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Scrollar invariants of singular curves on toric surfaces

This paper generalizes a combinatorial formula for calculating scrollar invariants of smooth curves to general integral curves of fixed geometric genus on a large class of toric surfaces, while also identifying specific cases where the expected behavior fails on certain components of the Severi variety.

Original authors: Karl Christ, Xiang He, Ilya Tyomkin

Published 2026-08-17

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CC BY 4.0

Original authors: Karl Christ, Xiang He, Ilya Tyomkin

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

Imagine you are standing in a vast, magical garden where the plants don't just grow; they follow strict, invisible blueprints drawn on a grid. This is the world of algebraic geometry, a branch of mathematics where shapes like curves and surfaces are defined by equations. In this garden, there are special "toric surfaces," which are like gardens built from a specific type of lattice (a grid of points) where the rules of symmetry make the plants grow in very predictable patterns. Now, imagine you want to take a picture of one of these complex, winding vines (a curve) and project its shadow onto a simple straight line. This process is called a "monomial projection."

When you project a complex curve onto a line, the curve doesn't just flatten out; it folds and twists in a specific way. Mathematicians call the "folding pattern" of this curve a set of "scrollar invariants." Think of these invariants as the unique fingerprint of how the curve is wrapped around the line. If the curve is smooth and perfect, we already have a recipe to predict this fingerprint. But what happens if the curve is knotted, broken, or has sharp corners (singularities)? Does the fingerprint change? Does the folding pattern become chaotic, or does it still follow a hidden order? This is the mystery the paper sets out to solve.

The authors, Karl Christ, Xiang He, and Ilya Tyomkin, dive into this question by looking at "singular curves" on these special grid-based surfaces. They ask: If we take a general curve from a family of such curves (some smooth, some knotted) and project it, what will its folding pattern look like?

Their main discovery is a bit like finding a "best-case scenario" for these folding patterns. They prove that for a large class of these grid-based gardens (specifically those called "h-transverse polygons"), there is always at least one family of curves where the folding pattern is as balanced and "tight" as mathematically possible. They call this the "minimal element." It's as if, no matter how messy the garden gets, there is always a way to arrange the vines so they fold in the most efficient, symmetrical way possible.

However, the story isn't a simple "everything is perfect." The authors also show that this perfect balance isn't guaranteed for every family of curves. If the garden's blueprint (the polygon) lacks a specific feature—a horizontal side—then some families of curves might fold in a less balanced, more "lopsided" way. They provide examples where the folding pattern is not the minimal one, proving that the "perfect balance" rule has exceptions.

To solve this, the team uses a clever trick called "tropical geometry." Imagine taking the complex, curved vines and melting them down into a simplified, stick-figure version made of straight lines and sharp corners. This "tropical" version is much easier to analyze. They build a specific stick-figure model that represents the most balanced folding pattern they are looking for. Then, using a powerful mathematical bridge, they show that this simple stick-figure model can be "lifted" back up into the real, complex garden. This proves that the perfect, balanced folding pattern actually exists in the real world of these curves.

The paper confirms that while the "perfect balance" isn't universal for every single curve family, it is always achievable for at least one family within the garden. Furthermore, if the garden's blueprint has a horizontal side, then every family of curves will fold in this perfect, balanced way. This work extends a previous formula that only worked for smooth, perfect curves, now applying it to the messy, knotted, and singular curves that make up the real complexity of the mathematical garden.

Technical Summary: Scrollar Invariants of Singular Curves on Toric Surfaces

Problem Statement
The paper investigates the scrollar invariants of finite covers $f: C^\nu \to \mathbb{P}^1$ induced by monomial projections from singular integral curves $C$ on polarized toric surfaces $(S_\Delta, \mathcal{O}(\Delta))$ . Given a curve $C$ in a linear system $|\mathcal{O}(\Delta)|$ , its normalization $C^\nu$ admits a degree- $d$ map to $\mathbb{P}^1$ via the $x_1$ -coordinate projection. The scrollar invariants $e(C, f) = (e_1, \dots, e_{d-1})$ are defined by the splitting type of the vector bundle $f_* \mathcal{O}_{C^\nu} = \mathcal{O}_{\mathbb{P}^1} \oplus \bigoplus_{i=1}^{d-1} \mathcal{O}_{\mathbb{P}^1}(-e_i)$ .

While the invariants for smooth curves were determined by Castryck and Cools [CC17] via a combinatorial formula involving the lattice width of $\Delta$ , the behavior for singular curves is more complex. On a given Severi variety $V^{irr}_{g, \Delta}$ (parametrizing integral curves of geometric genus $g$ ), the scrollar invariants are not necessarily constant. The central question is to determine the scrollar invariants for a general curve in an irreducible component of the Severi variety, particularly for $h$ -transverse polygons, and to understand how these invariants vary across different components.

Methodology
The authors employ tropical geometry, specifically the framework of parametrized tropical curves and floor diagrams developed in their previous works [CHT22, CHT23, CHT26a, CHT26b]. The proof strategy relies on the following steps:

Semicontinuity and Minimality: The scrollar invariants are upper semicontinuous with respect to the specialization relation (a partial order on sequences). To prove that a specific set of invariants $e_{g, \Delta}$ is realized by a general curve in a component $V$ , it suffices to find a single curve $C \in V$ whose invariants are "minimal" (i.e., $e(C, f) \preceq e_{g, \Delta}$ ).
Tropical Rank Calculation: The authors construct a specific parameterized tropical curve $\Gamma$ (a simple stretched floor-decomposed or ssfd curve) and a divisor $D_{trop}$ such that the Baker-Norine rank of multiples of $D_{trop}$ is bounded by the values prescribed by the target invariants. This establishes an upper bound on the invariants for any algebraic curve tropicalizing to $(\Gamma, D_{trop})$ .
Liftability via Relative Approach: The core technical challenge is lifting the tropical curve $(\Gamma, D_{trop})$ $(Γ, D_{t r o p})$ to an algebraic curve in the Severi variety. The authors utilize a relative approach to liftability [CHT26a, CHT26b]. They demonstrate that the constructed tropical curve is "sw-equivalent" (via specific modifications of floor diagrams) to a curve that is known to be liftable.
- A key innovation is the reformulation of liftability conditions in terms of floor diagrams. They prove that for $h$ -transverse polygons, the relevant floor diagrams are mod-equivalent to "prime-to- $p$ " diagrams (where edge multiplicities are not divisible by the characteristic of the base field).
- By results of Tyomkin [Tyo12], prime-to- $p$ tropical curves are liftable to algebraic curves.
Handling Non-Primitive Covers: The authors address cases where the projection is not primitive (i.e., factors through an intermediate cover). They show that if the polygon $\Delta$ lacks a horizontal side, the scrollar invariants may not be minimal on all components of the Severi variety, providing explicit counterexamples.

Key Contributions and Results

Generalization of Castryck-Cools Formula: The paper extends the Castryck-Cools formula [CC17] to singular curves and arbitrary characteristics, providing an upper bound on the scrollar invariants of $f: C^\nu \to \mathbb{P}^1$ in terms of the lattice width invariants of $\Delta$ (Corollary 4.4).
Main Theorem (Theorem 5.4): For an $h$ $h$ -transverse polygon $\Delta$ $Δ$ of height $d \ge 2$ $d \geq 2$ :
1. There exists at least one irreducible component $V \subseteq V^{irr}_{g, \Delta}$ such that the scrollar invariants of a general curve in $V$ are equal to the unique minimal element $e_{g, \Delta}$ of the set of admissible invariants.
2. If $\Delta$ possesses a horizontal side, then every irreducible component $V \subseteq V^{irr}_{g, \Delta}$ has scrollar invariants equal to $e_{g, \Delta}$ .
Counterexamples to Universality: The paper provides examples (Example 5.1, Remark 5.2) showing that if $\Delta$ has no horizontal side, the scrollar invariants can vary between components of the Severi variety, and the minimal invariants may not be realized on all components. This highlights the necessity of the "horizontal side" assumption for the second part of the Main Theorem. The authors note that while the second claim can fail without a horizontal side, they are not aware of any example where the first claim (existence of at least one component with minimal invariants) fails.
Realizability: As a consequence, the paper establishes the realizability of the minimal scrollar invariants $e_{g, \Delta}$ , recovering and extending realizability results from Vakil and Vemulapalli [VV26] to all characteristics and singular curves.

Significance
This work represents the first application of tropical methods to determine scrollar invariants for general singular curves on toric surfaces. It bridges the gap between the combinatorial understanding of smooth curves and the geometric complexity of singular curves in Severi varieties.

The paper claims significance in several specific areas:

Extension of Known Results: It generalizes the Castryck-Cools formula to singular curves and arbitrary characteristic, and extends Redigolo's [Red26] results on Hirzebruch surfaces to the broader class of $h$ -transverse polygons.
Tropical-Liftability Framework: It demonstrates the power of the "relative approach" to liftability, showing that even in small characteristics where previous realizability results (e.g., [Tyo12], [ABBR15]) do not apply, liftability can be established via sw-equivalence to prime-to- $p$ diagrams.
Clarification of Moduli Behavior: It clarifies the behavior of Maroni loci (loci of covers with fixed scrollar invariants) in the context of toric surfaces, showing that while a minimal invariant is always realizable, the uniformity of invariants across the entire Severi variety depends on the geometry of the polarization (specifically the presence of horizontal sides).

The authors note that the interaction between the stratification by scrollar invariants (Maroni loci) and the stratification by splitting types of line bundles (Brill-Noether loci) remains an open question, which this paper addresses as a first step in a planned series.

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