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Derived category of coherent systems on curves and stability conditions

This paper establishes an open locus of Bridgeland stability conditions on the derived category of coherent systems over a smooth projective curve of genus $g>0$ , demonstrating that the resulting stability manifold encodes the curve's Brill--Noether theory.

Original authors: Soheyla Feyzbakhsh, Aliaksandra Novik

Published 2026-08-03

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CC0 1.0

Original authors: Soheyla Feyzbakhsh, Aliaksandra Novik

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 a detective trying to solve a mystery, but the crime scene is a piece of math called a "curve." In the world of algebraic geometry, these curves are smooth, looping shapes that exist in higher dimensions. For decades, mathematicians have tried to understand the "DNA" of these shapes by studying the objects living on them, like bundles of strings or sheets of fabric. To do this, they use a powerful tool called a "stability condition." Think of a stability condition as a special set of rules or a ruler that tells you which objects are "well-behaved" (stable) and which are messy or chaotic.

Usually, when you look at a simple curve, the rules are so strict that there is only one possible way to measure stability. It's like trying to find a new way to measure the length of a table when you only have one ruler that fits perfectly; there's no room to wiggle, no room to discover anything new. This made it very hard to use these rules to learn deep secrets about the curve's shape. However, mathematicians have found a clever trick: instead of looking at the curve alone, they can look at a "coherent system." Imagine this not just as a single sheet of fabric, but as a sheet of fabric attached to a specific set of instructions or a small team of workers (a vector space) that are trying to hold it together. This combination creates a richer, more complex world where the old, rigid rules break down, and suddenly, there is a whole landscape of new ways to measure stability.

This paper, written by Soheyla Feyzbakhsh and Aliaksandra Novik, explores this new landscape. The authors show that if you switch your focus from simple curves to these "coherent systems," you unlock a vast, open space of stability conditions. They prove that this new space isn't just random; it is deeply connected to the curve's own history and geometry. Specifically, they discovered that the shape of this new "stability map" is controlled by a famous mathematical puzzle called the Brill–Noether theory, which asks how many ways you can draw lines or curves on a shape. The paper demonstrates that by moving through this new space, you can actually "see" the answers to these old puzzles. They didn't just guess this; they built a rigorous mathematical map showing exactly how these different stability rules fit together, proving that the geometry of the curve dictates the shape of the stability world.

The Story of the Curve and the System

To understand what the authors did, let's start with the basics. Imagine a smooth, closed loop drawn on a piece of paper. In math, this is a "smooth projective curve." For a long time, mathematicians tried to study these curves by looking at the "derived category" of coherent sheaves on them. In plain English, this is a giant library of all the possible shapes and bundles you can put on that curve. To make sense of this library, you need a "stability condition," which acts like a sorting machine. It decides which items in the library are "stable" (solid, unchanging) and which are "unstable" (likely to fall apart).

Here is the problem: for a simple curve, this sorting machine is incredibly boring. A previous result showed that no matter how you try to tweak the rules, there is essentially only one way to sort these items. It's like having a library where every book is exactly the same size and weight; you can't arrange them in any interesting way. Because there is only one way to sort them, you can't use the "walls" between different sorting methods (called wall-crossing) to learn new things about the curve.

The authors' big idea was to change the library. Instead of just looking at the bundles (the sheaves), they looked at "coherent systems." A coherent system is a triple: a bundle of strings (the sheaf), a team of workers (a vector space), and a map showing how the workers are trying to hold the strings. It's like taking a kite (the bundle) and attaching a specific set of strings and handles (the vector space) to it. The math of these systems is more complex, and the authors show that this complexity creates a whole new world of possibilities.

The Two Ways to Sort the World

The paper's main discovery is that this new world of stability conditions has a specific shape, and it can be described by two main types of "sorting rules." The authors proved that any valid way to sort these systems (that meets certain basic criteria) must fall into one of two categories.

Type A: The Glued Method
Imagine you have two separate rooms. One room contains only the "workers" (the vector spaces), and the other room contains the "bundles" (the sheaves on the curve). In Type A, the authors show that you can "glue" the stability rules from the workers' room and the bundles' room together to create a rule for the whole system. This works like a puzzle where you take a piece from one box and a piece from another and snap them together. However, there is a catch: you can only snap them together if the "angle" of the workers' room is tilted just right (mathematically, a specific number must be less than 1/2). If the angle is wrong, the glue doesn't hold, and this type of stability doesn't exist.

Type B: The Tilted Method
The second type of rule comes from "tilting" the whole system. Imagine you have a flat table with objects on it. If you tilt the table, some objects slide to the left, and some slide to the right. In math, this is called "tilting," and it changes the definition of what is "stable." The authors found that for these coherent systems, you can tilt the table at any angle you want, as long as you stay above a certain "ceiling." This ceiling is defined by a function called the Brill–Noether function. Think of this function as a mountain range that rises and falls depending on the curve's shape. You can only build your stability rules on the flat ground above this mountain. If you try to build below the mountain, the rules break.

The Map and the Mountain

The most exciting part of the paper is how these two types of rules connect to the curve itself. The authors show that the "mountain" (the Brill–Noether function) isn't just a random barrier; it is a direct reflection of the curve's geometry. The Brill–Noether theory is a classic area of math that studies how many independent ways you can map a curve to a line or a plane. The paper proves that the shape of the stability world is literally controlled by this theory.

They describe the stability world as a complex map made of two overlapping open areas, which they call $U_A$ and $U_B$ .

$U_A$ is the region where the "glued" rules work.
$U_B$ is the region where the "tilted" rules work.

These two regions overlap, meaning there are some stability rules that can be described in both ways. The authors provide a precise mathematical description of this map, showing exactly where the boundaries are. They also show that as you move to the "large volume limit" (a specific direction on the map where the numbers get very big), these new, complex rules slowly turn back into the old, classical rules that mathematicians have used for decades. This confirms that their new map is a natural extension of the old one, not a replacement.

Why It Matters

The paper doesn't just draw a pretty map; it solves a specific problem about how to study curves. By showing that the stability manifold (the map of all possible rules) is so closely tied to the Brill–Noether theory, the authors open the door to using these new stability tools to solve old problems. For example, they suggest that by walking across the "walls" in this new map, mathematicians can learn about the geometry of vector bundles on curves in ways that were impossible before.

The authors are very careful to state that they have proved the existence of this open locus and the classification of these two types of stability. They haven't just suggested it; they have built a rigorous framework that covers the entire open set of stability conditions they are interested in. They also note that there are other parts of the stability world they haven't explored yet (the "complement" of their open set), which they plan to tackle in future work. But for the part they studied, the picture is complete: the stability of coherent systems on a curve is a rich, two-dimensional landscape, shaped by the curve's own history, and accessible through either gluing or tilting.

In short, Feyzbakhsh and Novik took a rigid, one-dimensional problem and expanded it into a flexible, two-dimensional world. They showed that by adding a little bit of "worker" to the "bundle," you gain a whole new perspective on the curve, one that reveals deep connections to the curve's geometry and offers new tools for future mathematical adventures.

Technical Summary: Derived Category of Coherent Systems on Curves and Stability Conditions

Problem Statement
The paper addresses the scarcity of Bridgeland stability conditions on the bounded derived category of coherent sheaves, $D^b(C)$ , for a smooth projective curve $C$ of genus $g > 0$ . As established in [Mac07], the space of stability conditions on $D^b(C)$ (modulo the $\widetilde{GL}^+(2, \mathbb{R})$ -action) consists of a single point. This rigidity prevents the use of wall-crossing techniques to extract geometric information, such as Brill–Noether theory, directly from $D^b(C)$ . While previous approaches have embedded curves into higher-dimensional varieties (e.g., K3 surfaces) to utilize wall-crossing there, these methods restrict attention to special curves and may lose data under pushforward.

This work proposes an alternative: keeping the curve $C$ fixed but enlarging the category from coherent sheaves to coherent systems. A coherent system is a triple $(V, E, \phi)$ , where $V$ is a finite-dimensional vector space, $E$ is a coherent sheaf on $C$ , and $\phi: \mathcal{O}_C \otimes V \to E$ is a sheaf morphism. The authors study the bounded derived category of these systems, denoted $D(T_C)$ , and construct an open locus of stability conditions on this category that reflects the Brill–Noether theory of the curve.

Methodology
The authors employ two primary constructions to generate stability conditions on $D(T_C)$ :

Tilting Stability Conditions: Analogous to Bridgeland's construction on surfaces [Bri08], the authors start with the natural heart $T_C$ and define a slope function $\mu(T) = \deg(E)/\text{rk}(E)$ for objects with non-zero rank. By tilting the abelian category $T_C$ at a real parameter $b$ , they construct a new heart $A(b) = \langle \mathcal{F}_b[1], \mathcal{T}_b \rangle$ . They define a stability function $Z_{b,w}$ on this heart involving a parameter $w$ . The existence of these stability conditions is governed by a Brill–Noether function $\Phi_C(b)$ , defined as the limit superior of $h^0(F)/\text{rk}(F)$ for semistable sheaves of slope $b$ . The stability conditions exist in the region $w > \Phi_C(b)$ .
Gluing Stability Conditions: Utilizing the semiorthogonal decompositions of $D(T_C)$ , the authors glue stability conditions from the components.
- First Type: Based on the decomposition $D(T_C) = \langle [\mathcal{O}_C \to 0], \perp[\mathcal{O}_C \to 0] \rangle$ , where the orthogonal component is equivalent to $D(C)$ . They glue a trivial stability condition on the vector space component with a stability condition on $D(C)$ .
- Second Type: Based on the decomposition $D(T_C) = \langle \perp[\mathcal{O}_C \to \mathcal{O}_C], [\mathcal{O}_C \to \mathcal{O}_C] \rangle$ .

The paper rigorously verifies the support property and the Harder–Narasimhan property for these constructions, often using quadratic forms and induction on the imaginary part of the central charge.

Key Contributions and Results

Classification of Stability Conditions (Theorem 1.1 / 5.1): The authors prove that any stability condition $\sigma$ in a specific open locus $\text{Stab}^\circ(D(T_C))$ —defined by the stability of objects $[\mathcal{O}_C \to 0]$ and $[0 \to \mathcal{O}_x]$ for all $x \in C$ —is, up to the $\widetilde{GL}^+(2, \mathbb{R})$ -action, of one of two types:
- Type A: A gluing construction $\text{gl}^{(1)}(\sigma_V, \sigma_g)$ where the stability condition on $D(C)$ satisfies a specific phase constraint ( $f(0) < 1/2$ ).
- Type B: A tilting construction $\sigma_{b,w}$ parameterized by $(b, w)$ with $w > \Phi_C(b)$ .
  The intersection of these two loci is non-empty and explicitly described.
Structure of the Stability Manifold (Corollary 1.2 / 5.8): The open locus $\text{Stab}^\circ(D(T_C))$ is described as the union of two open sets, $U_A$ and $U_B$ . $U_B$ is identified with the region $\{b + iw \in \mathbb{C} \mid w > \Phi_C(b)\}$ modulo the group action, demonstrating that the stability manifold is directly controlled by the Brill–Noether function $\Phi_C$ .
Wall-Chamber Decomposition and Large Volume Limit (Section 6): The authors analyze the wall-and-chamber structure within the two-dimensional slice of Type B stability conditions. They show that as $w \to \infty$ (the large volume limit), the stability conditions recover the classical notion of $\alpha$ -stability for coherent systems (as studied in [LP93, BGPMN03]). Furthermore, they derive a Bogomolov-type inequality for $\mu_\alpha$ -semistable objects using the convexity of the stability region.
Second Open Locus (Theorem 1.3 / 7.1): The paper describes a second open locus $\widetilde{\text{Stab}}^\circ(D(T_C))$ where $[0 \to \mathcal{O}_C]$ and $[0 \to \mathcal{O}_x]$ are stable. They prove that stability conditions in this locus are either of Type A (gluing) or a second type of gluing $\text{gl}^{(2)}(\sigma_g, \sigma_V)$ where the phase constraint is $f(0) \geq 1/2$ .

Significance
The paper establishes that the derived category of coherent systems on a curve provides a natural framework for defining a rich space of stability conditions that encodes the Brill–Noether theory of the curve. Unlike the derived category of sheaves on the curve itself, which admits no deformations of stability, $D(T_C)$ admits a two-dimensional family of stability conditions. The geometry of this stability manifold is intimately tied to the curve's Brill–Noether function $\Phi_C$ .

The authors note that this work lays the groundwork for applying wall-crossing techniques in this two-dimensional slice to study the Brill–Noether theory of vector bundles on curves and the stability of kernels of evaluation maps (Butler's conjecture), which will be addressed in subsequent work. The results also connect to recent developments involving Kuznetsov components of blow-ups of Fano threefolds [AK25], identifying the category of coherent systems with these components.

The Story of the Curve and the System

The Two Ways to Sort the World

The Map and the Mountain

Why It Matters

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