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arxiv: 2604.01331 · v2 · submitted 2026-04-01 · 🧮 math.AG · math.RT

Non-reduced components of global nilpotent cones

David Zhiyuan Bai , David Fang This is my paper

Pith reviewed 2026-05-13 21:52 UTC · model grok-4.3

classification 🧮 math.AG math.RT
keywords global nilpotent coneHitchin fibrationmoduli spaceone-dimensional sheavesBeauville-Mukai systemGIT stratificationnon-reduced schemeLagrangian fibration
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The pith

Under coprimality conditions, global nilpotent cones for Hitchin fibrations and moduli of one-dimensional sheaves are nowhere reduced.

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

The paper shows that in key examples from algebraic geometry, certain global nilpotent cones have no reduced points at all. For L-twisted GL_r-Hitchin fibrations on high-genus curves, when the rank and degree are coprime and L satisfies degree conditions, the entire cone is non-reduced. The same non-reduced property holds for the global nilpotent cones in moduli spaces of one-dimensional sheaves on K3, abelian, and del Pezzo surfaces. For Beauville-Mukai systems on K3 surfaces, a general fiber has primitive homology class only when the multiple of the primitive class is exactly one. These findings matter because they reveal hidden multiplicities in the geometry of these moduli spaces, which are used to study integrable systems and sheaf moduli.

Core claim

We determine the non-reduced components of global nilpotent cones in various cases of interest. In particular, under the appropriate coprimality conditions, the global nilpotent cone for an L-twisted GL_r-Hitchin fibration associated to a curve C of genus g≥2 is nowhere reduced, where L is either the canonical bundle or has degree greater than 2g-2; the global nilpotent cone for a moduli space of one-dimensional sheaves on a K3, abelian, or del Pezzo surface is nowhere reduced; and a general fiber of a Beauville-Mukai system for the class rℓ has primitive homology class if and only if r=1. Our methods include group scheme actions on Lagrangian fibrations, a GIT-stratification of global nilpo

What carries the argument

Group scheme actions on Lagrangian fibrations together with a GIT-stratification of the global nilpotent cones and deformation to the normal cone to detect non-reduced structure.

If this is right

  • The global nilpotent cone fails to be reduced at every point when the coprimality conditions hold for the Hitchin fibration.
  • The same complete non-reducedness applies to the nilpotent cones in one-dimensional sheaf moduli on the listed surfaces.
  • For Beauville-Mukai systems, only the r=1 case yields a general fiber with primitive homology class.

Where Pith is reading between the lines

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

  • This non-reducedness likely requires adjustments in computing intersection numbers or Euler characteristics on these spaces.
  • The result may extend to other classes of surfaces or fibrations if similar group actions can be constructed.
  • Such structure could influence the study of the singular fibers in the associated integrable systems.

Load-bearing premise

The coprimality of rank and degree (or primitivity of the class) together with the degree bounds on L or the surface conditions, which enable the group scheme actions and GIT stratification to show non-reducedness everywhere.

What would settle it

An explicit calculation in a low-rank, low-genus example finding a point where the scheme is reduced would disprove the nowhere-reduced claim.

read the original abstract

We determine the non-reduced components of global nilpotent cones in various cases of interest. In particular, under the appropriate coprimality conditions, we show: (1) the global nilpotent cone for an $L$-twisted $\operatorname{GL}_r$-Hitchin fibration associated to a curve $C$ of genus $g\ge 2$ is nowhere reduced, where $L$ is either the canonical bundle or has degree greater than $2g-2$; (2) the global nilpotent cone for a moduli space of one-dimensional sheaves on a K3, abelian, or del Pezzo surface is nowhere reduced; (3) suppose $\ell$ is a primitive, basepoint-free, big and nef class on a K3 surface, then a general fiber of a Beauville-Mukai system for the class $r\ell$ has primitive homology class if and only if $r=1$. Our methods include group scheme actions on Lagrangian fibrations, a GIT-stratification of global nilpotent cones of Hitchin fibrations, and deformation to the normal cone.

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.

Referee Report

2 major / 2 minor

Summary. The paper determines the non-reduced components of global nilpotent cones in several settings of interest in algebraic geometry. Under appropriate coprimality conditions on rank and degree (or primitivity of the class ℓ), it claims: (1) the global nilpotent cone of an L-twisted GL_r-Hitchin fibration on a curve C of genus g≥2 is nowhere reduced, for L the canonical bundle or with deg(L)>2g−2; (2) the global nilpotent cone for moduli spaces of one-dimensional sheaves on K3, abelian, or del Pezzo surfaces is nowhere reduced; (3) a general fiber of a Beauville-Mukai system for class rℓ on a K3 has primitive homology class if and only if r=1. The proofs rely on group-scheme actions on Lagrangian fibrations, a GIT stratification of the nilpotent cones, and deformation to the normal cone.

Significance. If the results hold, they provide a precise description of the non-reduced loci in these important moduli spaces and integrable systems, clarifying the geometry of Hitchin fibrations and Beauville-Mukai systems beyond the reduced case. This has potential implications for the study of singularities in moduli spaces, the structure of Lagrangian fibrations, and related questions in enumerative geometry and mirror symmetry. The use of GIT stratifications and group-scheme actions to detect non-reducedness everywhere is a technically interesting approach that could be applied more broadly.

major comments (2)
  1. [Introduction and the GIT-stratification section] The central argument for claims (1) and (2) uses a GIT stratification of the global nilpotent cone together with a group-scheme action whose stabilizers are supposed to force non-reduced structure on every stratum. However, the manuscript does not explicitly verify that the stratification is exhaustive or that no closed stratum admits stabilizers large enough to make the action ineffective for detecting non-reducedness (see the stress-test concern). This gap is load-bearing for the 'nowhere reduced' conclusion and is most acute for claim (1) when deg L > 2g−2, where the stability condition differs from the canonical case.
  2. [Section on Beauville-Mukai systems] For claim (3), the statement that the general fiber has primitive homology class iff r=1 relies on the primitivity of ℓ and the deformation-to-the-normal-cone technique. It is not clear from the provided outline whether the argument rules out the possibility that non-primitive classes appear in the homology for r>1 even when ℓ is primitive, basepoint-free, big and nef; a concrete computation or reference to the relevant homology computation is needed to confirm this direction.
minor comments (2)
  1. [Abstract] The abstract lists the methods but does not indicate the precise sections where the GIT stratification is constructed or where the group-scheme action is defined; adding forward references would improve readability.
  2. [Introduction] Notation for the twisting bundle L and the class ℓ should be made consistent between the statements of claims (1) and (3) to avoid confusion for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its significance. The comments highlight areas where the exposition can be clarified, and we address each major point below with the revisions we will incorporate.

read point-by-point responses

  1. Referee: [Introduction and the GIT-stratification section] The central argument for claims (1) and (2) uses a GIT stratification of the global nilpotent cone together with a group-scheme action whose stabilizers are supposed to force non-reduced structure on every stratum. However, the manuscript does not explicitly verify that the stratification is exhaustive or that no closed stratum admits stabilizers large enough to make the action ineffective for detecting non-reducedness (see the stress-test concern). This gap is load-bearing for the 'nowhere reduced' conclusion and is most acute for claim (1) when deg L > 2g−2, where the stability condition differs from the canonical case.

    Authors: We agree that an explicit verification of exhaustiveness and stabilizer sizes would strengthen the argument. The GIT stratification is constructed by partitioning according to the Jordan type of the nilpotent section, adapted to the stability condition induced by L. For deg L > 2g−2 the stability is governed by the positive degree of L, which ensures that the strata remain the same as in the canonical case up to a shift in the numerical invariants. In the revised manuscript we will add a lemma in the GIT-stratification section that (i) proves the stratification is exhaustive by showing every point lies in one of the defined strata and (ii) computes the stabilizers explicitly on each closed stratum, confirming they remain proper subgroups of the acting group scheme and therefore detect non-reducedness via the infinitesimal action. This addresses the concern uniformly for both the canonical and higher-degree cases. revision: yes

  2. Referee: [Section on Beauville-Mukai systems] For claim (3), the statement that the general fiber has primitive homology class iff r=1 relies on the primitivity of ℓ and the deformation-to-the-normal-cone technique. It is not clear from the provided outline whether the argument rules out the possibility that non-primitive classes appear in the homology for r>1 even when ℓ is primitive, basepoint-free, big and nef; a concrete computation or reference to the relevant homology computation is needed to confirm this direction.

    Authors: We thank the referee for requesting a more explicit verification of this direction. The deformation-to-the-normal-cone argument identifies the homology class of a general fiber with r times the class of the central fiber; since ℓ is primitive, the resulting class is non-primitive precisely when r > 1. In the revised version we will insert a short computation (in the Beauville-Mukai section) that tracks the homology class through the deformation, together with a reference to the standard computation of the Beauville-Mukai system homology (e.g., the fact that the fiber class is a multiple of the primitive class ℓ). This makes the implication for r > 1 fully explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on standard geometric techniques without self-referential reduction

full rationale

The paper's central results on non-reduced components of global nilpotent cones rely on group scheme actions on Lagrangian fibrations, GIT-stratification of Hitchin fibrations, and deformation to the normal cone. These are invoked under explicit coprimality and degree assumptions on rank/degree or classes ℓ, which are standard in moduli theory and do not reduce the conclusions to fitted parameters or self-citations by construction. No equations in the provided abstract or description equate a 'prediction' to an input fit, nor does any load-bearing step collapse to a prior self-citation whose content is unverified. The derivation chain remains self-contained against external benchmarks in algebraic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on standard facts about moduli spaces of sheaves, Hitchin fibrations, and Lagrangian fibrations; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Standard properties of moduli spaces of vector bundles, one-dimensional sheaves, and their Hitchin or Beauville-Mukai fibrations hold under the stated coprimality and positivity conditions.
    The non-reducedness conclusions and the homology primitivity criterion presuppose these background results from algebraic geometry.

pith-pipeline@v0.9.0 · 5490 in / 1582 out tokens · 45216 ms · 2026-05-13T21:52:40.010168+00:00 · methodology

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