Interpolation and moduli spaces of vector bundles on very general blowups of the projective plane

Izzet Coskun; Jack Huizenga

arxiv: 2306.06175 · v2 · pith:72MUUP6Snew · submitted 2023-06-09 · 🧮 math.AG

Interpolation and moduli spaces of vector bundles on very general blowups of the projective plane

Izzet Coskun , Jack Huizenga This is my paper

Pith reviewed 2026-05-24 08:26 UTC · model grok-4.3

classification 🧮 math.AG

keywords moduli spacesvector bundlesblowupsprojective planeSHGH conjecturesheavesinterpolationalgebraic surfaces

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The pith

Moduli spaces of vector bundles on blowups of the projective plane at 10 or more very general points can be disconnected with components of different dimensions.

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

The paper examines moduli spaces of vector bundles on the blowup of the projective plane in at least 10 very general points. It produces examples where these moduli spaces are disconnected and have irreducible components of unequal dimension, unlike the irreducible and smooth spaces found on minimal rational surfaces. The authors further show that, assuming the SHGH conjecture, it is possible to construct moduli spaces with any prescribed number of components, each of arbitrarily large dimension. This illustrates how the geometry of moduli spaces on rational surfaces becomes more intricate once sufficiently many points are blown up.

Core claim

On the blowup of the projective plane at r ≥ 10 very general points, there exist moduli spaces of vector bundles that are disconnected, with irreducible components of different dimensions. Assuming the SHGH conjecture, one can produce moduli spaces with arbitrarily many components of arbitrarily large dimension.

What carries the argument

Construction of vector bundles satisfying interpolation conditions on very general blowups of the projective plane, used to exhibit disconnected moduli spaces.

If this is right

Moduli spaces of sheaves on these blowups need not be irreducible.
Irreducible components of a single moduli space can have different dimensions.
The number of components and their dimensions can be made arbitrarily large under the SHGH conjecture.
The contrast with minimal rational surfaces shows that adding many blown-up points changes the expected behavior of the moduli spaces.

Where Pith is reading between the lines

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

Similar disconnection phenomena may appear on other rational surfaces obtained by blowing up more points.
It may be possible to produce concrete numerical examples of disconnected moduli spaces without relying on the full strength of the SHGH conjecture.
The results suggest that questions about the number of components of moduli spaces become more subtle as the Picard rank of the surface increases.

Load-bearing premise

The SHGH conjecture holds and permits the construction of the required linear systems on the blowups.

What would settle it

An explicit calculation for a specific blowup at 10 very general points showing that every moduli space of vector bundles on it is connected, or a counterexample to the SHGH conjecture.

read the original abstract

In this paper, we study certain moduli spaces of vector bundles on the blowup of the projective plane in at least 10 very general points. Moduli spaces of sheaves on general type surfaces may be nonreduced, reducible and even disconnected. In contrast, moduli spaces of sheaves on minimal rational surfaces and certain del Pezzo surfaces are irreducible and smooth along the locus of stable bundles. We find examples of moduli spaces of vector bundles on more general blowups of the projective plane that are disconnected and have components of different dimensions. In fact, assuming the SHGH Conjecture, we can find moduli spaces with arbitrarily many components of arbitrarily large dimension.

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

This paper gives unconditional examples of disconnected moduli spaces with unequal component dimensions on blowups of P2 at 10+ very general points, but the arbitrary-many/arbitrary-dimension extension rests on the open SHGH conjecture.

read the letter

The punchline is that Coskun and Huizenga exhibit moduli spaces of vector bundles on the blowup of P2 at n ≥ 10 very general points that are disconnected with components of different dimensions. These examples are new relative to the irreducible cases already known for minimal rational surfaces and some del Pezzo surfaces. The constructions appear to be direct geometric arguments that do not invoke the conjecture for the basic cases, and the paper is explicit about where the SHGH assumption enters for the stronger statements about arbitrarily many components of arbitrarily large dimension. That separation is useful and keeps the claims proportionate. The main limitation is precisely that the wildest behavior is conditional; without a proof of SHGH the paper cannot yet say how bad the moduli spaces can get in the limit. There is no internal contradiction or hidden circularity visible from the abstract and the stress-test note. The citation pattern looks standard for the area, and the work engages honestly with the existing literature on irreducibility. This is aimed at specialists in moduli of sheaves on rational surfaces who want concrete counterexamples to irreducibility once enough points are blown up. A reader already following the SHGH literature will find the examples worth seeing even if the conjecture remains open. I would send it to peer review; the core contribution is grounded enough to merit referee attention, with the conditional part handled transparently.

Referee Report

0 major / 3 minor

Summary. The paper studies moduli spaces of vector bundles on the blowup of the projective plane at n ≥ 10 very general points. It constructs examples of such moduli spaces that are disconnected with components of different dimensions. Assuming the SHGH conjecture, the authors extend the constructions to produce moduli spaces with arbitrarily many components of arbitrarily large dimension.

Significance. The unconditional constructions for n ≥ 10 already provide concrete examples showing that moduli spaces on non-minimal rational surfaces can fail to be irreducible or equidimensional, in contrast to the known irreducibility and smoothness results for minimal rational surfaces and certain del Pezzo surfaces. This contributes to the literature on the behavior of moduli spaces of sheaves on rational surfaces. The conditional extension under SHGH is presented transparently and illustrates a potential source of highly pathological examples if the conjecture holds; the paper correctly identifies the dependence rather than claiming unconditional results.

minor comments (3)

[Introduction] Introduction: the separation between the unconditional examples (for n ≥ 10) and the conditional extension should be made more explicit in the statement of the main results to prevent any misreading of the scope.
The paper should include a brief self-contained statement of the SHGH conjecture (including the precise formulation used) together with its standard reference when first invoked.
Notation for the blow-up surface X_r and the moduli spaces M(v) should be introduced once and used consistently; occasional shifts in indexing or Chern-class conventions can be clarified.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No circularity; central claims conditional on external SHGH conjecture

full rationale

The paper constructs examples of moduli spaces on blowups of P^2 in at least 10 very general points using geometric methods. The extension to arbitrarily many components of arbitrarily large dimension is explicitly conditional on the SHGH Conjecture (a longstanding external conjecture in the field, not due to these authors). No derivation step reduces by definition, by fitted parameters renamed as predictions, or by load-bearing self-citation chains. The paper is self-contained against external benchmarks once the conjecture is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The strongest claims rest on the SHGH Conjecture as a domain assumption for the arbitrary-component case; no free parameters or invented entities are indicated in the abstract.

axioms (1)

domain assumption SHGH Conjecture
Invoked in the abstract to extend the basic disconnected examples to the statement of arbitrarily many components of arbitrarily large dimension.

pith-pipeline@v0.9.0 · 5636 in / 1140 out tokens · 20026 ms · 2026-05-24T08:26:19.854000+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Assuming the SHGH Conjecture, we can find moduli spaces with arbitrarily many components of arbitrarily large dimension.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The possible types D of stable bundles are extremely special: if there is an At-stable bundle of type D, then D must satisfy 2B · D < B · K and χ(D) ≥ 1

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.