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Semiorthogonal indecomposability for Hilbert schemes of points on integral locally planar curves

This paper proves that for an integral projective curve with locally planar singularities, the categories of perfect complexes and bounded derived coherent sheaves on its Hilbert schemes of points are semiorthogonally indecomposable for all $1 \leq n \leq g-1$ , a result that extends to relative families over a connected base and holds in arbitrary characteristic.

Original authors: Qingyuan Jiang, Xun Lin

Published 2026-08-13

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CC BY 4.0

Original authors: Qingyuan Jiang, Xun Lin

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 a detective trying to solve a mystery about the hidden structure of shapes. In the world of mathematics, specifically a branch called algebraic geometry, scientists study shapes defined by equations. These aren't just simple circles or squares; they are complex, multi-dimensional landscapes that can twist, turn, and even have "kinks" or sharp points where they aren't perfectly smooth. One of the biggest questions mathematicians ask is: "Is this shape a single, unified whole, or is it secretly made of smaller, independent building blocks glued together?"

To answer this, mathematicians use a powerful tool called a "derived category." Think of this not as a picture of the shape, but as a massive library containing every possible way to describe the shape using algebra. If a shape is "decomposable," it means this library can be split into two separate, non-interacting sections, like a bookshelf where the left side only holds mystery novels and the right side only holds cookbooks, with no overlap. If a shape is "indecomposable," the library is a chaotic, interconnected mess where you can't split it apart without breaking the rules of the universe. This paper focuses on a specific type of shape: the "Hilbert scheme of points." If you imagine a curve (like a twisted wire) and ask, "How many ways can I pick $n$ dots on this wire?" the answer forms a new, complex shape. The question is: when you have a curve with some rough spots (singularities), does the shape formed by picking $n$ dots break apart into smaller pieces, or does it stay as one solid, indivisible unit?

The authors, Qingyuan Jiang and Xun Lin, tackle this puzzle for curves that are "integral" (they don't fall apart into separate pieces) and have "locally planar singularities" (their rough spots look like crumpled sheets of paper rather than tangled knots). They prove a very specific and strict rule: if you pick a number of dots, $n$ , that is between 1 and the curve's "genus" minus 1 (where genus is a measure of how many holes or loops the curve has), then the shape formed by these dots is indecomposable. In other words, the library of descriptions for this shape cannot be split into two separate, non-interacting sections. It is a single, unified whole.

The paper doesn't just say this is true for one specific curve; it proves it for any such curve, no matter what the rough spots look like, and it works even if the math is done in "arbitrary characteristic" (a technical way of saying the proof holds true regardless of the specific number system used, even if it's not the standard one we use in school). They also show that this holds true even when you have a whole family of these curves changing smoothly over time, rather than just looking at a single frozen moment.

To understand how they found this, imagine the shape formed by the dots as a landscape. The mathematicians looked for "obstructions" that would force the shape to break apart. They used a clever trick involving "paracanonical sections," which you can think of as special flashlights that can shine on different parts of the landscape. If the shape were breakable, these flashlights would be forced to shine only on specific, limited areas, leaving other parts in total darkness (a "base locus"). However, the authors showed that by moving these flashlights around—specifically by shifting the "theta divisor" (a special boundary line on the curve) and the "incidence divisor" (the area where the dots touch a specific point)—they could always find a way to shine a light on any point they chose. They proved that for the range $1 \le n \le g-1$ , there is no spot on the landscape that is permanently in the dark. Because every point can be illuminated, the shape cannot be split into separate, isolated islands.

The paper is very confident in this result; it is a rigorous mathematical proof, not a guess or a simulation. They explicitly rule out the possibility that these shapes are decomposable in the range they studied. In fact, they note that if you pick a number of dots $n$ that is larger than $g-1$ (specifically $n \ge g$ ), the story changes completely: the shape does break apart into a known collection of smaller pieces. But for the "sweet spot" where $n$ is between 1 and $g-1$ , the shape remains stubbornly, beautifully whole. This result settles a question that had been open for a while, confirming that the "categorical minimality" (the idea that the shape is a fundamental, unbreakable unit) holds true even when the underlying curve is rough and imperfect.

Technical Summary: Semiorthogonal Indecomposability for Hilbert Schemes of Points on Integral Locally Planar Curves

Problem Statement
The paper addresses the question of semiorthogonal decomposability for the derived categories of Hilbert schemes of points, $\text{Hilb}_n(C)$ , where $C$ is an integral projective curve with locally planar singularities. Specifically, it investigates whether the categories $\text{Perf}(\text{Hilb}_n(C))$ and $D^b_{\text{coh}}(\text{Hilb}_n(C))$ admit nontrivial semiorthogonal decompositions. This extends previous work on smooth curves and symmetric products to the singular setting, where the Hilbert scheme is generally singular. The study focuses on the range $1 \le n \le g-1$ , where $g$ is the arithmetic genus of the curve, a range where the variety is expected to exhibit categorical minimality.

Methodology
The authors employ a geometric obstruction approach based on paracanonical base loci, refining techniques introduced by Kawatani and Okawa [KO18] and Lin [Lin26]. The methodology proceeds through three main technical steps:

Derived Projectivization and Canonical Bundle Formula:
The authors utilize the identification of $\text{Hilb}_n(C)$ with the derived projectivization of a perfect complex $\mathcal{E}_n$ over the compactified Jacobian $J = \text{Pic}^0(C)$ . This construction, based on [Jia25b], allows for the derivation of an explicit formula for the dualizing line bundle $\omega_{\text{Hilb}_n(C)}$ :
$\omega_{\text{Hilb}_n(C)} \simeq \mathcal{O}_{\text{Hilb}_n(C)}((g-1-n)D_p) \otimes a_n^* \mathcal{O}_J(\Theta_p)$
Here, $D_p$ is the divisor of subschemes containing a fixed smooth point $p$ , $a_n$ is the Abel map, and $\Theta_p$ is a theta divisor. This formula holds even when the virtual rank of $\mathcal{E}_n$ is negative, provided $C$ is locally planar.
Cohomological Shifting of Divisors:
To prove the emptiness of the paracanonical base locus, the authors adapt an argument from [AK80]. For any point $Z \in \text{Hilb}_n(C)$ , they choose a smooth point $p \notin \text{Supp}(Z)$ . Using Lemma 2.2, they demonstrate the existence of a line bundle $M \in \text{Pic}^0(C)$ such that twisting the relevant sheaf by $M$ eliminates global sections (specifically, ensuring $H^0(C, F_Z \otimes M) = 0$ ). This allows the theta divisor $\Theta_p$ to be translated via the action of $M$ so that it avoids the image of $Z$ under the Abel map.
Singular Paracanonical Criterion:
The paper establishes a singular version of Lin's criterion (Proposition 3.1). It asserts that for an integral projective Gorenstein variety $X$ , if the paracanonical base locus $\text{PBs}|\omega_X|$ is empty, then neither $\text{Perf}(X)$ nor $D^b_{\text{coh}}(X)$ admits a nontrivial semiorthogonal decomposition. The proof relies on Spence's Cohen–Macaulay argument [Spe22] and the rigidity of semiorthogonal components under algebraically trivial twists, valid over algebraically closed fields of arbitrary characteristic.

Key Contributions and Results

Main Theorem (Theorem 1.1): For an integral projective curve $C$ of arithmetic genus $g$ with locally planar singularities over an algebraically closed field, and for every integer $1 \le n \le g-1$ , the categories $\text{Perf}(\text{Hilb}_n(C))$ and $D^b_{\text{coh}}(\text{Hilb}_n(C))$ are semiorthogonally indecomposable.
Relative Statement (Theorem 4.2): The result extends to flat families of such curves over a connected base $S$ . Specifically, $\text{Perf}(\text{Hilb}_n(C/S))$ admits no nontrivial $S$ -linear semiorthogonal decomposition, and $D^b_{\text{coh}}(\text{Hilb}_n(C/S))$ admits no nontrivial $S$ -linear decomposition with admissible components.
Characteristic Independence: The proofs hold in arbitrary characteristic, avoiding assumptions of characteristic zero often found in classical results on compactified Jacobians and Abel maps.
Sharpness: The paper notes that the result is sharp; for $n \ge g$ , semiorthogonal decompositions do exist, as demonstrated by the locally planar specialization of results in [Jia25b].

Significance
The paper settles the absolute case of the indecomposability question for Hilbert schemes on singular curves, a problem previously posed in [Jia25b, Section 4.2.3]. By establishing that the paracanonical base locus is empty for these singular schemes, the authors provide a robust geometric obstruction to decomposability that applies to singular varieties. This work unifies the categorical behavior of Hilbert schemes on smooth and singular curves within the range $1 \le n \le g-1$ , reinforcing the connection between categorical minimality and the birational geometry of the underlying variety. The relative version further extends these insights to families, providing a tool for studying semiorthogonal decompositions in moduli-theoretic contexts.

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