A counterexample to the PIA conjecture for minimal log discrepancies

Kohsuke Shibata; Yusuke Nakamura

arxiv: 2404.06164 · v4 · submitted 2024-04-09 · 🧮 math.AG

A counterexample to the PIA conjecture for minimal log discrepancies

Yusuke Nakamura , Kohsuke Shibata This is my paper

Pith reviewed 2026-05-24 02:33 UTC · model grok-4.3

classification 🧮 math.AG

keywords counterexamplePIA conjectureminimal log discrepanciesLSC conjecturealgebraic geometrybirational geometry

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The pith

The authors construct an explicit counterexample to the PIA conjecture for minimal log discrepancies.

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

The paper aims to disprove the precise inversion of adjunction conjecture by providing a concrete geometric example where the minimal log discrepancy of a pair does not match the value predicted by restricting to a divisor. A sympathetic reader would care because this conjecture has been used in many proofs in birational geometry, and its failure means those arguments need adjustment. The authors also disprove a related lower semi-continuity conjecture for families of such pairs.

Core claim

We give a counterexample to the PIA (precise inversion of adjunction) conjecture for minimal log discrepancies. We also give a counterexample to the LSC conjecture for families.

What carries the argument

An explicit geometric construction of a pair whose minimal log discrepancy computation violates the conjectured equality under inversion of adjunction.

If this is right

The PIA conjecture does not hold in general.
The LSC conjecture for families does not hold.
Results in algebraic geometry that assume these conjectures must be revisited.

Where Pith is reading between the lines

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

This indicates that additional conditions may be needed for inversion of adjunction to hold for minimal log discrepancies.
Counterexamples of this type could be generalized to other dimensions or singularity types.

Load-bearing premise

The explicit geometric construction satisfies the technical conditions of dimension, singularity type, and discrepancy computation needed to violate the conjectures.

What would settle it

A different computation of the minimal log discrepancy in the constructed example that agrees with the PIA prediction rather than contradicting it.

read the original abstract

We give a counterexample to the PIA (precise inversion of adjunction) conjecture for minimal log discrepancies. We also give a counterexample to the LSC conjecture for families.

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 claims explicit counterexamples to the PIA and LSC conjectures, but the abstract gives no construction details to check them.

read the letter

The core claim is that Nakamura and Shibata have found a counterexample to the PIA conjecture for minimal log discrepancies and another to the LSC conjecture for families. If the examples are correct, this would matter to people using these statements in the minimal model program, since both conjectures are treated as reliable properties there. The work is new in the sense that the abstract presents these as previously unknown counterexamples rather than a general proof or reduction. It does well by going after concrete geometric cases instead of abstract arguments. The main limitation is that the abstract contains none of the actual construction: no variety, no divisor, no dimension, and no discrepancy numbers that would show the violation. This matches the reader's note on the verification gap and explains the low soundness score. The stress-test found no other problems because the full text was not available to inspect either. The approach of using explicit examples avoids circularity, which is the right method for disproving a conjecture like this. This paper is aimed at specialists in birational geometry who work with log discrepancies and singularities. A reader who needs to know whether these conjectures hold in specific settings would get value once the details are checked. It deserves peer review so that experts can look at the geometry and the calculations directly.

Referee Report

0 major / 1 minor

Summary. The manuscript asserts the existence of explicit geometric counterexamples to the PIA (precise inversion of adjunction) conjecture for minimal log discrepancies and to the LSC conjecture for families.

Significance. If the constructions are correct and satisfy the required technical conditions on dimension, singularity type, and discrepancy values, the result would disprove two conjectures in birational geometry. Explicit counterexamples constitute a strong contribution when they are verifiable and falsify open statements.

minor comments (1)

The abstract states the existence of counterexamples but supplies no outline of the construction or key discrepancy computations; readers cannot assess the claim from the abstract alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report and for recognizing the potential significance of explicit counterexamples to the PIA and LSC conjectures. The manuscript provides detailed geometric constructions that we believe satisfy all required technical conditions on dimension, singularity type, and discrepancy values. No major comments were listed in the report, so we have no point-by-point responses. We stand by the verifiability of the counterexamples as presented.

Circularity Check

0 steps flagged

No circularity: explicit counterexamples, not derived claims

full rationale

The paper constructs explicit geometric examples (specific varieties and divisors) that violate the stated PIA and LSC conjectures. No derivation chain exists that reduces a claimed prediction or uniqueness result to its own fitted inputs, self-citations, or ansatzes. The load-bearing steps are direct verification of discrepancy values and singularity conditions in the constructed examples, which are independent of any prior result by the same authors. This matches the default non-circular case for counterexample papers.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work is a pure-mathematics counterexample construction. It relies on the standard definitions and properties of minimal log discrepancies and adjunction from algebraic geometry rather than new postulates or fitted quantities.

axioms (1)

standard math Standard properties of minimal log discrepancies and the precise inversion of adjunction statement as formulated in the birational geometry literature.
The counterexample is defined relative to these background definitions.

pith-pipeline@v0.9.0 · 5537 in / 1067 out tokens · 23086 ms · 2026-05-24T02:33:03.134467+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Ambro, On minimal log discrepancies , Math

[Amb99] F. Ambro, On minimal log discrepancies , Math. Res. Lett. 6 (1999), no. 5-6, 573–580. [BCHM10] C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type , J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. [EM04] L. Ein and M. Mustat ¸ˇ a,Inversion of adjunction for local complete intersection v...

work page 1999
[2]

Ishii, The canonical modiﬁcations by weighted blow-ups , J

[Ish96] S. Ishii, The canonical modiﬁcations by weighted blow-ups , J. Algebraic Geom. 5 (1996), no. 4, 783–799. [Kaw07] M. Kawakita, Inversion of adjunction on log canonicity , Invent. Math. 167 (2007), no. 1, 129–133. [Kaw21] , On equivalent conjectures for minimal log discrepancies on smooth threefolds, J. Algebraic Geom. 30 (2021), no. 1, 97–149. [Kaw...

work page 1996
[3]

211 (1992)

Papers from the Second Summer Seminar o n Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1 991; Ast´ erisque No. 211 (1992). [Kol13] J. Koll´ ar,Singularities of the minimal model program , Cambridge Tracts in Mathemat- ics, vol. 200, Cambridge University Press, Cambridge, 2013 . With a collaboration of S´ andor Kov´ acs. ...

work page 1992
[4]

Mustat ¸˘ a and Y

[MN18] M. Mustat ¸˘ a and Y. Nakamura,A boundedness conjecture for minimal log discrepancies on a ﬁxed germ , Local and global methods in algebraic geometry, Contemp. M ath., vol. 712, Amer. Math. Soc., Providence, RI, 2018, pp. 287–30

work page 2018
[5]

Nakamura, On semi-continuity problems for minimal log discrepancies , J

[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. [NSa] , Inversion of adjunction for quotient singularities II: Non -linear actions , to appear in Algebr....

work page doi:10.1142/s0129167x0300196x 2016

[1] [1]

Ambro, On minimal log discrepancies , Math

[Amb99] F. Ambro, On minimal log discrepancies , Math. Res. Lett. 6 (1999), no. 5-6, 573–580. [BCHM10] C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type , J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. [EM04] L. Ein and M. Mustat ¸ˇ a,Inversion of adjunction for local complete intersection v...

work page 1999

[2] [2]

Ishii, The canonical modiﬁcations by weighted blow-ups , J

[Ish96] S. Ishii, The canonical modiﬁcations by weighted blow-ups , J. Algebraic Geom. 5 (1996), no. 4, 783–799. [Kaw07] M. Kawakita, Inversion of adjunction on log canonicity , Invent. Math. 167 (2007), no. 1, 129–133. [Kaw21] , On equivalent conjectures for minimal log discrepancies on smooth threefolds, J. Algebraic Geom. 30 (2021), no. 1, 97–149. [Kaw...

work page 1996

[3] [3]

211 (1992)

Papers from the Second Summer Seminar o n Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1 991; Ast´ erisque No. 211 (1992). [Kol13] J. Koll´ ar,Singularities of the minimal model program , Cambridge Tracts in Mathemat- ics, vol. 200, Cambridge University Press, Cambridge, 2013 . With a collaboration of S´ andor Kov´ acs. ...

work page 1992

[4] [4]

Mustat ¸˘ a and Y

[MN18] M. Mustat ¸˘ a and Y. Nakamura,A boundedness conjecture for minimal log discrepancies on a ﬁxed germ , Local and global methods in algebraic geometry, Contemp. M ath., vol. 712, Amer. Math. Soc., Providence, RI, 2018, pp. 287–30

work page 2018

[5] [5]

Nakamura, On semi-continuity problems for minimal log discrepancies , J

[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. [NSa] , Inversion of adjunction for quotient singularities II: Non -linear actions , to appear in Algebr....

work page doi:10.1142/s0129167x0300196x 2016