Prym-Brill-Noether Theory for General Covers

Abstract: We bound the dimension of the Prym-Brill-Noether variety for an open subset of the moduli space of étale double covers of k-elliptic curves. We also obtain new bounds on the dimension of the Prym-Brill-Noether variety for general étale double covers of k-gonal curves, disproving a conjecture of Creech, Len, Ritter, and Wu, and provide a new conjecture for its dimension. To do this, we completely describe the Prym-Brill-Noether variety of a double cover of a certain tropical curve known as the loop of loops. We use the combinatorics of Coxeter groups to establish several topological properties of these tropical Prym-Brill-Noether varieties, and prove a lifting result when the edge lengths are generic.

• The paper establishes new optimal bounds for the dimensions of Prym-Brill-Noether varieties for k-gonal and k-elliptic covers.
• It employs tropical geometry using the innovative ‘loop of loops’ approach and Coxeter group combinatorics to analyze the structure of the Prym loci.
• The results refute a previous conjecture and extend lifting theorems, bridging classical and tropical Prym theory with practical implications.

Prym-Brill-Noether Theory for General Covers: An Expert Analysis

Introduction and Context

The paper "Prym-Brill-Noether Theory for General Covers" (2607.01173) undertakes a detailed study of the dimensions and structural properties of Prym-Brill-Noether varieties associated with étale double covers of algebraic curves, extending classical results to $k$-gonal and $k$-elliptic loci. This research operates at the intersection of algebraic, tropical, and combinatorial Brill-Noether theory, and makes substantial use of tropical techniques—particularly via the analysis of double covers of so-called "loop of loops" tropical curves. The author provides optimal or improved upper bounds for the dimensions of the associated Prym-Brill-Noether loci, derives new structural theorems using Coxeter group combinatorics, and disproves a recent conjecture from the literature, presenting a new, more precise conjectural framework.

Structure of Prym-Brill-Noether Varieties and Classical Results

Given a smooth curve $C$ of genus $g$ and an étale double cover $f : \widetilde{C} \to C$, the Prym variety arises as the connected component of the norm-zero divisor classes in $\mathrm{Pic}^0(\widetilde{C})$ that support the canonical class under the norm map. The associated Brill-Noether loci $V^r(\widetilde{C},f)$ parameterize divisor classes with at least $r$ sections, subject to a parity constraint dictated by the cover. Classical results (Welters, Bertram) show that for a general étale double cover in $\mathcal{R}_g$, the dimension of these Brill-Noether loci satisfies

$\dim V^r(\widetilde{C}, f) = g-1 - \binom{r+1}{2}.$

The present work gives a new proof of this statement using tropical and combinatorial tools.

Bound for the Dimension of Prym-Brill-Noether Varieties on $k$0-Gonal and $k$1-Elliptic Loci

A primary contribution of the paper is the derivation of sharp upper bounds for the dimension of $k$2 for general covers where $k$3 is $k$4-gonal or $k$5-elliptic, that is, when $k$6 admits a degree $k$7 map to $k$8 or to an elliptic curve, respectively.

The crucial theorems establish that for a general étale double cover in the $k$9-elliptic locus (non-isogenous, degree $C$0 map to a genus 1 curve), the Prym-Brill-Noether variety satisfies

$C$1

with equality for $C$2. For general double covers in the $C$3-gonal locus, a piecewise-defined bound is given, refining those in previous literature and disproving a conjecture in [CLRW20].

The proof strategy hinges on a fine analysis of tropicalizations of the relevant varieties, leveraging combinatorics of Coxeter groups to encode the geometry of the tropical Prym-Brill-Noether loci.

Tropical Approach via the “Loop of Loops”

The research replaces the "chain of loops" (ubiquitous in tropical Brill-Noether theory) with the "loop of loops," a metric graph whose combinatorics underlie the tropicalization of $C$4-elliptic and $C$5-gonal loci. Notably, while the chain of loops serves classical Brill-Noether calculations, the author shows the loop of loops is better suited for Prym-Brill-Noether settings, particularly due to more tractable combinatorial models and coordinate systems on the Prym locus. These are indexed by “Prym words” (special lingering words in type $C$6 Coxeter groups subject to symmetry relations).

A central result is that the tropical Prym-Brill-Noether variety $C$7 for a loop of loops $C$8 with antipodal involution $C$9 decomposes as a union of subtori, indexed by lingering words of length $g$0 in $g$1 containing a reduced word for $g$2, the maximal Coxeter group element. The structure and inclusion relations among these tori are captured by a “lingering subword poset”. The paper establishes the purity and connectivity in codimension one of the tropical Prym-Brill-Noether variety, and identifies cases of finite tori corresponding to the number of reduced words for $g$3, with a precise count given by Stanley’s enumeration via the hook-length formula.

Lifting, Degenerations, and Scheme-Theoretic Properties

An important technical accomplishment is the establishment of a lifting theorem: Every divisor class in $g$4 lifts to a divisor class in $g$5 for a specialization $g$6 of the double cover, generalizing known results to this broader context. The proof combines a scheme-theoretic analysis of degeneracy loci (of type D) and an application of Rabinoff’s lifting theorem, made possible by verifying the necessary local complete intersection structure of the tropical variety.

As corollaries, the work proves that the classical Prym-Brill-Noether loci for general curves in these loci are reduced and has the anticipated dimension, and in the finite case, the tropicalization map is bijective.

Combinatorial and Theoretical Implications

The tropological viewpoint grounded in Coxeter group combinatorics yields new insight into the structure and potential pathologies of Prym-Brill-Noether loci, especially in the presence of special coverings. The approach systematically describes the combinatorial indices for cohomology positions in the Prym variety, exposing intricate connections to representation theory (standard Young tableaux, Schubert calculus, and symmetric functions). These results open avenues for the application of symmetric group and Coxeter combinatorics in the context of algebraic and tropical Prym theory.

Moreover, the disproof of [CLRW20, Conj 3.9] motivates a more nuanced conjecture on the dimension of Prym-Brill-Noether loci for general $g$7-gonal covers, formulated precisely in this paper (Conjecture 1.2), and justified by a detailed analysis of orthogonal splitting types via pushforward vector bundles.

Contrasts, Generalizations, and Special Cases

The analysis extends to the bielliptic ($g$8) case, which is shown to be governed by Coxeter groups of type $g$9 (dihedral groups), and all main topological features (purity, finiteness, enumeration) are deduced in this low-degree setting. The techniques also allow fine bounds for $f : \widetilde{C} \to C$0-uniform, non-generic torsion profiles, highlighting subtle differences in the behavior of Prym-Brill-Noether loci beyond the generic regime.

Speculation and Future Developments

The methodology used in this work suggests several directions for expansion. The combinatorial techniques lend themselves to the potential classification of other degeneracy loci in tropical Prym theory, with specific conjectures on dimension and local equations for small $f : \widetilde{C} \to C$1 or non-generic edge length profiles. Furthermore, the interplay between orthogonal bundle splitting types and tropical geometry described here could illuminate further relationships between classical and tropical moduli problems (e.g., Schubert-type problems for moduli of covers, degeneration of abelian varieties), as well as applications in arithmetic Brill-Noether theory.

Additionally, there is scope for extending comprehensive lifting theorems to more general degenerations, further linking the combinatorial structure of tropical Prym varieties to scheme-theoretic and logarithmic geometry, especially as tropical period maps become better understood.

Conclusion

"Prym-Brill-Noether Theory for General Covers" realizes an overview of tropical and algebraic geometry with deep combinatorial representation theory, producing new effective bounds, structural theorems, and conjectures for the dimension and geometry of Prym-Brill-Noether varieties associated to general étale double covers. The use of the loop of loops and Coxeter group indices provides a powerful toolkit for future research on degenerations of abelian varieties and special moduli loci, and introduces precise counterexamples and sharp conjectural reformulations for classical problems in Prym theory.