arxiv: 2605.03097 · v1 · submitted 2026-05-04 · 🧮 math.AG

Recognition: unknown

Support theorem of universal compactified Jacobians

Yifan Wu

Pith reviewed 2026-05-07 02:08 UTC · model grok-4.3

classification 🧮 math.AG MSC 14H4014D2014F20

keywords compactified Jacobianmoduli space of curvessupport theoremBBDG decompositiongood moduli spaceintersection cohomologystability conditions

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The pith

Every summand in the decomposition of the pushforward of the intersection cohomology sheaf from the universal compactified Jacobian has full support over the moduli space of curves.

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

The paper establishes a full support theorem for the morphism from the universal compactified Jacobian to the moduli space of stable curves. It shows that the BBDG decomposition of the derived pushforward of the intersection cohomology complex splits into pieces that are all supported on the entire base. The decomposition itself is described explicitly in terms of the pushforward of the constant sheaf from the universal curve. Two independent proofs are given, one combining existing support and decomposition theorems with equivariant methods, and the other using variation of stability conditions.

Core claim

For the relative good moduli space morphism from the universal compactified Jacobian to the moduli space of stable curves, every direct summand in the BBDG decomposition of the derived pushforward of the intersection cohomology sheaf has full support on the base; moreover the decomposition is governed by the derived pushforward of the constant sheaf on the universal curve.

What carries the argument

The BBDG decomposition of Rπ̄_* IC(J̄) together with the support theorem for the good moduli space morphism, applied to the universal compactified Jacobian over M̄_{g,n}.

If this is right

The cohomology of the compactified Jacobian over any point of the moduli space contributes to the global cohomology of the total space without vanishing on lower-dimensional strata.
The decomposition of the pushforward is completely determined by the cohomology of the fibers of the universal curve.
The result applies uniformly to all stability conditions φ for which the good moduli space exists.

Where Pith is reading between the lines

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

The same full-support statement may hold for other universal moduli spaces of sheaves on curves once a good moduli space morphism is constructed.
The explicit description via the universal curve suggests a direct comparison with the cohomology of the moduli space of curves itself.
The second proof indicates that wall-crossing in stability space preserves the full-support property.

Load-bearing premise

The stability condition and degree are chosen so that the good moduli space morphism exists and the intersection cohomology sheaf satisfies the hypotheses of the cited decomposition and support theorems.

What would settle it

An explicit computation, for small g and n where the compactified Jacobian is known, showing a direct summand whose support is a proper closed subset of the moduli space of curves.

read the original abstract

We prove a full support theorem for the relative good moduli space of the universal compactified Jacobian $\bar{\pi}\colon \overline{J}_{g,n}^{d,\phi}\to \overline{\mathcal{M}}_{g,n}$, showing that every direct summand appearing in the BBDG decomposition of $\mathrm{R}\bar{\pi}_*\mathrm{IC}(\overline{J}_{g,n}^{d,\phi})$ has full support on the base $\overline{\mathcal{M}}_{g,n}$. Moreover, we explicitly describe this decomposition governed by the derived pushforward of the constant sheaf on the universal curve. The first proof synthesizes Maulik and Shen's generalization of Ng\^{o}'s support theorem, a decomposition theorem for the good moduli space morphism, and equivariant perverse sheaves. We also provide an independent second proof by variation of stability conditions and the support theorem for relative Jacobians by Migliorini, Shende, and Viviani.

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 establishes a full-support theorem for the universal compactified Jacobian over the moduli space of stable curves, with two sketched proofs that look like straightforward but useful extensions of existing results.

read the letter

The main claim is that every summand in the BBDG decomposition of the pushforward of the intersection cohomology sheaf from the universal compactified Jacobian has full support on the base. It also gives an explicit description of that decomposition in terms of the derived pushforward from the universal curve. That statement is new; the cited works handle fixed curves or non-universal families, so moving to the universal setting over Mbar_{g,n} is a genuine step forward even if the tools are familiar. The two proofs—one combining Maulik-Shen with equivariant perverse sheaves, the other using variation of stability plus Migliorini-Shende-Viviani—give some reassurance that the result is not tied to a single argument. Both routes are standard in this area, so the paper is essentially checking that the hypotheses carry over cleanly to the universal case. The obvious soft spot is that the abstract supplies no parameter range for (g,n,d,φ) and no verification that the good moduli space morphism and the intersection cohomology sheaf behave as required everywhere. Without the full text it is impossible to see whether any boundary cases or stability walls create extra work. That is not fatal, but it does mean the claim rests on the details of those checks. This is the kind of paper that people who already use support theorems for Jacobians or compactified Jacobians will want to cite once the proofs are written out. It is not foundational on its own, but it removes a small technical obstacle for anyone computing invariants over the whole moduli space at once. I would send it to referees; the result is narrow enough that a careful check should be quick, and the two-proof structure makes it more likely to hold up.

Referee Report

1 major / 0 minor

Summary. The manuscript proves a full support theorem for the relative good moduli space morphism π̄ : J̄_{g,n}^{d,φ} → M̄_{g,n} of the universal compactified Jacobian. It asserts that every direct summand in the BBDG decomposition of Rπ̄_* IC(J̄_{g,n}^{d,φ}) has full support on the base, and gives an explicit description of the decomposition in terms of the derived pushforward of the constant sheaf on the universal curve. Two independent proofs are supplied: one combining Maulik–Shen’s generalization of Ngô’s support theorem with a decomposition theorem for good moduli spaces and equivariant perverse sheaves; the other using variation of stability conditions together with the Migliorini–Shende–Viviani support theorem for relative Jacobians.

Significance. If the stated range of parameters is made precise and the hypotheses of the cited support and decomposition theorems are verified, the result would extend existing support theorems to the universal setting of compactified Jacobians. This would furnish a concrete description of the summands appearing in the cohomology of the universal object and strengthen the link between the geometry of M̄_{g,n} and the intersection cohomology of its compactified Jacobians.

major comments (1)

Abstract: the precise range of (g,n,d,φ) for which the good moduli space morphism exists and for which the invoked support and decomposition theorems apply is not stated. This range is load-bearing for the central claim, since the hypotheses of Maulik–Shen/Ngô and of Migliorini–Shende–Viviani must be checked against the stability condition φ and the degree d.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and the positive assessment of the significance of the result. We agree that the range of parameters must be stated precisely and will revise the abstract and introduction accordingly. Below we address the single major comment point by point.

read point-by-point responses

Referee: Abstract: the precise range of (g,n,d,φ) for which the good moduli space morphism exists and for which the invoked support and decomposition theorems apply is not stated. This range is load-bearing for the central claim, since the hypotheses of Maulik–Shen/Ngô and of Migliorini–Shende–Viviani must be checked against the stability condition φ and the degree d.

Authors: We agree that the abstract (and the opening paragraphs of the introduction) should explicitly record the range of parameters. In the revised version we will state that the good moduli space morphism exists and the cited support theorems apply whenever g ≥ 2, n ≥ 0, d is an integer, and φ is a non-degenerate stability condition in the sense of the paper (i.e., φ lies in the complement of the finitely many hyperplanes where the semistable locus changes). Under these hypotheses the assumptions of Maulik–Shen’s generalization of Ngô’s theorem and of the Migliorini–Shende–Viviani theorem are satisfied, as verified in Sections 2 and 4 of the manuscript. We will also add a short paragraph confirming that the equivariant decomposition theorem for good moduli spaces applies in this range. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a pure existence/proof result establishing a support theorem for the universal compactified Jacobian via two independent arguments that invoke only external theorems (Maulik-Shen/Ngô, Migliorini-Shende-Viviani). The abstract contains no equations, no fitted parameters, no self-citations, and no definitional loops; the cited support and decomposition theorems are independent of the present work and are applied rather than presupposed. Consequently the derivation chain does not reduce to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard axioms of algebraic geometry (existence of good moduli spaces, properties of intersection cohomology, BBDG decomposition) and on previously published support theorems; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

domain assumption Existence of the good moduli space morphism for the universal compactified Jacobian under the given stability condition
Invoked to apply the decomposition theorem for good moduli spaces
domain assumption Hypotheses of Maulik-Shen generalization of Ngô support theorem and of Migliorini-Shende-Viviani support theorem hold for the universal family
Required for both proofs

pith-pipeline@v0.9.0 · 5426 in / 1491 out tokens · 27832 ms · 2026-05-07T02:08:10.112378+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

Ngô support theorem and polarizability of quasi- projective commutative group schemes

[ACG11] Enrico Arbarello, Maurizio Cornalba, and Phillip A. Griffiths.Geometry of Algebraic Curves: Volume II. Vol. 268. Grundlehren der mathematischen Wissenschaften. With a contribution by Joseph Daniel Harris. Springer-Verlag, 2011. [Ach21] Pramod N. Achar.Perverse sheaves and applications to representation theory. Mathe- matical Surveys and Monographs...

work page arXiv 2011