Inversion of adjunction for quotient singularities III: semi-invariant case

Kohsuke Shibata; Yusuke Nakamura

arxiv: 2312.05808 · v2 · submitted 2023-12-10 · 🧮 math.AG

Inversion of adjunction for quotient singularities III: semi-invariant case

Yusuke Nakamura , Kohsuke Shibata This is my paper

Pith reviewed 2026-05-24 04:40 UTC · model grok-4.3

classification 🧮 math.AG

keywords inversion of adjunctionquotient singularitiessemi-invariant equationscomplete intersectionsminimal log discrepanciesfinite group actionsbirational geometry

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The pith

The precise inversion of adjunction formula holds for finite linear group quotients of complete intersections defined by semi-invariant equations.

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

This paper establishes the precise inversion of adjunction formula in the setting of quotient singularities arising from finite linear group actions on complete intersection varieties whose equations are semi-invariant. It extends earlier results that covered only the invariant equation case. A key application is the proof of semi-continuity for minimal log discrepancies in these settings. Sympathetic readers would care because inversion of adjunction relates singularities of a variety to those of a hypersurface section, which is fundamental for studying birational geometry and singularity theory.

Core claim

We prove the precise inversion of adjunction formula for finite linear group quotients of complete intersection varieties defined by semi-invariant equations. As an application, we prove the semi-continuity of minimal log discrepancies for them. These results extend the results in our first paper, where we prove the same results for complete intersection varieties defined by invariant equations.

What carries the argument

The precise inversion of adjunction formula, relating log discrepancies on the quotient to those on the original variety for semi-invariant complete intersections.

If this is right

Semi-continuity of minimal log discrepancies holds for these quotient varieties.
The inversion formula extends from the invariant-equation case to the semi-invariant case.
The results apply to a wider class of quotient singularities than previously covered.

Where Pith is reading between the lines

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

The method may adapt to study adjunction in settings beyond complete intersections.
Explicit low-dimensional examples with cyclic groups could test whether new discrepancy patterns emerge under semi-invariance.
The semi-continuity result might connect to deformation theory questions left open in the invariant case.

Load-bearing premise

The complete intersection varieties have defining equations that are semi-invariant with respect to the finite linear group action.

What would settle it

A concrete finite linear group action on affine space, together with semi-invariant polynomials defining a complete intersection, where direct computation of a minimal log discrepancy at a point on the quotient fails to match the value predicted by the inversion formula.

read the original abstract

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 handles the semi-invariant case for inversion of adjunction on quotient singularities, extending the authors' prior work on invariant equations with an application to mld semi-continuity.

read the letter

The core advance is the proof of the precise inversion of adjunction formula for finite linear group quotients of complete intersections cut out by semi-invariant equations, plus the resulting semi-continuity statement for minimal log discrepancies. The abstract positions this as a direct follow-up to the authors' first paper on the invariant case, with the semi-invariant setting treated as a separate but parallel track rather than a minor tweak. That separation is the main new element, and the application to semi-continuity follows in the expected way once the formula is in hand. The work stays tightly focused and builds on the earlier results without apparent reinvention. The scope is narrow by design—complete intersections under finite linear actions with the semi-invariant condition—which keeps the claims manageable but also limits how far the result travels outside that setting. If the proofs rely mostly on adapting arguments from the first paper, that is standard for a series and not a flaw, though it does mean the paper's value is highest for readers already following the sequence. No signs of circularity or loose parameters appear in the stated claims. This is for people working on birational geometry and singularities who already use inversion of adjunction or track quotient singularities. A specialist in that subfield will see the utility immediately; a general algebraic geometer probably will not. The series looks worth finishing, so I would send it to peer review.

Referee Report

0 major / 0 minor

Summary. The manuscript proves the precise inversion of adjunction formula for finite linear group quotients of complete intersection varieties defined by semi-invariant equations. As an application, it establishes the semi-continuity of minimal log discrepancies for these varieties. The results extend the authors' prior work on the invariant-equation case.

Significance. If the central claims are established, the work would extend inversion-of-adjunction techniques to a wider class of quotient singularities, supplying a tool for minimal log discrepancies in the semi-invariant setting and yielding the semi-continuity statement as a direct corollary.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript, which correctly describes the extension of our prior results on invariant equations to the semi-invariant case. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; proof extends prior results independently

full rationale

The paper is a mathematical proof extending inversion of adjunction results from the authors' prior work on invariant equations to the semi-invariant case for quotient singularities. The abstract and description present this as a direct proof with an application to semi-continuity of minimal log discrepancies. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claim to unverified inputs are present. The derivation chain is a standard theorem-proof structure in algebraic geometry and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no equations, lemmas, or explicit assumptions beyond the statement of the setting; ledger entries cannot be populated.

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Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

2 11 (1992) (1992)

Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991; Ast´ erisque No. 2 11 (1992) (1992). [Amb99] F. Ambro, On minimal log discrepancies , Math. Res. Lett. 6 (1999), no. 5-6, 573–580. [DL02] J. Denef and F. Loeser, Motivic integration, quotient singularities and the McKay corre- spo...

work page 1991
[2]

Part 2, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., P rovidence, RI, 2009, pp. 505–546. [EMY03] L. Ein, M. Mustat ¸˘ a, and T. Yasuda, Jet schemes, log discrepancies and inversion of adjunction, Invent. Math. 153 (2003), no. 3, 519–535. [Eis95] D. Eisenbud, Commutative algebra , Graduate Texts in Mathematics, vol. 150, Springer- Verlag, New York,

work page 2009
[3]

Translated from the Japan- ese by M. Reid. [Nak16] Y. Nakamura, On semi-continuity problems for minimal log discrepancies , J. Reine Angew. Math. 711 (2016), 167–187. [NS22] Y. Nakamura and K. Shibata, Inversion of adjunction for quotient singularities , Algebr. Geom. 9 (2022), no. 2, 214–251. [NS] , Inversion of adjunction for quotient singularities II: ...

work page arXiv 2016

[1] [1]

2 11 (1992) (1992)

Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991; Ast´ erisque No. 2 11 (1992) (1992). [Amb99] F. Ambro, On minimal log discrepancies , Math. Res. Lett. 6 (1999), no. 5-6, 573–580. [DL02] J. Denef and F. Loeser, Motivic integration, quotient singularities and the McKay corre- spo...

work page 1991

[2] [2]

Part 2, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., P rovidence, RI, 2009, pp. 505–546. [EMY03] L. Ein, M. Mustat ¸˘ a, and T. Yasuda, Jet schemes, log discrepancies and inversion of adjunction, Invent. Math. 153 (2003), no. 3, 519–535. [Eis95] D. Eisenbud, Commutative algebra , Graduate Texts in Mathematics, vol. 150, Springer- Verlag, New York,

work page 2009

[3] [3]

Translated from the Japan- ese by M. Reid. [Nak16] Y. Nakamura, On semi-continuity problems for minimal log discrepancies , J. Reine Angew. Math. 711 (2016), 167–187. [NS22] Y. Nakamura and K. Shibata, Inversion of adjunction for quotient singularities , Algebr. Geom. 9 (2022), no. 2, 214–251. [NS] , Inversion of adjunction for quotient singularities II: ...

work page arXiv 2016