Realizable Tropical Canonical Divisors

• Realizable tropical canonical divisors are effective divisors obtained by tropicalizing canonical divisors on smooth algebraic curves, linking algebraic and tropical geometry.
• A combinatorial criterion using enhanced level graphs tests realizability by ensuring every inconvenient vertex and horizontal edge lies on a simple cycle without lower-level nodes.
• Techniques such as tropical modifications, reduced-divisor maps, and projection theory reveal the intricate polyhedral and matroidal structure of the realizability locus.

Realizable tropical canonical divisors are effective divisors in the tropical canonical linear system that arise as tropicalizations of canonical divisors on smooth algebraic curves. Concretely, for a stable tropical curve $\Gamma$, the canonical divisor is

$$K_\Gamma=\sum_{v\in V(\Gamma)}(2h(v)+\mathrm{val}(v)-2)\cdot v,$$

and an effective tropical canonical divisor has the form $D=K_\Gamma+(f)$ for a rational function $f$ on $\Gamma$. The realizability problem asks whether there exists a smooth curve $X$ over a non-Archimedean field whose stable reduction has dual tropical curve $\Gamma$, together with an effective canonical divisor on $X$ specializing to $D$. The subject lies at the intersection of divisor theory on metric graphs, tropical moduli, harmonic morphisms, and tropicalizations of algebraic linear series (Moeller et al., 2017).

1. Divisors, reduced divisors, and very ampleness

A foundational viewpoint treats tropical canonical divisors through reduced divisors and the complete linear system $|D|$. For any divisor $$K_\Gamma=\sum_{v\in V(\Gamma)}(2h(v)+\mathrm{val}(v)-2)\cdot v,$$0 on a tropical curve $$K_\Gamma=\sum_{v\in V(\Gamma)}(2h(v)+\mathrm{val}(v)-2)\cdot v,$$1 and any point $$K_\Gamma=\sum_{v\in V(\Gamma)}(2h(v)+\mathrm{val}(v)-2)\cdot v,$$2, there exists a unique $$K_\Gamma=\sum_{v\in V(\Gamma)}(2h(v)+\mathrm{val}(v)-2)\cdot v,$$3-reduced divisor $$K_\Gamma=\sum_{v\in V(\Gamma)}(2h(v)+\mathrm{val}(v)-2)\cdot v,$$4 equivalent to $$K_\Gamma=\sum_{v\in V(\Gamma)}(2h(v)+\mathrm{val}(v)-2)\cdot v,$$5. This defines the reduced-divisor map

$$K_\Gamma=\sum_{v\in V(\Gamma)}(2h(v)+\mathrm{val}(v)-2)\cdot v,$$6

and this map is integral affine and continuous. A divisor $$K_\Gamma=\sum_{v\in V(\Gamma)}(2h(v)+\mathrm{val}(v)-2)\cdot v,$$7 is very ample if and only if $$K_\Gamma=\sum_{v\in V(\Gamma)}(2h(v)+\mathrm{val}(v)-2)\cdot v,$$8 is injective, so point separation in the tropical linear system is the tropical analogue of an embedding criterion (Amini, 2010).

For canonical divisors, the distinction between very ampleness and failure of separation is especially sharp. On a compact tropical curve of genus $$K_\Gamma=\sum_{v\in V(\Gamma)}(2h(v)+\mathrm{val}(v)-2)\cdot v,$$9 without 1-valent vertices, the canonical divisor is not very ample if and only if the underlying graph has one of four explicit shapes denoted C.I, C.II, C.II', and C.III. In particular, the banana graph provides a basic obstruction: the reduced-divisor map fails to distinguish the two vertices. Outside these classified cases, the canonical divisor is very ample. In the non-very-ample cases, the canonical divisor is associated, up to linear equivalence, to $D=K_\Gamma+(f)$0 for a degree-$D=K_\Gamma+(f)$1 divisor $D=K_\Gamma+(f)$2 of rank $D=K_\Gamma+(f)$3, so the obstruction is tied to tropical hyperellipticity rather than to a deficiency of the full linear system in generic combinatorial types (Amini, 2010).

This divisor-theoretic perspective isolates a central principle that persists throughout the realizability problem: tropical existence inside $D=K_\Gamma+(f)$4 is weaker than algebraic realizability, but failures of realizability are often first detected by failures of point separation.

2. The combinatorial realizability criterion

A complete solution for effective tropical canonical divisors in equicharacteristic zero is given by a combinatorial criterion formulated in terms of an enhanced level graph. Starting from

$D=K_\Gamma+(f)$5

one refines $D=K_\Gamma+(f)$6 so that the support of $D=K_\Gamma+(f)$7 is visible, assigns each vertex the level $D=K_\Gamma+(f)$8, and decorates half-edges by integers derived from outgoing slopes of $D=K_\Gamma+(f)$9. This produces the enhanced level graph $f$0, whose vertical and horizontal edges encode the slope structure relevant to degeneration of abelian differentials (Moeller et al., 2017).

The realizability criterion has two parts. First, every inconvenient vertex must lie on a simple cycle that does not pass through any node at a lower level. Second, every horizontal edge must lie on a simple cycle that does not pass through any node at a lower level. If either condition fails, the pair $f$1 is not realizable. If both hold, then $f$2 is realizable. The criterion is purely combinatorial, and it converts a priori analytic lifting data into a graph-theoretic test (Moeller et al., 2017).

The same work embeds this criterion in a moduli-theoretic framework. There is a tropicalization map

$f$3

from the analytification of the moduli space of smooth curves with effective divisors to the tropical moduli of pairs $f$4. The realizability locus of tropical canonical divisors is a generalized rational polyhedral cone complex of pure dimension $f$5. This provides both a structural statement and an algorithmic one: realizability is not merely decidable case-by-case, but organized by a global polyhedral geometry (Moeller et al., 2017).

A later generalization treats pluri-canonical divisors $f$6 for all $f$7. The key new object is a tropical normalized cover, and realizability reduces to the existence of a realizable cover with prescribed enhancements. The criterion requires: no illegal vertices; every horizontal edge lies in an effective cycle; and every inconvenient vertex is redeemed by an admissible cycle or an independent pair of cycles. For $f$8, every tropical normalized cover is the identity map, and the conditions recover exactly the canonical divisor criterion above. The tropical $f$9-Hodge bundle has maximal cone dimension $\Gamma$0, while the realizability locus has dimension $\Gamma$1 for $\Gamma$2 and dimension $\Gamma$3 for $\Gamma$4 (Röhrle et al., 2021).

3. Genus $\Gamma$5, tropical quartics, and realizably hyperelliptic curves

The genus-$\Gamma$6 case is the most developed explicit realization problem for tropical canonical divisors because canonically embedded non-hyperelliptic curves are plane quartics. Brodsky, Joswig, Morrison and Sturmfels showed that not all abstract tropical curves of genus $\Gamma$7 can be realized as tropicalizations of quartics in the Euclidean plane. A precise classification on the interiors of maximal cones in $\Gamma$8 identifies the exact realizable locus among maximal tropical curves (Hahn et al., 2018).

The central distinction is between realizably hyperelliptic (rh) and non-rh curves. A tropical curve is rh if it is the tropicalization of an algebraic hyperelliptic curve. The main theorem states that any maximal non-rh tropical curve of genus $\Gamma$9 can be embedded as a faithfully tropicalized quartic in a possibly linearly modified tropical plane, whereas no maximal rh tropical curve of genus $X$0 can be embedded in this way. Thus, within maximal cones, realizability as a tropical plane quartic is equivalent to not being realizably hyperelliptic (Hahn et al., 2018).

Curve type	Realizability as tropical quartic	Mechanism
Maximal non-rh	Yes, after modifications	Explicit quartics and linear modifications
Maximal rh	No	Canonical sections do not separate points

The constructive side uses quartic polynomials together with sequences of tropical modifications along lines. These modifications enlarge edges and unfold hidden lollipop or banana cycles from higher-multiplicity edges, yielding a faithful tropicalization matching the prescribed graph. The constructions are case-by-case for the maximal combinatorial types $X$1, $X$2, $X$3, $X$4, and $X$5, and are phrased as tropical refinements of the plane (Hahn et al., 2018).

The obstruction on the rh locus is expressed in terms of realizable sections of the tropical canonical divisor. For a maximal rh tropical curve $X$6 and any piecewise linear function $X$7 satisfying

$X$8

the realizable tropical canonical linear system does not separate certain pairs of points. In type $X$9 with equal bridge lengths, one has

$\Gamma$0

for all canonical $\Gamma$1. Analogous failures occur in type $\Gamma$2 when the two cut-edges have equal length. In type $\Gamma$3, the full canonical system is very ample, but the linear system of realizable tropical canonical sections is not. This is the decisive distinction: abstract tropical very ampleness does not by itself guarantee realizability of the canonical embedding (Hahn et al., 2018).

4. Geometry of the realizability locus

The realizability locus inside $\Gamma$4 has a geometry of its own. It is not merely a subset cut out by existential lifting data; it carries tropical convex, polyhedral, and matroidal structures. One approach shows that the set of realizable canonical divisors $\Gamma$5 is tropically convex, definable by polyhedral conditions, and closed. It therefore admits the structure of an abstract polyhedral complex. Moreover, maximal cells of dimension $\Gamma$6 always occur inside the realizability locus. There is also a graph-theoretic dichotomy: if $\Gamma$7 has no two disjoint cycles, then every divisor in $\Gamma$8 is realizable, whereas if $\Gamma$9 has disjoint cycles, then non-realizable canonical divisors exist (Dupraz, 26 Jun 2025).

A complementary framework describes tropical linear series through the Baker–Norine rank and the independence rank. For canonical divisors, the locus

$X$0

is a polyhedral submodule, and near a nondegenerate divisor its local structure is the Bergman fan of a matroid. At the canonical divisor itself, the local matroid is the cographic matroid of the graph. The main criterion in this direction states that for a curve $X$1 with skeleton $X$2,

$X$3

Hence the equality between tropicalized canonical linear series and the full realizable locus is controlled exactly by the expected dimension condition (Chang et al., 27 Aug 2025).

This local-to-global picture clarifies a recurrent phenomenon. The realizable locus can be strictly larger than the tropicalization of a single algebraic canonical linear series. The cited work gives a genus-$X$4 hyperelliptic chain where $X$5 contains a $X$6-dimensional face although the canonical series has dimension $X$7. A plausible implication is that realizability of individual divisors is more flexible than tropicalization of an entire linear series, and the excess flexibility is encoded by matroidal local structure rather than by the intrinsic rank of $X$8 alone (Chang et al., 27 Aug 2025).

5. Modifications, projections, and canonical maps

Tropical modifications are central to explicit realizability. In genus $X$9, modifications along tropical lines repair naive planar tropicalizations by revealing combinatorial features that do not fit in $D$0 before re-embedding. This mechanism explains how non-rh maximal curves can be realized as quartics after passing to a modified tropical plane, even when no direct planar faithful tropicalization exists (Hahn et al., 2018).

A different but related mechanism arises from reduced-divisor maps and tropical projections. The reduced-divisor map embeds a very ample divisor into its complete linear system, and tropical projection theory extends the notion of reduced divisors from complete linear systems to arbitrary compact tropical convex subsets. In this framework, the $D$1-reduced divisor in a tropical convex set is the tropical projection of $D$2 to that set, and reduced-divisor maps to dominant tropical trees correspond to harmonic morphisms to metric trees. This provides a geometric language for rank-one linear systems and for the interaction between divisor theory and morphisms, which is relevant to canonical divisors because canonical realizability is ultimately governed by whether realizable sections separate points and assemble into suitable tropical linear systems (Luo, 2018).

These projection-theoretic methods do not replace the cycle criteria for canonical realizability, but they clarify the ambient geometry in which those criteria live. The resulting picture is that realizable canonical divisors are simultaneously governed by chip-firing, tropical convexity, and the combinatorics of cycles and slopes.

6. Broader obstructions and adjacent realizability phenomena

Realizable tropical canonical divisors sit inside a larger lifting landscape where realizability may fail for reasons not visible from abstract divisor theory alone. One source of failure is superabundance. New genus-$D$3 and genus-$D$4 superabundant tropical curves in $D$5 and $D$6 have properties resembling canonical embeddings of algebraic curves, including a genus-$D$7 degree-$D$8 planar tropical curve and a genus-$D$9 example in the product of a tropical line and a tropical conic. Their deformation spaces have excess dimension $|D|$0, and generic non-realizability follows from a dimension comparison with spaces of logarithmic curves. The genus-$|D|$1 example is contained in a tropical plane and mimics the canonical embedding of a non-hyperelliptic genus-$|D|$2 curve, but it is non-realizable. This shows that canonical resemblance does not imply liftability once superabundance enters (Koyama, 2023).

Another source of subtlety is arithmetic. For rank-$|D|$3 divisors constructed from matroids, liftability can depend on the residue characteristic or on the field of definition. There are graphs and rank-$|D|$4 divisors lifting over $|D|$5 exactly in prescribed characteristics, and others lifting exactly when the ground field contains a chosen number field. These examples are not usually canonical divisors, but they demonstrate that lifting tropical divisors can encode matroid realizability and Mnëv universality. This suggests that the relative tractability of the canonical realizability problem relies on special features of canonical linear systems and on the geometry of differentials, rather than on a general simplification of tropical lifting questions (Cartwright, 2015).

The contrast with principal divisors is also instructive. Tropical double ramification theory distinguishes the locus of principal divisors from the smaller locus arising from finite effective harmonic morphisms to trees. The principal divisor locus has cones of codimension zero in $|D|$6, while the tropical double ramification locus has the expected codimension $|D|$7. This separates broad tropical existence from algebraically meaningful realizability and helps explain why tropical linear systems are often larger than their algebraic shadows (Ulirsch et al., 2019).

Taken together, these developments place realizable tropical canonical divisors in a precise position within tropical geometry: the canonical case admits a complete combinatorial criterion and rich constructive results, yet it remains sensitive to hyperellipticity, superabundance, dimension excess, and the distinction between an abstract tropical linear system and the genuinely realizable part of that system.