Pith. sign in

REVIEW 3 major objections 4 minor 27 references

Covers of rational double points in mixed characteristic

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read In mixed characteristic $(0,p>5)$, every Gorenstein rational surface singularity is a rational double point with a split finite regular cover.

desk verdict Solid mixed-characteristic classification paper; the E8 descent worry dissolves on close reading, but the finiteness step before reduction deserves a clearer proof. read the letter →

arxiv 1908.01416 v2 pith:RNNTYWG3 submitted 2019-08-04 math.AG math.AC

Javier Carvajal-Rojas , Linquan Ma , Thomas Polstra , Karl Schwede , Kevin Tucker This is my paper · ORCID
classification math.AGmath.AC MSC 14J1714B0513H10
keywords rationalsurfacesingularitiesdoublepointsmixedcharacteristicsplitfinitecoverscyclicDynkindiagramsBCM-regularclassificationof
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper extends the classical story of rational surface singularities to mixed characteristic $(0,p>5)$. Its main theorem asserts that every excellent, strictly Henselian, 2-dimensional Gorenstein rational singularity is a rational double point—one of the hypersurface types $A_n$, $D_n$, $E_6$, $E_7$, $E_8$—and that each such singularity admits a finite cover by a regular local ring for which the structure map splits as a map of modules. Splitting means the singular ring is a direct summand of a regular ring, the same behavior already known in equicharacteristic $0$ and in prime characteristic $p>5$. The authors first classify all defining equations (Theorem B), then build the covers type by type: cyclic covers for $A_n$, $D_n$, $E_6$, and $E_7$, and an explicit degree-120 cover for $E_8$. They apply the result to show that 2-dimensional BCM-regular singularities in this mixed-characteristic setting are finite direct summands of regular rings.

What carries the argument

The load-bearing objects are the rational double point equations and the cyclic cover construction attached to the divisor class group. For each type except $E_8$, the paper identifies a pure height-one prime ideal whose class has finite order in the divisor class group; the cyclic cover of that order is a domain by the Tomari--Watanabe cyclic cover theorem, and the authors verify directly that it is regular and that the structure map splits. For $E_8$, whose divisor class group is trivial, the cover is instead the explicit subring $W(k)[[u,v]]$ mapped by the icosahedral polynomial triple $f_1,f_2,f_3$; the trace map of this degree-120 finite extension, divided by the degree, gives the splitting. A general descent claim (Claim 4.12) is used to pass the finiteness and degree of this extension from characteristic zero down to the mixed-characteristic Witt vector setting.

What would settle it

Take $p=7$ and compute the module structure of $W(k)[[u,v]]$ over $W(k)[[x,y,z]]/(x^2+y^3+z^5)$ under the map sending $x,y,z$ to the three explicit polynomials $f_1,f_2,f_3$; concretely, test whether some power of the maximal ideal annihilates the cokernel. If the extension is not finite, the trace-splitting proof of Proposition 4.11 fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that rational double points in mixed characteristic $(0,p>5)$ behave like their equicharacteristic counterparts not only in classification but also in covering: they are covered by regular schemes in a split way. Theorem B states that if $(S,\mathfrak{n},k)$ is a 3-dimensional complete regular local ring of mixed characteristic with separably closed residue field and $f \in \mathfrak{n}^2-\mathfrak{n}^3$ defines a rational double point, then after choosing minimal generators and multiplying by a unit, $f$ must be exactly one of $x^2+y^2+z^{n+1}$, $x^2+y^2z+z^{n-1}$, $x^2+y^3+z^4$, $x^2+y^3+yz^3$, or $x^2+y^3+z^5$. Theorem A then asserts that a strictly Henselian excellent Gorenstein rational singularity of dimension 2 is such a hypersurface and that there is a finite cover $Y \to \operatorname{Spec}(R)$ with $Y$ regular such that $\mathcal{O}_R \to \pi_*\mathcal{O}_Y$ splits. The covers are constructed explicitly, and in the $E_8$ case the construction exhibits three polynomials $f_1,f_2,f_3$ in two variables satisfying $f_1^2+f_2^3+f_3^5=0$ and generating a finite split extension of degree 120.

Load-bearing premise

The heart of the proof is a descent step: a degree-$120$ cover that is finite over the characteristic-zero ring is asserted to remain finite after reducing modulo the mixed-characteristic prime; if the reduction loses finiteness, the trace map that splits the $E_8$ cover may not exist.

Editorial extensions

If this is right

  • Every $A_n$ singularity in this setting has a cyclic cover of index $n+1$ that is regular, since the divisor class of $(x+iy,z)$ has order $n+1$.
  • Every $D_n$ singularity has a degree-2 cyclic cover of type $A_{2n-5}$; $E_6$ has a degree-3 cyclic cover of type $D_4$; and $E_7$ has a degree-2 cyclic cover of type $E_6$, so the covers chain down to a regular ring.
  • For $E_8$, an explicit finite split regular cover is given by the polynomials $f_1,f_2,f_3$, and the cover is etale on the punctured spectrum whenever the residual characteristic is $p>5$.
  • As an application, every 2-dimensional BCM-regular local singularity of mixed characteristic $(0,p>5)$ is a finite direct summand of a regular ring (Theorem 5.2).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The explicit $E_8$ cover polynomials are written with rational coefficients, so it is worth testing whether the same triple yields a split regular cover in equicharacteristic $p>5$ and possibly at boundary primes where the current hypotheses stop applying.
  • The chain of cyclic covers $A \leftarrow D \leftarrow E_6 \leftarrow E_7$, capped by the degree-120 $E_8$ cover, suggests that the split covers might assemble into a tower whose Galois group is related to the Weyl group or icosahedral group; the paper does not pursue this structural statement.
  • Because the classification is of equations up to unit multiplication rather than of rings up to isomorphism, the same Dynkin type can hide genuinely different mixed-characteristic singularities; the explicit covers may depend on the chosen $p$-adic parameter, so tracking that dependence is a natural next step.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper classifies, in mixed characteristic (0,p>5), the possible defining equations of a 2-dimensional rational double point inside a 3-dimensional regular local ring, proving the normal forms listed in Theorem B (An, Dn, E6, E7, E8). It then uses this classification to prove Theorem A: every excellent strictly Henselian 2-dimensional Gorenstein rational singularity is a rational double point and admits a finite cover by a regular scheme that splits as a map of O_X-modules. The final section applies Theorem A to show that 2-dimensional BCM-regular singularities of mixed characteristic (0,p>5) are finite direct summands of regular rings.

Significance. The classification and the split-cover theorem are a substantial advance: they extend Artin's equicharacteristic classification and Lipman's E8 treatment to mixed characteristic, with explicit cyclic covers for types A, D, E6, E7 and an explicit degree-120 cover for E8. The proof is organized by singularity type and includes detailed coordinate-change arguments. If the gaps in the E8 descent are repaired, the results would be a significant step for the theory of rational singularities and for mixed-characteristic BCM-regularity.

major comments (3)
  1. [§4.11, Claim 4.12] Claim 4.12 is false as stated. Take A = k[t], B = k[t,s]/(s^2 - t), and p = (t). Then A -> B is a finite extension of domains of generic degree 2, B is Cohen-Macaulay, and pd_A(A/p) is finite, but B/pB = k[s]/(s^2) is not reduced; the degree of the reduced generic fiber of A/p -> B/pB is 1, not 2. The proof of the claim identifies the rank of the finite free localization B_p over A_p with the residual degree, but when B/pB has nilpotents the length of the generic fiber exceeds the sum of the degrees of its reduced components. In the application, A = B0/pB0 is asserted to be regular and hence a domain, but that assertion is not part of Claim 4.12 and is not proved at the point where it is used. Therefore the trace-map argument in Proposition 4.11 does not currently establish that R -> A is a finite extension of degree 120.
  2. [§4.11, before Claim 4.12] The finiteness of W(k)[[x,y,z]]/(x^2+y^3+z^5) -> W(k)[[u,v]] over W(k) is justified only by inverting p and invoking the equicharacteristic zero case. Localization at the element p is not faithfully flat because p lies in the maximal ideal, so finiteness of A0[1/p] -> B0[1/p] does not imply finiteness of A0 -> B0. The authors need a direct proof of integral finiteness over W(k), for example by showing that the f_i generate the invariant subring for a finite group action that is defined over W(k), or by proving that u and v are integral over the image of the hypersurface ring.
  3. [§4.11, definition of A] The ring A := W(k)[[u,v]]/(p - Q(f1,f2,f3)) is called a regular local ring without proof. This is true because p - Q(f1,f2,f3) has class p in m/m^2 (as Q(f1,f2,f3) is in (u,v)), so it is a regular parameter in the regular ring W(k)[[u,v]]; but the argument should be included, since the claim that A is a domain is needed to identify the generic degree of the reduced fiber with the localization rank. As written, the proof of the E8 case is incomplete at this point.
minor comments (4)
  1. [Table of contents / Section 2] The section title 'Prelimaries' should be spelled 'Preliminaries'.
  2. [Lemma 5.1] The phrase 'Assume noe p | n' should read 'Assume now p | n'.
  3. [Proposition 3.8] The phrase 'one can check easily, with for instance Macaulay2' appears for a finite polynomial identity used in the E7 reduction. The authors should either display the expanded identity or state that it is a direct finite expansion, since the surrounding proof is otherwise computational.
  4. [Abstract] The sentence 'The classification of such functions are used' should use the singular verb 'is used'.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: classification and covers are derived from rationality constraints and explicit constructions; self-citations are auxiliary tools. The E8 descent gap in Claim 4.12 is a proof-gap, not circularity.

full rationale

Theorem B is obtained by normal-form manipulations (Propositions 3.2–3.8) that reduce f using the hypothesis that the singularity is rational and that quadratic transforms preserve normality; no target equation is assumed. Theorem A's cyclic-cover cases are explicit: for A_n, D_n, E6, and E7 the covers are constructed and proved regular by direct computation of maximal ideals and relations; for E8, the polynomials f1,f2,f3 are verified to satisfy f1^2+f2^3+f3^5=0 and are traced to the icosahedral invariant ring (Klein), not to the theorem being proved. The use of [Car17, Proposition 4.21] and [MS18] is as technical tools (locality of cyclic covers, BCM test ideals); these citations do not assume Theorem A or Theorem B and therefore do not make the argument circular. The one genuinely questionable point is not circular: in Proposition 4.11 the descent to characteristic p is justified by Claim 4.12, which identifies the residual degree of the finite free extension A_p→B_p with the generic degree of A/p→B/pB; this identification requires B/pB to be reduced or a domain, whereas the regularity of A=W(k)[[u,v]]/(p−Q(f1,f2,f3)) is only asserted in the course of the proof. That is a missing-support or correctness issue in the E8 descent, not a reduction of the conclusion to its hypotheses. No fitted parameter is later renamed as a prediction, and no uniqueness theorem from the authors' prior work is invoked to force the construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims introduce no free parameters fitted to data and no new postulated entities. The proofs rest on prior structural results in singularity theory (Lipman, Artin, Andre, Ma-Schwede) and on standard commutative algebra. These are listed as axioms above.

assumptions (6)
  • standard math Excellent surfaces admit resolutions of singularities; quadratic transforms preserve rationality ([Lip69, Lip78]).
    Used in Section 2 to characterize rational singularities and throughout the classification proofs to test rationality by blowing up.
  • standard math The divisor class group of a strictly Henselian rational surface singularity is unchanged under completion ([Lip69, Proposition 17.1, Proposition 16.3, and correction on page 279]).
    Used in Corollaries 4.6 and 4.13 to transfer the complete-case cyclic cover constructions to the Henselian case.
  • domain assumption The BCM test ideal and B-regularity framework, including the purity criterion and Theorem 6.17, is taken from [MS18].
    Section 5 applies the paper's main theorem to BCM-regular singularities; these definitions and results are cited from the authors' earlier work.
  • standard math Direct summands of regular rings are splinters in mixed characteristic ([And18]); in equal characteristic p>5 rational double points are F-regular and splinters ([HH94]).
    Used to compare with known equicharacteristic results and in Example 4.14 to show failure of splitting in small characteristic.
  • standard math The cyclic cover of a normal domain associated to a Weil divisor of order n is a domain ([TW92, Corollary 1.9]).
    Used in Proposition 2.1 and in the construction of all cyclic covers in Section 4.
  • standard math Artin's equal-characteristic classification of rational double points for p>5 [Art77] and Lipman's E8 analysis [Lip69, Section 25] are used as background and comparison.
    The introduction frames the mixed characteristic result as a generalization of Artin's classification; the classification in Section 3 is independent of Artin's list except as comparison.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Covers of rational double points in mixed characteristic." pith.science (2026). https://pith.science/paper/RNNTYWG3

@misc{pith2026190801416,
  author       = {Pith},
  title        = {Pith review of: Covers of rational double points in mixed characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RNNTYWG3}},
  note         = {Machine review of arXiv:1908.01416}
}
abstract

We further the classification of rational surface singularities. Suppose $(S, \mathfrak{n}, \mathcal{k})$ is a strictly Henselian regular local ring of mixed characteristic $(0, p > 5)$. We classify functions $f$ for which $S/(f)$ has an isolated rational singularity at the maximal ideal $\mathfrak{n}$. The classification of such functions are used to show that if $(R, \mathfrak{m}, \mathcal{k})$ is an excellent, strictly Henselian, Gorenstein rational singularity of dimension $2$ and mixed characteristic $(0, p > 5)$, then there exists a split finite cover of $\mbox{Spec}(R)$ by a regular scheme. We give an application of our result to the study of $2$-dimensional BCM-regular singularities in mixed characteristic.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 26 canonical work pages

  1. [1]

    Andr\' e : La conjecture du facteur direct, Publ

    Y. Andr\' e : La conjecture du facteur direct, Publ. Math. Inst. Hautes \' E tudes Sci. 127 (2018), 71--93. 3814651

    work page 2018

  2. [2]

    Artin : Coverings of the rational double points in characteristic p , Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp

    M. Artin : Coverings of the rational double points in characteristic p , Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 11--22. 0450263

    work page 1977

  3. [3]

    Artin : On isolated rational singularities of surfaces, Amer

    M. Artin : On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129--136. 0199191

    work page 1966

  4. [4]

    M. F. Atiyah and I. G. Macdonald : Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR0242802 (39 \#4129)

    work page 1969

  5. [5]

    Bhatt , O

    B. Bhatt , O. Gabber , and M. Olsson : Finiteness of \'etale fundamental groups by reduction modulo p , ArXiv e-prints (2017)

    work page 2017

  6. [6]

    Finite torsors over strongly $F$-regular singularities

    J. Carvajal-Rojas : Finite torsors over strongly F -regular singularities , ArXiv e-prints (2017), arXiv: 1710.06887

    work page Pith review arXiv 2017

  7. [7]

    P. Du Val : On isolated singularities of surfaces which do not affect the conditions of adjunction (part i.), Mathematical Proceedings of the Cambridge Philosophical Society 30 (1934), no. 4, 453–459

    work page 1934

  8. [8]

    P. Du Val : On isolated singularities of surfaces which do not affect the conditions of adjunction (part ii.), Mathematical Proceedings of the Cambridge Philosophical Society 30 (1934), no. 4, 460–465

    work page 1934

Show all 27 references
  1. [9]

    P. Du Val : On isolated singularities of surfaces which do not affect the conditions of adjunction (part iii.), Mathematical Proceedings of the Cambridge Philosophical Society 30 (1934), no. 4, 483–491

    1934

  2. [10]

    Elkik : Solutions d'\' e quations \`a coefficients dans un anneau hens\' e lien , Ann

    R. Elkik : Solutions d'\' e quations \`a coefficients dans un anneau hens\' e lien , Ann. Sci. \' E cole Norm. Sup. (4) 6 (1973), 553--603 (1974). 0345966

    1973

  3. [11]

    Fedder : F -purity and rational singularity , Trans

    R. Fedder : F -purity and rational singularity , Trans. Amer. Math. Soc. 278 (1983), no. 2, 461--480. MR701505 (84h:13031)

    1983

  4. [12]

    D. R. Grayson and M. E. Stillman : Macaulay2, a software system for research in algebraic geometry

  5. [13]

    Hartshorne : Generalized divisors on G orenstein schemes , Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), vol

    R. Hartshorne : Generalized divisors on G orenstein schemes , Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), vol. 8, 1994, pp. 287--339. MR1291023 (95k:14008)

    1992

  6. [14]

    Hawes : InvariantRing: A Macaulay2 package

    T. Hawes : InvariantRing: A Macaulay2 package. Version 1.1.0

  7. [15]

    Hawes : Computing the invariant ring of a finite group, J

    T. Hawes : Computing the invariant ring of a finite group, J. Softw. Algebra Geom. 5 (2013), 15--19. 3073718

    2013

  8. [16]

    Hochster and C

    M. Hochster and C. Huneke : Tight closure, invariant theory, and the B rian con- S koda theorem , J. Amer. Math. Soc. 3 (1990), no. 1, 31--116. MR1017784 (91g:13010)

    1990

  9. [17]

    Hochster and C

    M. Hochster and C. Huneke : Tight closure of parameter ideals and splitting in module-finite extensions, J. Algebraic Geom. 3 (1994), no. 4, 599--670. 1297848 (95k:13002)

    1994

  10. [18]

    F. Klein : Lectures on the icosahedron and the solution of equations of the fifth degree, revised ed., Dover Publications, Inc., New York, N.Y., 1956, Translated into English by George Gavin Morrice. 0080930

    1956

  11. [19]

    Koll\' a r : Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol

    J. Koll\' a r : Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013, With a collaboration of S\' a ndor Kov\' a cs. 3057950

    2013

  12. [20]

    Lipman : Rational singularities, with applications to algebraic surfaces and unique factorization, Inst

    J. Lipman : Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes \'Etudes Sci. Publ. Math. (1969), no. 36, 195--279. MR0276239 (43 \#1986)

    1969

  13. [21]

    Lipman : Desingularization of two-dimensional schemes, Ann

    J. Lipman : Desingularization of two-dimensional schemes, Ann. Math. (2) 107 (1978), no. 1, 151--207. 0491722 (58 \#10924)

    1978

  14. [22]

    Lipman and B

    J. Lipman and B. Teissier : Pseudorational local rings and a theorem of B rian con- S koda about integral closures of ideals , Michigan Math. J. 28 (1981), no. 1, 97--116. MR600418 (82f:14004)

    1981

  15. [23]

    Ma and K

    L. Ma and K. Schwede : Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras , ArXiv e-prints (2018)

    2018

  16. [24]

    Prill : Local classification of quotients of complex manifolds by discontinuous groups, Duke Math

    D. Prill : Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J. 34 (1967), 375--386. 0210944

    1967

  17. [25]

    Roman : Field theory, second ed., Graduate Texts in Mathematics, vol

    S. Roman : Field theory, second ed., Graduate Texts in Mathematics, vol. 158, Springer, New York, 2006. 2178351

    2006

  18. [26]

    Tomari and K

    M. Tomari and K. Watanabe : Normal Z_r -graded rings and normal cyclic covers , Manuscripta Math. 76 (1992), no. 3-4, 325--340. 1185023 (93j:13002)

    1992

  19. [27]

    write newline

    " write newline "" before.all 'output.state := FUNCTION output.nonempty.mrnumber duplicate missing pop "" 'skip if duplicate empty 'pop " " swap * " " * write if FUNCTION fin.entry add.period write mrnumber output.nonempty.mrnumber newline INTEGERS nameptr namesleft numnames F...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.