refined graph \(G_D\), and then introducing one vertex \(t_x\) in \(T_D\) for each maximal representative \(D_x^{\max}\). Whenever \(D_x^{\max}\) and \(D_y^{\max}\) are consecutive, one adds an edge \(t_xt_y\) to \(T_D\). The map \(\varphi_D\) sends the support points of \(D_x^{\max}\) to \(t_x\), and each source edge whose endpoints lie in supports of consecutive maximal divisors maps to \(t_xt_y\) with an index dictated by the cut type. | Source configuration | Edge indices | Target-edge length | |---|---:|---:| | \(3\)-edge cut | \(1,1,1\) | preserved | | \(2\)-edge cut \(\{\text{short }e_1,\text{ long }e_2\}\) | \(2\) on \(e_1\), \(1\) on \(e_2\) | \(l_G(e_2)=2l_G(e_1)\) | | bridge \(b\) | \(3\) on \(b\) | \(3l_G(b)\) | These assignments make \[ m_{\varphi_D}(v)=\sum_{\substack{e\in E_v(G_D)\\ \varphi_D(e)=e'}} \mu_{\varphi_D}(e) \] independent of the chosen tangent direction, so vertex-wise harmonicity holds. The total degree is \(3\), and since there are no contractions in the refined model, non-degeneracy holds. It is further checked that \(T_D\) is a tree because each edge of \(T_D\) corresponds to a cut in the source that disconnects the target when removed [2509.09502]. The final step glues the morphism on the complement to the morphisms on the \(D\)-hyperelliptic halves. Because intersections of distinct halves are single points by \((\mathrm{H}2)\), 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 \(3\) [2509.09502]. In the earlier \(3\)-edge-connected loopless case, the same procedure takes a simpler form. For each vertex \(x\), there is a unique admissible representative \(D_x=x+x_1+x_2\); consecutive representatives are separated by a \(3\)-edge cut of equal-length edges; all non-contracted edges carry index \(1\); and no tropical modification is needed. When loops are present, one adds a leaf at a point \(y\) determined by a relation \(D_x=2x+y\), and maps the two half-edges of the loop together with the added leaf to a leaf of the target, restoring the degree-\(3\) count [2501.03903]. ## 4. Necklaces, hyperelliptic blocks, and exceptional behavior Lower edge connectivity introduces phenomena absent in the \(3\)-edge-connected case. Bridges force edges of index \(3\), while separating vertices produce path-uniqueness and \(2\)-edge cuts that constrain chip motion. These effects destroy the disjoint-support behavior of admissible representatives that underlies the rigid \(3\)-edge-connected construction. The general construction resolves this by separating off \(D\)-hyperelliptic halves and treating the complement with maximal admissible representatives and precise index-length relations [2509.09502]. The principal obstruction is the necklace. A necklace is a connected graph with a cycle \(\gamma\) whose vertices are all separating, with at least three such vertices. On \(\gamma\), any two edges form a \(2\)-edge cut, and chips can slide freely around the cycle. This flexibility prevents the construction of a non-degenerate degree-\(3\) harmonic morphism to a tree in general; Luo’s example in ABBR is presented precisely as an instance where a divisor of degree \(3\) and rank \(1\) exists but no degree-\(3\) harmonic morphism to a tree exists [2509.09502]. 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 \(\Gamma\), divisorial trigonality is equivalent to the existence of a non-degenerate harmonic morphism \(\varphi':\Gamma'\to T_\Delta\) of degree \(3\), where \(\Gamma'\) is a tropical modification of \(\Gamma\) and \(T_\Delta\) 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 \(T_\Delta\) is the corresponding cycle in \(\Gamma\) with edge lengths divided by \(3\) [2509.09502]. Hyperelliptic substructures occupy the opposite extreme. If \(\Gamma\) is hyperelliptic and \(|V(G^{-})|\ge 3\), then \(\Gamma\) is divisorially trigonal and trigonal. The construction begins with the degree-\(2\) hyperelliptic morphism \(\psi:\Gamma\to T\) and attaches, at a preimage of a leaf \(t\), a copy of \(T\) mapped identically to \(T\); this raises the total degree to \(3\) while preserving harmonicity and non-degeneracy [2509.09502]. Cycle geometry imposes additional restrictions. A cycle cannot admit a non-degenerate degree-\(3\) harmonic morphism to a tree that has three vertices mapping to distinct leaves with local degree \(3\). Even after allowing tropical modifications, there are precise distance and balancing constraints on the positions of points of local multiplicity \(2\). These cycle constraints explain why necklace obstructions are global rather than merely local [2509.09502]. ## 5. Maps to a line and low-genus embedded models A complementary formulation of tropical trigonality, developed for genera \(3\) and \(4\), replaces the target tree by a line. A connected metric graph \(\Gamma\) is called tropical trigonal if there exists a continuous, piecewise-linear map \(\varphi:\Gamma\to \mathbb R\) with integer slopes, harmonic with non-degenerate fibers, of degree \(3\). Equivalently, \(\varphi\) is a well-contracted tropical cover of degree \(3\): a non-degenerate tropical cover to a path such that no loops are contracted. For such maps, tropical Riemann–Hurwitz gives \[ K_\Gamma=\varphi^*(K_T)+R,\qquad \deg(R)=2g(\Gamma)-2+2\deg(\varphi)=2g+4, \] so \(\deg(R)=10\) in genus \(3\) and \(12\) in genus \(4\) [2602.02257]. The embedding-theoretic realization uses Hirzebruch polygons. A tropical plane realization of a trigonal curve embedded in a Hirzebruch surface \(F_n\) is a smooth tropical plane curve dual to a unimodular triangulation of \[ \Delta=\operatorname{conv}\left\{(0,0),(3,0),\left(0,\frac{g+3n+2}{2}\right),\left(3,\frac{g-3n+2}{2}\right)\right\}. \] The projection \(pr:F_n\to \mathbb P^1\) tropicalizes to projection \(\mathbb R^2\to \mathbb R\), and for a smooth tropical plane curve dual to \(\Delta\), the restriction \(pr|_\Gamma\) is a realizable, well-contracted degree-\(3\) tropical cover. At each \(3\)-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 [2602.02257]. In genus \(3\), the paper gives a complete criterion for maximal combinatorial types. An abstract tropical curve of genus \(3\) and maximal combinatorial type is realizable in \(F_1\) if and only if its combinatorial type is one of \((000)\), \((020)\), \((111)\), \((212)\). For type \((000)\), with edge lengths labeled as in Figure 6 of the paper, the conditions are \[ \max\{x,y\}\le u,\qquad \max\{x,z\}\le v,\qquad y+z\le w. \] For types \((020)\), \((111)\), and \((212)\), the constraints coincide with the plane degree-\(4\) case [2602.02257]. In genus \(4\), the Appendix gives necessary edge-length conditions for types realizable in \(F_0\) or \(F_2\). For example, for type \((000)\)A one has in \(F_0\) \[ wz+vw0}^{|E(G)/\le_\varphi|}\subset \sigma_G=\mathbb R_{>0}^{|E(G)|}, \] cut out by the linear relations implied by \(\sim_\varphi\) and the inequalities implied by \(\le_\varphi\) [2501.03903]. The moduli space of tropical trigonal covers is the colimit \[ H^{\trop}_{g,3}:=\operatorname{colim}_{(G,w,\varphi)} \bar{\tau}_{(G,w,\varphi)}, \] taken over trigonal types of genus \(g\), with face maps induced by \(\varphi\)-contractions. Forgetting the morphism yields the moduli of tropical trigonal curves as a locus inside \(M^{\trop}_g\); the \(3\)-edge-connected subspaces are defined analogously by restricting to \(3\)-edge-connected types [2501.03903]. Maximal cells in the \(3\)-edge-connected locus are parameterized by \(3\)-ladders. Given a tree \(T\) with \(n\) vertices, one forms \(G_T\) by taking three disjoint copies \(T^{(1)},T^{(2)},T^{(3)}\) and connecting the copies at each vertex by prescribed “vertical” edges depending on whether the original valence is \(1\) or \(2\). The natural map \(\varphi_T:G_T\to T\) is a non-degenerate harmonic morphism of degree \(3\), and after stabilization the resulting graph is \(3\)-edge connected and trigonal. These are precisely the graphs parameterizing the maximal cells of the \(3\)-edge-connected tropical trigonal locus [2501.03903]. The combinatorics yield the expected dimension. If \(n=|V(T)|\), then \(g(G_T)=n\) and \(|E(G_T)/\sim_{\varphi_T}|=2n+1\). After stabilization the number of independent edge-length parameters remains \(2n+1\) for \(g>3\), and equals \(6\) for \(g=3\). Consequently, the maximal cones have dimension \(2g+1\) for \(g\ge 4\) and \(6\) for \(g=3\). This matches the dimension of the algebraic trigonal locus in \(M_g\). Moreover, after passing to tropical modifications with no contractions, the local Riemann–Hurwitz equalities hold at every vertex, so the degree-\(3\) harmonic morphisms define tropical admissible covers in the sense of Cavalieri–Markwig–Ranganathan [2501.03903]. Source: https://www.emergentmind.com/topics/tropical-trigonal-construction