---
title: Tropical Prym Variety
url: https://www.emergentmind.com/topics/tropical-prym-variety
type: topic
---

# Tropical Prym Variety

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The tropical Prym variety is the tropical analogue of the classical Prym variety attached to a cover of curves. In the foundational double-cover setting, it is constructed from a harmonic morphism of metric graphs by taking the connected component of the identity in the kernel of the norm map on Jacobians, and it can also be described as the anti-invariant part under the deck involution. Subsequent work has enlarged this picture from free double covers to dilated double covers, from divisorial to continuous and principalized models, and from double covers to unramified cyclic Galois covers. Across these variants, tropical Prym varieties appear as real tori or principally polarized tropical abelian varieties whose geometry is governed by divisor theory, harmonic morphisms, matroidal data, Abel–Prym maps, and non-Archimedean skeletons [1606.02282], [1902.09410], [2604.01574].

## 1. Definition and basic structure

For an unramified double cover of metric graphs
\[
\varphi:\widetilde{\Gamma}\to\Gamma,
\]
the tropical Prym variety is defined as the connected component of the kernel of pushforward on Jacobians containing \(0\):
\[
P(\varphi):=(\mathrm{Id}-\iota)\Pic^0(\widetilde{\Gamma}),
\]
where \(\iota\) is the involution exchanging the two sheets [1606.02282]. In the equivalent norm-map formulation used throughout later work, for a free double cover \(\pi:\widetilde{\Gamma}\to\Gamma\),
\[
\Prym(\widetilde\Gamma/\Gamma):=(\Ker \Nm)_0,
\]
with \(\Nm:\Pic^0(\widetilde\Gamma)\to\Pic^0(\Gamma)\) induced by pushforward of divisor classes [2012.15235].

This object is the tropical analogue of the classical Prym variety attached to a double cover of algebraic curves. In the free case, \(\ker\Nm\) has two connected components, and the identity component is the even Prym variety; the other is the odd Prym variety. Every Prym divisor class is linearly equivalent to one of the form \(E-\iota(E)\), and the parity of \(\deg(E)\) determines the connected component [2210.14060]. For a free double cover of a base graph of genus \(g\), the source has genus \(2g-1\), and the Prym has dimension \(g-1\) [2012.15235].

A lattice-theoretic description is also standard. In one formulation,
\[
\Prym(\widetilde{\Gamma}/\Gamma)=\bigl((\mathrm{Coker}\,\pi^*)^{tf},\,\Ker\pi_*,\,[\cdot,\cdot]_2\bigr),
\]
so the tropical Prym is realized as a tropical abelian variety built from the torsion-free part of the cokernel of pullback and the anti-invariant homology lattice [2412.06971]. In the free double-cover case this is a principally polarized tropical abelian variety.

## 2. Free and dilated covers, polarization, and refined notions

A harmonic double cover of metric graphs may be free or dilated. In a double cover, vertices and edges can be undilated or dilated, and the dilation subgraph \(G_{\dil}\subset G\) is the subgraph made of all dilated vertices and dilated edges. The number of connected components of this subgraph, denoted \(d(G_{\dil})\), is a basic invariant in the dilated theory [2303.03904].

The free case is the closest tropical analogue of an étale double cover. In that case the kernel of the norm has two connected components, and the identity component carries the principal polarization used to define the Prym variety. The dilated case is subtler. One line of development defines the tropical Prym variety for a dilated double cover as a connected torus with a natural principal polarization, even though the induced polarization from \(\Jac(\widetilde{\Gamma})\) behaves differently from the free case [2303.03904]. Another line of development distinguishes the divisorial Prym variety
\[
\Prym_d(\widetilde{\Gamma}/\Gamma)=(\Ker \pi)_0
\]
from the continuous Prym variety
\[
\Prym_c(\widetilde{\Gamma}/\Gamma)=\Big( \Omega^1_{\widetilde{\Gamma}}(\widetilde{\Gamma})/\Ker(\Id-\iota^*),\ \Im(\Id-\iota_*),\ [\cdot,\cdot]_P \Big),
\]
precisely because \(\Prym_d\) may fail to be principally polarized for general dilated covers, while \(\Prym_c\) always carries a natural principal polarization \(\zeta_c\) [2210.02267].

In this refined framework there is a canonical factorization
\[
\Jac(\widetilde{\Gamma}) \xrightarrow{\epsilon} \Prym_c(\widetilde{\Gamma}/\Gamma) \xrightarrow{\gamma} \Prym_d(\widetilde{\Gamma}/\Gamma) \hookrightarrow \Jac(\widetilde{\Gamma}),
\]
where \(\gamma\) is a free isogeny of degree \(2^{d-1}\), with \(d\) the number of connected components of the dilation subgraph. The induced polarization on \(\Prym_d\) has type \((1,\dots,1,2,\dots,2)\), with \(d-1\) ones, and the induced polarization on \(\Prym_c\) is exactly twice a principal polarization [2210.02267]. A related construction is the principalization \(\Prym_p(\widetilde{\Gamma}/\Gamma)\), used when one wants a principally polarized tropical abelian variety reconstructed directly from combinatorial data [2311.09872].

These distinctions matter whenever dilation changes under degeneration. The divisorial Prym behaves discontinuously under edge contractions when the number of connected components of the dilation subgraph changes, which is one reason the continuous Prym was introduced [2210.02267].

## 3. Combinatorics, volume formulas, and matroidal reconstruction

The tropical Prym variety admits explicit combinatorial volume formulas. For a free double cover of metric graphs, the analogue of Kirchhoff’s matrix-tree theorem is
\[
\Vol^2(\Prym(\widetilde\Gamma/\Gamma)) = \sum_{F} 4^{r(F)-1} w(F),
\]
where the sum runs over odd genus one decompositions \(F\subset E(\Gamma)\), \(r(F)\) is the rank of the decomposition, and
\[
w(F)=\prod_{e\in F}\ell(e)
\]
is the product of the corresponding edge lengths [2012.15235]. The factor \(4^{r(F)-1}\) records the Prym-specific multiplicity coming from the combinatorics of the cover.

For dilated double covers this formula acquires an additional prefactor determined by the dilation subgraph:
\[
\Vol^2(\Prym(\widetilde{\Gamma}/\Gamma))=2^{1-d(\widetilde{\Gamma}_{\dil})}\sum_{F\subset E(\Gamma)} 4^{r(F)-1}\prod_{e\in F}\ell(e),
\]
where the sum is over \(h\)-element ogods \(F\), with \(h=g(\widetilde{\Gamma})-g(\Gamma)\) [2303.03904]. Here an ogod is a set of undilated edges such that every connected component of \(\Gamma\setminus F\) has connected preimage in \(\widetilde{\Gamma}\), together with the stated genus constraints. This formula exhibits two separate combinatorial inputs: the connectivity of complements of edge sets and the topology of the dilation locus itself.

The matroidal formulation makes this structure more systematic. To a harmonic double cover \(\pi:\widetilde{\Gamma}\to\Gamma\) one associates a signed cographic matroid \(M^*(\widetilde{\Gamma}/\Gamma)\) and its dual signed graphic matroid \(M(\widetilde{\Gamma}/\Gamma)\). The bases of the dual matroid are exactly the ogods, and the principalization \(\Prym_p(\widetilde{\Gamma}/\Gamma)\) can be reconstructed from the signed graphic matroid together with an index function, an orientation, and edge lengths [2311.09872]. In this framework, the index function is
\[
\ind(F)= \text{number of connected components of } G \backslash F,
\]
and the pairing on the reconstructed ppav is
\[
[\gamma_{C_1}, \gamma_{C_2}] = \sum_{e\in E} 2\gamma_{C_1}(e) \gamma_{C_2}(e) \ell(e).
\]
The simplification theorem shows that natural matroid simplification does not change \(\Prym_p\) [2311.09872].

Low-genus metric invariants can be pushed further. For every double cover \(\pi:\widetilde{\Gamma}\to\Gamma\) with \(g(\Gamma)\le 4\), the second moment satisfies
\[
I_2(\Prym(\widetilde{\Gamma}/\Gamma)) = \frac{p(\widetilde{\Gamma}/\Gamma)+q(\widetilde{\Gamma}/\Gamma)}{12\sqrt{w_0(\widetilde{\Gamma}/\Gamma)}},
\]
where \(p\) is a polynomial term and \(q\) is a piecewise-polynomial term controlled by \(FS_n\)-sets and Friedman–Smith-type degenerations [2507.06401]. The appearance of \(q\), which has no Jacobian analogue, reflects wall-crossing of the Prym Voronoi polytope.

## 4. Abel–Prym maps and tropical \(n\)-gonal constructions

The Abel–Prym map is the basic geometric morphism from the cover into its Prym. In the degree-one form for a free double cover it is
\[
\Psi:\widetilde{\Gamma}\longrightarrow \Prym(\widetilde{\Gamma}/\Gamma),\qquad p\longmapsto [p-\iota(p)],
\]
and in degree \(d\) one has
\[
\Psi_d(p_1+\cdots +p_d) = [p_1+\cdots + p_d - \iota p_1 - \ldots - \iota p_d]
\]
from \(\Sym^d(\widetilde{\Gamma})\) to the appropriate Prym torsor [2412.06971], [2210.14060]. The degree-\(g-1\) Abel–Prym map is surjective, and its global degree is
\[
\deg(\Psi)=2^{g-1}
\]
in the free double-cover case [2012.15235].

Its local behavior is highly sensitive to graph combinatorics. For a top-dimensional cell indexed by a set \(F\) of distinct edges, the restriction has maximal rank exactly when \(F\) is an odd genus one decomposition, and then the cell degree is
\[
\deg_{\Psi}(F)=2^{r(F)-1};
\]
otherwise it vanishes [2012.15235]. A complementary formulation states that the Abel–Prym map of degree \(d\) is finite on a cell if and only if the complementary subgraph is relatively connected under the cover [2210.14060].

Hyperelliptic sources give a particularly rigid case. If \(\widetilde{\Gamma}\) is hyperelliptic, then
\[
\Psi:\widetilde{\Gamma}\to \Prym(\widetilde{\Gamma}/\Gamma)
\]
is harmonic of degree \(2\). There is also a partial converse: if \(\Psi\) is a finite harmonic morphism of degree \(2\), then \(\widetilde{\Gamma}\) is hyperelliptic [2412.06971]. In that situation the Abel–Prym graph \(\Psi(\widetilde{\Gamma})\) is a hyperelliptic metric graph of genus \(g_{\Gamma}-1\), and
\[
\Prym(\widetilde{\Gamma}/\Gamma)\cong \Jac(\Psi(\widetilde{\Gamma}))
\]
as principally polarized tropical abelian varieties [2412.06971].

The tropical \(n\)-gonal constructions place Pryms in a broader correspondence theory. For \(n=2\), the bigonal construction produces another tower of the same type and yields a duality
\[
\Prym_d(\widetilde{\Gamma}/\Gamma)^\vee \cong \Prym_d(\widetilde{\Pi}/\Pi)
\]
of polarized tropical abelian varieties, with the proof using \(\Prym_c\) in the dilated case [2210.02267]. For \(n=3\), the tropical trigonal construction takes a free double cover of a trigonal tropical curve to a tetragonal tropical curve \(\Pi\) and proves
\[
\Prym(\widetilde{\Gamma}/\Gamma)\cong \Jac(\Pi)
\]
as principally polarized tropical abelian varieties [2210.02267]. For \(n=4\), the tropical tetragonal construction yields a triality: under the orientability and fiber-type hypotheses of the tropical Donagi theorem, three towers have canonically isomorphic Prym varieties [2507.05987]. A direct consequence is non-injectivity of the tropical Prym–Torelli morphism [2507.05987].

## 5. Tropical Prym–Brill–Noether loci

Tropical Prym varieties support a Prym–Brill–Noether theory parallel to the algebraic theory. For a free double cover of metric graphs
\[
\pi:\widetilde{\Gamma}\to\Gamma,
\]
the tropical Prym divisor set is
\[
P(\Gamma,\pi):=\{[D]\in \mathrm{Pic}(\widetilde{\Gamma})\mid \mathrm{Nm}_\pi[D]=[K_\Gamma]\},
\]
and the tropical Prym–Brill–Noether locus is
\[
V^r(\Gamma,\pi):=\{[D]\in P(\Gamma,\pi)\mid \mathrm{rk}(D)\ge r \text{ and } \mathrm{rk}(D)\equiv r \pmod 2\}
\]
[2607.01173]. The model case is the loop of loops \(\Gamma\), a genus-\(g\) metric graph built from \(g-1\) loops connected in a cyclic pattern, together with its antipodal double cover.

For this cover, every Prym class has a canonical representative in one of two tori,
\[
P(\Gamma,\pi)=P^0\sqcup P^1,
\]
each a real torus of dimension \(g-1\). In explicit coordinates \(\xi_j(D)\), the Prym condition is
\[
\xi_{g-1+j}(D)=-1-\xi_j(D)
\]
[2607.01173]. When the edge lengths are generic, the main dimension statement is
\[
\dim V^r(\Gamma,\pi)=g-1-\binom{r+1}{2},
\]
the tropical analogue of the classical expected dimension.

The internal combinatorics of \(V^r(\Gamma,\pi)\) are encoded by Coxeter data. A divisor class \([D]\in P^r\) determines a Prym word \(Q_D\), and the tropical Prym–Brill–Noether locus decomposes as
\[
V^r(\Gamma,\pi)=\bigcup T(Q),
\]
where the union runs over lingering words \(Q\) of length \(g-1\) in \(A_r\) that contain a reduced subword for the longest element \(w_0\). Each \(T(Q)\) is a real torus, and
\[
T(Q)\subseteq T(Q') \iff Q' \text{ is a lingering subword of }Q
\]
[2607.01173]. This yields pure dimensionality, connectedness in codimension \(1\) when the dimension is positive, and a zero-dimensional point count equal to the number of reduced words for \(w_0\), hence to the number of standard Young tableaux of staircase shape \(\Delta_r\).

The tropical description is not merely internal to graph theory. A lifting theorem states that if \(\Gamma\) is an \(r\)-generic loop of loops and \(f:C\to C\) specializes to \(\pi:\widetilde{\Gamma}\to\Gamma\), then every divisor class in \(V^r(\Gamma,\pi)\) lifts to a divisor class in \(V^r(C,f)\). Consequences include equality of tropical and algebraic dimensions in the \(r\)-generic case, reducedness of \(V^r(C,f)\), and, in the extremal case \(g-1=\binom{r+1}{2}\), a bijection
\[
\mathrm{Trop}\colon V^r(C,f)\to V^r(\Gamma,\pi)
\]
[2607.01173]. This extends the earlier strategy in which tropical Prym–Brill–Noether loci on folded chains of loops were used to give dimension bounds and a tropical proof of the classical Prym–Brill–Noether theorem [1902.09410].

## 6. Tropicalization, moduli behavior, and extensions beyond double covers

The tropical Prym variety is compatible with non-Archimedean skeletons. For an unramified double cover of smooth projective curves over a non-Archimedean field,
\[
\pi:\widetilde X\to X,
\]
the skeleton of the algebraic Prym is canonically isomorphic to the tropical Prym of the induced tropical double cover:
\[
\mu_{X,\pi}:\Pr(\Gamma_X,\pi^{trop})\xlongrightarrow{\simeq} \Sigma\big(\Pr(X,\pi)\big).
\]
This is an isomorphism of principally polarized tropical abelian varieties, and it commutes with the Abel–Prym maps [1902.09410]. In this sense tropical Prym formation commutes with tropicalization just as Jacobian formation does.

At the same time, the tropical theory has moduli-theoretic pathologies not present in the simplest Jacobian setting. The volume of a dilated tropical Prym depends explicitly on the number of connected components of the dilation subgraph, so it can jump under deformations that change this number; this is a direct discontinuity statement for the tropical Prym variety [2303.03904]. The second moment exhibits a piecewise-polynomial correction term whose support is controlled by Friedman–Smith-type degenerations and linked to the indeterminacy of the Prym–Torelli map toward the second Voronoi compactification [2507.06401]. The tropical \(n\)-gonal construction is also poorly behaved under edge contractions, which prevents any immediate moduli-theoretic extension by continuity [2507.05987].

The double-cover formalism extends to unramified cyclic Galois covers. For an unramified cyclic cover
\[
\varphi:\widetilde{\Gamma}\to\Gamma
\]
of degree \(n\), with generator \(\tau\) of the cyclic Galois group, the tropical Prym variety is
\[
\Prym(\widetilde{\Gamma}/\Gamma):=\bigl(\ker \mathrm{Nm}_\varphi\bigr)^\circ,
\]
and the equivalent descriptions are
\[
\Prym(\widetilde{\Gamma}/\Gamma)=\ker(\mathrm{Nm}_G)^\circ = \operatorname{Im}(1-\tau).
\]
If the homology and 1-forms are decomposed into invariant and non-invariant parts, then
\[
\Prym(\widetilde{\Gamma}/\Gamma) = \frac{\Omega(\widetilde{\Gamma})^-{}^*}{H_1(\widetilde{\Gamma},\mathbb{Z})^-}
\]
and
\[
\Jac(\widetilde{\Gamma}) \sim \Prym(\widetilde{\Gamma}/\Gamma)\times \mu^*\Jac(\Gamma)
\]
up to isogeny [2604.01574]. In the free \(\mathbb Z_3\)-case one has the explicit volume identity
\[
\vol^2\!\bigl(\Prym(\widetilde{\Gamma}/\Gamma)\bigr) = \frac{\vol^2\!\bigl(\Jac(\widetilde{\Gamma})\bigr)}{3\,\vol^2\!\bigl(\Jac(\Gamma)\bigr)},
\]
which generalizes the double-cover story [2604.01574].

Taken together, these results show that the tropical Prym variety is not a single rigid construction but a family of closely related tropical abelian objects. In the free double-cover case it is a principally polarized tropical abelian variety with a remarkably explicit divisor theory. In the dilated case it requires refined choices of polarization and connected component. In special graph models it admits Coxeter-theoretic and matroidal descriptions, and under tropicalization it captures the skeleton of the algebraic Prym exactly [2311.09872], [2607.01173], [1902.09410].

Source: https://www.emergentmind.com/topics/tropical-prym-variety