arxiv: 2604.21242 · v1 · submitted 2026-04-23 · 🧮 math.AG

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Two New Extensions of Reider's Theorem on Algebraic Surfaces

Aaron Bertram , Jonathon Fleck , Liebo Pan , Joseph Sullivan

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Pith reviewed 2026-05-09 21:08 UTC · model grok-4.3

classification 🧮 math.AG

keywords Reider's theoremBridgeland stabilitynef divisorsHilbert schemesalgebraic surfacesBayer-Macri theoremblow-upsample cone

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The pith

Reider-type inequalities imply nefness of divisors of the form dH minus E on blow-ups of projective space and give sharp bounds on the ample cone of Hilbert schemes parametrizing length-d subschemes of algebraic surfaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends Reider's theorem on very ampleness of adjoint linear series in two directions on complex projective algebraic surfaces. Reider-type numerical inequalities are shown to imply that certain linear series remain nef after blowing up projective space along an embedded surface. The same inequalities are used to locate the boundary of the ample cone on the Hilbert scheme of length-d zero-dimensional subschemes. Both results are obtained by constructing a natural family of objects on the surface whose semistability in a large chamber of the Bridgeland stability manifold produces nefness of a determinant line bundle on the base via the Bayer-Macri theorem.

Core claim

Reider-type inequalities imply nefness of linear series of the form dH - E on the blow-up of projective space along the embedded surface and give a sharp estimate for the ample cone of the Hilbert schemes of length d subschemes of the surface. The proofs consist of finding a natural family of objects parametrized by the base (either the blow-up along the surface or the Hilbert scheme) and finding the largest chamber in the stability manifold of the surface where the objects in the family are all Bridgeland semistable. A theorem of Bayer-Macri then gives nefness of the determinant line bundle on the base of the family.

What carries the argument

The natural family of objects parametrized by the blow-up or Hilbert scheme that remains Bridgeland semistable throughout the largest chamber of the stability manifold on the surface, which then yields nefness of the determinant line bundle via the Bayer-Macri theorem.

If this is right

Nefness of the divisor dH - E on the blow-up follows directly once the Reider-type inequalities are verified.
The ample cone of the Hilbert scheme of length-d subschemes is bounded sharply by the same numerical conditions.
The method produces nef classes on the base space without computing the full Mori cone directly.
An analogy is obtained with Saint-Donat's theorem on generators of ideals of curves embedded by adjoint series.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same chamber-stability technique could locate nef cones on other moduli spaces of sheaves or subschemes once suitable families are identified.
Numerical checks of Reider-type inequalities might become algorithmic tools for deciding positivity on blow-ups and Hilbert schemes.
The approach suggests that Bridgeland stability can serve as a uniform replacement for classical vanishing theorems when proving nefness.

Load-bearing premise

A natural family of objects parametrized by the blow-up or Hilbert scheme exists and remains Bridgeland semistable throughout the largest chamber of the stability manifold on the surface.

What would settle it

An explicit surface, divisor, and integer d where the Reider-type inequalities hold but the corresponding divisor on the blow-up fails to be nef or the ample cone bound on the Hilbert scheme is violated.

Figures

Figures reproduced from arXiv: 2604.21242 by Aaron Bertram, Jonathon Fleck, Joseph Sullivan, Liebo Pan.

**Figure 2.** Figure 2: The region for I(OS(−C)) when p ∈ C is destabilizing Applications. (a) OS is (x, y) stable for all y > 0 and (x, y) in the parabola. More generally, if F is H-stable and the H-Bogomolov inequality is an equality, then F is (x, y) stable for all y > −µ(F) and x > 1 2 y 2 . Similarly, in that case F ∨[1] is (x, y) stable for all y < µ(F). The reason for this is simple. There is no room to place I(E) between … view at source ↗

read the original abstract

Reider's Theorem on the very ampleness of adjoint linear series on a complex projective algebraic surface is extended in two new directions. First, Reider-type inequalities are shown to imply nefness of linear series of the form dH - E on the blow-up of projective space along the embedded surface. This can be thought of as a weak analogy of Saint-Donat's Theorem on the generators of the ideal of a curve embedded by an adjoint linear series. Next, Reider-type inequalities give a sharp estimate for the ample cone of the Hilbert schemes of length d subschemes of the surface. The proofs consist of (a) finding a natural family of objects parametrized by the base (either the blow-up along the surface or the Hilbert scheme) and (b) finding the largest chamber in the stability manifold of the surface where the objects in the family are all Bridgeland semistable. A Theorem of Bayer-Macri then gives nefness of the determinant line bundle on the base of the family.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives two explicit extensions of Reider's theorem to nefness on blow-ups and ample cones on Hilbert schemes via Bridgeland chambers and Bayer-Macri, with the semistability verification as the main point to check.

read the letter

The paper extends Reider's theorem in two directions using stability conditions on surfaces. It shows that Reider-type inequalities imply nefness of dH - E on the blow-up of P^3 along the surface, and it gives a sharp estimate for the ample cone on the Hilbert scheme of d points on the surface. Both results come from the same strategy: build a natural family of objects parametrized by the base space, locate the largest stability chamber where the whole family stays Bridgeland semistable, and apply the Bayer-Macri theorem to conclude that the determinant line bundle is nef on the base. This is new for these particular moduli spaces. The classical Reider bounds are fed in as input, and the stability machinery turns them into positivity statements without reducing to earlier results. The approach is direct and organizes the positivity questions in terms of chamber walls rather than ad-hoc calculations. The potential soft spot is the claim that the family remains semistable throughout the largest chamber. If any object in the family destabilizes at an earlier wall because of a subsheaf or complex with a different Chern character, the nefness statement only holds on a strictly smaller region of the base. The abstract asserts that the maximal chamber works, so the paper must contain the explicit wall-crossing data and heart checks; that calculation is the part that needs close reading. This is for people working on positivity of moduli spaces of sheaves on surfaces and on Bridgeland stability. Readers already comfortable with Bayer-Macri will follow the arguments most easily. It deserves a serious referee because the applications are concrete and the method connects classical surface geometry to current stability techniques in a usable way.

Referee Report

2 major / 2 minor

Summary. The paper extends Reider's theorem on very ampleness of adjoint linear series on complex projective surfaces in two directions. First, Reider-type inequalities are shown to imply nefness of divisors of the form dH - E on the blow-up of projective space along an embedded surface. Second, the same inequalities are used to obtain a sharp description of the ample cone of the Hilbert scheme Hilb^d(S) of length-d subschemes of the surface S. Both results are obtained by constructing a natural family of objects on S parametrized by the base (blow-up or Hilbert scheme), identifying the largest chamber in the Bridgeland stability manifold of S in which all objects of the family remain semistable, and then invoking the Bayer-Macri theorem to deduce nefness of the associated determinant line bundle on the base.

Significance. If the semistability statements are verified, the work supplies concrete geometric applications of Bridgeland stability conditions and the Bayer-Macri theorem to classical questions about linear series and ample cones. It yields new nefness results on blow-ups and sharp ample-cone bounds on Hilbert schemes that are not available from classical methods alone, thereby illustrating how stability chambers can produce optimal positivity statements in algebraic geometry.

major comments (2)

[Section 4 (stability chamber analysis)] The central step in both extensions is the claim that the natural family remains Bridgeland semistable throughout the largest chamber whose wall is cut out by the Reider-type inequalities. The manuscript must supply the explicit wall-crossing calculation or the precise argument that no subsheaf or complex with different Chern character destabilizes any member of the family before the Reider bound is reached; without this verification the application of Bayer-Macri only guarantees nefness on a possibly smaller subcone.
[Section 3 (construction of the family)] The heart membership of the parametrized objects (whether they lie in the heart of the t-structure for every stability condition inside the claimed maximal chamber) is not independently verifiable from the outline. This membership is load-bearing for the semistability assertion and must be checked explicitly for the family over the blow-up and over Hilb^d(S).

minor comments (2)

[Section 2] Notation for the stability conditions and the precise form of the Reider-type inequalities should be collected in a single preliminary subsection for easy reference.
[Introduction] The statement of the Bayer-Macri theorem is invoked without recalling its precise hypotheses; a short reminder of the required conditions on the family would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the positive aspects of the work. We agree that the semistability and heart-membership arguments require more explicit verification than is currently provided. We address each major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

Referee: [Section 4 (stability chamber analysis)] The central step in both extensions is the claim that the natural family remains Bridgeland semistable throughout the largest chamber whose wall is cut out by the Reider-type inequalities. The manuscript must supply the explicit wall-crossing calculation or the precise argument that no subsheaf or complex with different Chern character destabilizes any member of the family before the Reider bound is reached; without this verification the application of Bayer-Macri only guarantees nefness on a possibly smaller subcone.

Authors: We agree that an explicit verification is necessary. In the revised version we will add a complete wall-crossing analysis in Section 4. For each family we will enumerate the possible Chern characters of potential destabilizing subsheaves or complexes, compute the corresponding walls, and show that none of these walls intersect the interior of the claimed maximal chamber before the Reider-type bound is attained. This will confirm that the family remains semistable throughout the chamber and that Bayer-Macri applies to the full cone described by the inequalities. revision: yes
Referee: [Section 3 (construction of the family)] The heart membership of the parametrized objects (whether they lie in the heart of the t-structure for every stability condition inside the claimed maximal chamber) is not independently verifiable from the outline. This membership is load-bearing for the semistability assertion and must be checked explicitly for the family over the blow-up and over Hilb^d(S).

Authors: We acknowledge that heart membership must be verified explicitly. We will insert a new subsection in Section 3 that checks, for every stability condition inside the maximal chamber, that the constructed objects lie in the heart of the t-structure. The verification will be carried out separately for the family on the blow-up of projective space and for the family on Hilb^d(S), using the explicit descriptions of the objects and the definition of the heart via the stability condition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external theorems to constructed objects.

full rationale

The paper's proof structure consists of constructing a natural family of objects on the surface, identifying the largest stability chamber where they remain Bridgeland semistable via direct calculation, and then invoking the external Bayer-Macri theorem to obtain nefness of the determinant line bundle. This chain does not reduce any claimed prediction or nefness statement to a fitted parameter, self-definition, or self-citation by construction. Reider-type inequalities are treated as input, and the Bayer-Macri application is a standard external result rather than a load-bearing self-reference. No equations or steps in the described derivation exhibit the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work assumes standard results on Bridgeland stability conditions and the Bayer-Macri theorem; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Existence and properties of Bridgeland stability conditions on the derived category of a projective surface
Used to define chambers where the constructed families are semistable.
standard math Bayer-Macri theorem that semistable families yield nef determinant line bundles
Directly invoked to conclude nefness on the base spaces.

pith-pipeline@v0.9.0 · 5477 in / 1344 out tokens · 43564 ms · 2026-05-09T21:08:34.690940+00:00 · methodology

discussion (0)

Reference graph

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