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Continuity of the tropical Prym-Torelli map

This paper presents a new intrinsic construction of the tropical Prym variety as an integral torus for harmonic double covers of graphs and proves the continuity of the associated tropical Prym-Torelli map.

Original authors: Giusi Capobianco

Published 2026-08-25

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CC BY 4.0

Original authors: Giusi Capobianco

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the landscape of modern mathematics, there is a field called tropical geometry that studies shapes made not of smooth curves and surfaces, but of networks of lines and points. Imagine a map of a city where the streets are the only things that matter, and the distance between intersections is measured by how long it takes to walk them. In this world, complex algebraic shapes from classical geometry are simplified into these skeletal networks, or graphs. This simplification allows mathematicians to study deep properties of shapes by looking at their underlying structure. One of the most important tools in this field is the Jacobian, a mathematical object that captures the essential loops and cycles within a graph, much like a fingerprint that identifies the shape's connectivity. When two such graphs are linked in a specific way, where one covers the other like a double layer of fabric, mathematicians can create a related object called a Prym variety. This object acts as a specialized fingerprint for the relationship between the two graphs, revealing hidden symmetries and connections.

For a long time, mathematicians believed they understood how to build these Prym varieties for these network shapes. However, a problem emerged when they tried to watch these shapes change. If you imagine a graph slowly shrinking, with some of its lines collapsing until they disappear, the mathematical object representing its Prym variety was supposed to change smoothly along with it. Instead, under the old rules, the object would suddenly jump or break, losing its continuity. It was as if a shape could morph perfectly until a critical moment, where it would snap into a completely different form without warning. This discontinuity meant that the map connecting the world of these double-covered graphs to the world of their Prym varieties was broken, leaving a gap in the mathematical understanding of how these structures behave under stress.

Giusi Capobianco, in this new work, has repaired that break by constructing a more robust version of the Prym variety. The author realized that the previous method was too rigid, failing to account for the specific weights and connections that appear when parts of the graph collapse. To fix this, Capobianco developed a new, intrinsic way to build the basis for these varieties. Instead of forcing a single, fixed method of construction that worked only for simple cases, the new approach allows for a flexible selection of paths within the graph. This flexibility is crucial because it adapts to the changing geometry of the graph as it degenerates. By carefully choosing which paths to include in the mathematical foundation, the author ensures that the resulting object remains stable and continuous, even as the graph shrinks and its structure simplifies.

The core of this achievement lies in how the new construction handles the "dilated" parts of the graph. In a double cover, some edges might stretch or compress in a way that creates a special kind of connection, known as a dilation. When these dilated sections are present, the old methods struggled to keep the mathematical object consistent. Capobianco's new method explicitly accounts for these dilated sections and the weights attached to the vertices. It does this by adding extra dimensions to the mathematical object whenever the graph loses complexity through contraction. This ensures that the total size and shape of the Prym variety adjust perfectly to the changes in the graph, preventing the sudden jumps that plagued earlier definitions.

The result is a continuous map that connects the space of all possible double-covered graphs to the space of their corresponding Prym varieties. This map, known as the tropical Prym–Torelli map, now works without interruption. Whether the graph is in its full, complex form or has been reduced to a simpler state through the contraction of edges, the map produces a consistent and well-defined result. The author proves that this new construction is not just a theoretical fix but a rigorous mathematical solution that holds true for all cases, including those with weighted vertices and complex dilation patterns. This continuity is essential because it allows mathematicians to study families of these shapes as they evolve, knowing that the underlying mathematical invariants will behave predictably.

This work resolves a significant issue that had been noticed by experts in the field, who had observed that the volume of the Prym variety did not vary continuously in families under the old definitions. By providing a construction that works for both free covers and those with dilations, the paper unifies the understanding of these objects. The new definition of the extended Prym variety, which includes additional dimensions to account for the weights lost during contraction, ensures that the dimension of the object remains correct throughout the process. This means that the mathematical fingerprint of the graph remains intact and reliable, no matter how much the graph changes.

The significance of this finding extends beyond just fixing a broken map. It provides a solid foundation for future research in tropical geometry, allowing mathematicians to explore the relationships between different types of graphs with confidence. The ability to track these objects continuously means that complex problems involving the deformation of shapes can now be approached with a reliable tool. The paper demonstrates that by understanding the intrinsic structure of the cycles within these graphs, one can build mathematical objects that are resilient to change. This resilience is key to understanding the deeper geometry of these networks, offering a clearer view of how these abstract shapes interact and transform.

In the end, the paper offers a complete and continuous picture of the tropical Prym variety. It shows that with the right construction, the mathematical objects associated with these double-covered graphs can be made to flow smoothly from one state to another. The work confirms that the tropical Prym–Torelli map is indeed continuous, bridging the gap between the moduli space of harmonic double covers and the moduli space of principally polarized tropical abelian varieties. This continuity is a fundamental property that was missing, and its restoration brings a new level of coherence to the field, allowing for a more unified and powerful exploration of tropical geometry.

Technical Summary: Continuity of the Tropical Prym–Torelli Map

Problem Statement
The paper addresses a fundamental discontinuity in the theory of tropical Prym varieties. While the classical Prym variety is associated with étale double covers of smooth algebraic curves, its tropical counterpart arises from harmonic morphisms of metric graphs. Previous work by Jensen and Len established a "standard" definition of the tropical Prym variety. However, it was shown in [GZ24] that under this definition, the volume of the Prym variety does not vary continuously in families, and consequently, the tropical Prym–Torelli map (assigning a Prym variety to a double cover) is not continuous.

Subsequent work by Röhrle and Zakharov ([RZ25]) introduced a "continuous" Prym variety ( $Prym_c$ ) and a "divisorial" Prym variety ( $Prym_d$ ). They demonstrated that $Prym_c$ varies continuously under edge contractions that increase the genus of the dilation subgraph, provided the double cover is either free or the dilation subgraph is connected. However, their construction relies purely on tropical combinatorics and does not account for the dimension drop that occurs when contracting multiple cycles in the presence of positive vertex weights (weighted metric graphs). Specifically, the standard dimension of the Prym torus ( $b_1(\tilde{\Gamma}) - b_1(\Gamma)$ ) is insufficient to maintain continuity when weights are involved, as the total genus of the cover changes in a way that requires an "extended" definition.

Methodology
The author proposes a novel, intrinsic construction of the continuous Prym variety for harmonic double covers of weighted metric graphs. The methodology proceeds through the following steps:

Definitions and Preliminaries: The paper formalizes harmonic morphisms for weighted metric graphs, defining the dilation subgraph ( $\Gamma_{dil}$ ) and the dilation index. It establishes the relationship between the weights of the base graph ( $w$ ) and the cover ( $\tilde{w}$ ), noting that the first Betti number of the cover is not constant but depends on the dilation structure.
Extended Prym Variety: The author defines the "extended Prym variety" ( $Prym_c(\pi_w)$ ) as an integral torus of dimension $g_{\tilde{\Gamma}} - g_{\Gamma}$ . This is achieved by taking the standard continuous Prym torus (defined via the image of $Id - \iota^*$ ) and adding $|\tilde{w}| - |w|$ dimensions to both the lattice of 1-forms and the lattice of homology cycles. This extension ensures the dimension matches the expected genus difference even when weights are present.
Intrinsic Basis Construction: A central technical contribution is the construction of an intrinsic basis for the lattice of anti-symmetric cycles ( $Im(Id - \iota^*)$ $I m (I d - ι^{*})$ ).
- Unlike previous constructions (e.g., [LZ22], [RZ25]) that degenerate bases from free covers or require specific spanning tree choices that may not degenerate well, this paper introduces Construction 1.
- This construction selects an "admissible" spanning tree $\tilde{T}$ for the cover $\tilde{\Gamma}$ relative to a spanning tree $T$ of the base $\Gamma$ . The choice of $\tilde{T}$ is flexible: it must contain the full preimage of the "positive" side of $T$ ( $\pi^{-1}(T)^+$ ) but allows for specific choices regarding the preimages of edges in $T$ that are not dilated.
- The basis elements $\beta_e$ are defined as anti-symmetric cycles ( $\tilde{\gamma}_e^- - \iota \tilde{\gamma}_e^-$ ) associated with edges $e$ not in $\tilde{T}$ .
Proof of Continuity: The author proves that this specific choice of basis behaves well under edge contractions. By showing that for any contraction of a subgraph $S$ , there exists a choice of admissible spanning tree in the pre-contraction graph that contracts exactly to the admissible spanning tree of the post-contraction graph, the author establishes that the basis of anti-symmetric cycles deforms continuously.

Key Contributions and Results

Theorem A (Continuity of the Map): The tropical Prym–Torelli map $P_g^{trop}: R_g^{trop} \to A_{g-1}^{trop, V}$ , which assigns to a harmonic double cover its extended continuous Prym variety, is continuous. This result holds for the general setting of weighted metric graphs, resolving the discontinuity issues found in previous definitions.
Theorem B (Theorem 3.1): The set of cycles constructed via Construction 1 forms a basis for the lattice defining the continuous Prym variety ( $Im(Id - \iota^*)$ ). This basis is proven to be independent and to generate the lattice for any choice of admissible spanning tree.
Intrinsic Construction: The paper provides a basis construction that is intrinsic to the double cover and does not rely on degenerating from a free cover. It explicitly handles the cases of dilated edges and $\pi$ -parallel loops, unifying the treatment of free and dilated covers.
Dimensional Consistency: The definition of the extended Prym variety correctly accounts for the change in dimension when contracting edges that increase the genus of the dilation subgraph, ensuring the resulting object remains a principally polarized tropical abelian variety of the correct dimension.

Significance and Claims
The paper claims to provide the first proof of the continuity of the tropical Prym–Torelli map in the general setting of weighted metric graphs. While the author notes that the result is "known to the experts," it had not yet appeared in the literature. The significance lies in:

Resolving Discontinuity: It fixes the failure of the standard Prym definition and extends the continuity results of Röhrle and Zakharov to cases involving positive weights and multiple cycle contractions.
Intrinsic Basis: The construction of the basis for anti-symmetric cycles is more flexible and robust under degeneration than previous methods, allowing for a direct proof of continuity without relying on limiting arguments from free covers.
Foundational Stability: By establishing the continuity of the Prym–Torelli map, the paper ensures that the moduli space of harmonic double covers maps continuously to the moduli space of principally polarized tropical abelian varieties, a necessary condition for the map to be a well-defined morphism in tropical geometry.

The work does not propose new applications or experimental setups but serves as a rigorous theoretical foundation for the study of Prym varieties in tropical geometry, ensuring that the geometric invariants behave consistently under the natural operations of the moduli space (edge contractions).

Technical Summary: Continuity of the Tropical Prym–Torelli Map

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