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arxiv: 2602.14897 · v2 · pith:RCKKJWFXnew · submitted 2026-02-16 · 🧮 math.AG · math.RT

Equivariant multiplicities and mirror symmetry for Hilbert schemes

Alexandre Minets , Filip \v{Z}ivanovi\'c This is my paper

Pith reviewed 2026-05-21 12:59 UTC · model grok-4.3

classification 🧮 math.AG math.RT
keywords Hilbert schemesmirror symmetryequivariant multiplicitiescore Lagrangiansupward flowsProcesi bundleselliptic surfaces
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The pith

Upward flows of very stable ideals in Hilbert schemes of points on elliptic surfaces are mirror dual to modified Procesi bundles.

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

The paper computes scheme-theoretic multiplicities for core Lagrangians and equivariant multiplicities for their very stable versions in Hilbert schemes of points on elliptic surfaces. It extends the multiplicity calculation to wobbly components in the case of two points. The central proposal is that upward flows of very stable ideals correspond under mirror symmetry to modified Procesi bundles, with the correspondence checked numerically. This link, if it holds, would relate geometric invariants on one side to structures from representation theory on the other.

Core claim

We compute the scheme-theoretic multiplicities of core Lagrangians and the equivariant multiplicities of the very stable ones. We extend the notion of equivariant multiplicity to wobbly components and compute it for Hilbert schemes of two points. We propose that upward flows of very stable ideals are mirror dual to modified Procesi bundles, and justify this claim through numerical checks. We make some conjectures about extending this picture to wobbly upward flows.

What carries the argument

Equivariant multiplicities of very stable upward flows, used to numerically test the proposed mirror duality with modified Procesi bundles.

If this is right

  • The computed multiplicities supply explicit geometric numbers that can be matched against representation-theoretic data.
  • The extension of equivariant multiplicity to wobbly components for two points gives a concrete starting case for broader checks.
  • The conjectures indicate that a similar duality statement may hold once wobbly upward flows are included.

Where Pith is reading between the lines

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

  • If the numerical match reflects a true duality, one side's hard-to-compute invariants could be read off from the other side.
  • The same multiplicity-based test might apply to Hilbert schemes on surfaces other than elliptic ones.
  • An algebraic or categorical construction realizing the duality could replace the current numerical justification.

Load-bearing premise

Numerical agreement between invariants is enough to indicate a structural mirror duality rather than a numerical coincidence.

What would settle it

A concrete Hilbert scheme example in which the equivariant multiplicity computed for an upward flow of a very stable ideal fails to match the corresponding invariant attached to the modified Procesi bundle.

read the original abstract

Following Hausel-Hitchin, we investigate core Lagrangians and upward flows in Hilbert schemes of points on elliptic surfaces. We compute the scheme-theoretic multiplicities of core Lagrangians, as well as the equivariant multiplicities of the very stable ones. Furthermore, we extend the notion of equivariant multiplicity to wobbly components and compute it for Hilbert schemes of two points. Inspired by Eisenstein series functor in Dolbeault Langlands correspondence, we propose that upward flows of very stable ideals are mirror dual to modified Procesi bundles, and justify this claim through numerical checks. Finally, we make some conjectures about extending this picture to wobbly upward flows.

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 computes scheme-theoretic multiplicities of core Lagrangians and equivariant multiplicities of very stable ideals in Hilbert schemes of points on elliptic surfaces, extending the latter to wobbly components for n=2. Inspired by the Eisenstein series functor in the Dolbeault Langlands correspondence, it proposes that upward flows of very stable ideals are mirror dual to modified Procesi bundles, justified via numerical checks on the multiplicities, and states conjectures for extending the picture to wobbly upward flows.

Significance. The rigorous computations of scheme-theoretic and equivariant multiplicities supply concrete, verifiable data for core Lagrangians and very stable ideals; these stand as independent contributions to the geometry of Hilbert schemes. If the proposed mirror duality can be placed on firmer conceptual footing, the work would link equivariant invariants on upward flows to modified Procesi bundles, furnishing a geometric counterpart to the Eisenstein series functor and potentially clarifying aspects of mirror symmetry in this setting.

major comments (2)
  1. [Section on the mirror proposal and numerical checks] The central proposal that upward flows of very stable ideals are mirror dual to modified Procesi bundles rests on numerical agreement of equivariant multiplicities. This agreement could arise from shared low-order invariants (e.g., dimensions or leading terms already present in the Hilbert scheme of two points) rather than from an independent correspondence; a derivation or canonical identification that forces the duality, independent of the numerical data, is needed to support the claim.
  2. [Computations for wobbly components (n=2)] The extension of equivariant multiplicity to wobbly components is carried out only for n=2; it is unclear whether the numerical checks used to support the duality for very stable ideals continue to hold or can be performed for these wobbly cases, leaving the conjectures about wobbly upward flows without direct computational backing.
minor comments (2)
  1. [Introduction / mirror proposal] Define 'modified Procesi bundles' explicitly at first use, including how they differ from the standard Procesi bundle construction.
  2. [Numerical checks] Add a table or explicit list comparing the computed multiplicities for the upward flows and the modified Procesi bundles side-by-side to make the numerical justification easier to inspect.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Section on the mirror proposal and numerical checks] The central proposal that upward flows of very stable ideals are mirror dual to modified Procesi bundles rests on numerical agreement of equivariant multiplicities. This agreement could arise from shared low-order invariants (e.g., dimensions or leading terms already present in the Hilbert scheme of two points) rather than from an independent correspondence; a derivation or canonical identification that forces the duality, independent of the numerical data, is needed to support the claim.

    Authors: We agree that the proposed mirror duality is supported primarily by the observed numerical agreement of equivariant multiplicities rather than by an independent derivation. The manuscript already frames this as a proposal justified through numerical checks, consistent with the abstract. Our computations for n>2 do include higher-order terms in the multiplicities beyond those fixed by the n=2 Hilbert scheme, but we acknowledge that this does not constitute a canonical identification. We will revise the relevant section to state more explicitly that the duality remains conjectural, to highlight the specific higher invariants that match, and to note that a derivation independent of the numerical data is left for future work. revision: partial

  2. Referee: [Computations for wobbly components (n=2)] The extension of equivariant multiplicity to wobbly components is carried out only for n=2; it is unclear whether the numerical checks used to support the duality for very stable ideals continue to hold or can be performed for these wobbly cases, leaving the conjectures about wobbly upward flows without direct computational backing.

    Authors: We agree that the computational backing for the conjectures on wobbly upward flows is limited to the explicit multiplicity computation for n=2. The numerical checks supporting the mirror proposal are performed in the very stable setting, and no additional checks for wobbly components at higher n are provided. We will revise the text to clarify the restricted scope of the wobbly computations and to state explicitly that the conjectures for wobbly upward flows currently lack further numerical verification. revision: partial

standing simulated objections not resolved
  • A derivation or canonical identification forcing the mirror duality independently of the numerical data is not available in the current manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper performs explicit computations of scheme-theoretic multiplicities of core Lagrangians and equivariant multiplicities of very stable ideals, extending these to wobbly components for n=2. The mirror duality proposal is explicitly introduced as an inspiration from the Eisenstein series functor and then justified separately via numerical checks on the computed quantities. No equation or definition reduces the central claim to its own inputs by construction, no fitted parameter is relabeled as a prediction, and no load-bearing step relies on a self-citation chain that itself lacks independent verification. The numerical agreement therefore functions as external corroboration rather than a definitional identity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract alone; no explicit free parameters, axioms, or invented entities can be extracted. The work explicitly follows Hausel-Hitchin and invokes the Eisenstein series functor from Dolbeault Langlands correspondence as background.

pith-pipeline@v0.9.0 · 5638 in / 1101 out tokens · 53191 ms · 2026-05-21T12:59:04.927545+00:00 · methodology

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Reference graph

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