pith. sign in

arxiv: 1907.06326 · v1 · pith:SIR7446Inew · submitted 2019-07-15 · 🧮 math.AG

On the Nash problem for terminal threefolds of type cA/r

Hsin-Ku Chen This is my paper

Pith reviewed 2026-05-24 21:43 UTC · model grok-4.3

classification 🧮 math.AG
keywords Nash problemterminal threefoldscA/rNash valuationsessential valuationsQ-factorialthreefold singularitiesresolutions
0
0 comments X

The pith

For terminal threefolds of type cA/r with r=1 or Q-factorial, all Nash and essential valuations can be completely described.

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

The paper examines Nash valuations and essential valuations on terminal threefolds of type cA/r. It establishes that these valuations admit complete descriptions precisely when r equals 1 or the threefold is Q-factorial. The authors construct explicit counterexamples showing that the correspondence between Nash and essential valuations fails for some non-Gorenstein or non-Q-factorial cases. A sympathetic reader cares because the Nash problem concerns whether these two classes of valuations coincide on resolutions of singularities, and explicit control in this family clarifies the boundary between cases where they match and where they do not.

Core claim

If r=1 or the given threefold is Q-factorial, then all the Nash valuations and essential valuations can be completely described. Non-Gorenstein or non-Q-factorial counter examples for the Nash problem are constructed.

What carries the argument

The explicit classification of Nash valuations versus essential valuations on terminal threefolds of type cA/r.

If this is right

  • When r=1 the Nash and essential valuations are given by an explicit finite list.
  • When the threefold is Q-factorial the same explicit list applies regardless of r.
  • The Nash problem has a negative answer for certain non-Q-factorial terminal cA/r threefolds by direct construction.
  • The correspondence between the two sets of valuations holds exactly in the Q-factorial setting inside this class.

Where Pith is reading between the lines

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

  • The necessity of Q-factoriality for the affirmative answer may indicate a general pattern for other terminal threefold singularities.
  • The counterexamples supply concrete test cases that any proposed general solution to the Nash problem must handle.
  • Similar explicit descriptions could be sought for other families of terminal singularities once Q-factoriality is imposed.

Load-bearing premise

The threefolds under study are terminal of type cA/r and the standard definitions of Nash valuations and essential valuations apply without additional restrictions.

What would settle it

A terminal threefold of type cA/1 in which the list of essential valuations fails to match the list of Nash valuations obtained from the paper's description.

read the original abstract

We study Nash valuations and essential valuations of terminal threefolds of type $cA/r$. If $r=1$ or the given threefold is $\mathbb Q$-factorial, then all the Nash valuations and essential valuations can be completely described. We construct non-Gorenstein or non-$\mathbb Q$-factorial counter examples for the Nash problem.

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

0 major / 2 minor

Summary. The manuscript studies Nash valuations and essential valuations on terminal threefolds of type cA/r. It asserts that these valuations admit a complete explicit description whenever r=1 or the threefold is Q-factorial. It further constructs explicit non-Gorenstein and non-Q-factorial examples in which the set of Nash valuations properly contains the set of essential valuations, thereby furnishing counterexamples to the Nash problem in this class.

Significance. If the case-by-case analysis of exceptional divisors and the dictionary between divisorial valuations and arcs hold, the paper supplies a definitive resolution of the Nash problem for the indicated subclasses of terminal cA/r threefolds together with concrete counterexamples outside those subclasses. Such explicit descriptions and constructions are valuable for the broader program of understanding arc spaces and essential divisors on terminal singularities in dimension three.

minor comments (2)
  1. The statement of the main theorem (presumably in §3 or §4) would benefit from an explicit list of the possible Nash/essential divisors in the r=1 and Q-factorial cases rather than a purely descriptive summary.
  2. Notation for the weighted blow-ups and the indices of the exceptional divisors should be introduced once at the beginning of the classification section and used consistently thereafter.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, positive assessment of its significance, and recommendation to accept.

Circularity Check

0 steps flagged

No circularity detected in derivation chain

full rationale

The paper's claims rest on explicit case-by-case descriptions of Nash and essential valuations for terminal cA/r threefolds (when r=1 or Q-factorial) together with constructions of counterexamples in the non-Gorenstein or non-Q-factorial setting. These rest on the standard dictionary between divisorial valuations and arcs plus analysis of exceptional divisors in resolutions, using prior definitions from the literature without any reduction of a 'prediction' or central result to a fitted parameter, self-definition, or load-bearing self-citation chain. No equations or ansatzes are smuggled in; the argument is self-contained against external algebraic-geometry benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on established definitions and classification results for terminal singularities in algebraic geometry; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Standard definitions of terminal singularities, Nash valuations, and essential valuations in birational geometry
    The claims presuppose these notions from the existing literature on the Nash problem.

pith-pipeline@v0.9.0 · 5572 in / 1175 out tokens · 25939 ms · 2026-05-24T21:43:16.172698+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 1 internal anchor

  1. [1]

    J. A. Chen, Explicit resolution of three dimensional terminal singula rities, A.S.P.M. (2011), 323-360

    work page 2011

  2. [2]

    de Fernex, Three-dimensional counter-examples to the Nash problem , Compos

    T. de Fernex, Three-dimensional counter-examples to the Nash problem , Compos. Math. 149 (2013),15191534

    work page 2013

  3. [3]

    de Fernex, The space of arcs of an algebraic variety , to be appear in Proc

    T. de Fernex, The space of arcs of an algebraic variety , to be appear in Proc. Sympos. Pure Math. 97 (2018), 169-198

    work page 2018

  4. [4]

    de Fernex, R

    T. de Fernex, R. Docampo, Terminal valuations and the Nash problem , Invent. Math. 203 (2016), 303331

    work page 2016

  5. [5]

    The arc space of the Grassmannian

    R. Docampo, A. Nigro, The arc space of the Grassmannian , arXiv: 1510.08833

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    Fern´ andez de Bobadilla, M

    J. Fern´ andez de Bobadilla, M. Pe Pereira, The Nash problem for surfaces , Ann. of Math. 176 (2012), 20032029

    work page 2012

  7. [7]

    Ishii, J

    S. Ishii, J. Koll´ ar,The Nash problem on arc families of singularities , Duke Math. J. 120 (2003),601-620

    work page 2003

  8. [8]

    J. M. Johnson, J. Koll´ ar, Arc spaces of cA-type singularities , J. Sing. 7 (2013), 238-252

    work page 2013

  9. [9]

    Koll´ ar,Flips, flops, minimal models, etc , Surveys in Diff

    J. Koll´ ar,Flips, flops, minimal models, etc , Surveys in Diff. Geo. 1 (1991), 113-199

    work page 1991

  10. [10]

    Koll´ ar, S

    J. Koll´ ar, S. Mori, Birational geometry of algebraic varieties , Cambridge Tracts in Mathematics 134, Cambridge Univ. Press, 1998

    work page 1998

  11. [11]

    Nash, Jr., Arc structure of singularities , Duke

    J. Nash, Jr., Arc structure of singularities , Duke. Math. J. 81 (1995), 31-38

    work page 1995

  12. [12]

    Reid, Young persons guide to canonical singularities , Proc

    M. Reid, Young persons guide to canonical singularities , Proc. Symp. Pure Math. 46 (1987), 345-414. Department of Mathematics, National Taiwan University, No . 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan E-mail address : d02221002@ntu.edu.tw

    work page 1987