REVIEW 2 major objections 1 minor 9 references
The one-step Shafarevich gap in embedding dimension five
T0 review · 2 major / 1 minor · reviewed 2026-06-26 · grok-4.3
Pith's one-line read The one-step loci with Hilbert function (1,5,r) in embedding dimension five are smoothable for every r except 3 and 5.
desk verdict Zhao finishes the classification for one-step loci in five variables but the dominance-to-containment link needs explicit checking. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
Dominance of the Erman-Velasco map GL(5)×(A^5)^r dashrightarrow Gr(r,Sym^2 k^5) that sends (g,a^(1),…,a^(r)) to the span of the images g·q(a^(i)), whose dominance implies the translated one-step locus lies inside the smoothable component.
What would settle it
An explicit one-step ideal I_Q for some r between 6 and 15 whose corresponding point in the Hilbert scheme lies outside the smoothable component would falsify the containment claim.
Extended reading notes
Core claim
We prove that the translated one-step locus defined by these ideals is contained in the smoothable component for every r in {6,7,…,15}. Combined with the known small cases and with the known elementary components for r=3 and r=5, this gives the complete one-step classification in embedding dimension five: the one-step loci with Hilbert function (1,5,r) are smoothable for all r≠3,5, and the cases r=3,5 are precisely the generically reduced elementary component cases.
Load-bearing premise
Dominance of the Erman-Velasco map implies that the translated one-step locus lies inside the smoothable component.
Editorial extensions
If this is right
- The one-step Shafarevich gap is resolved in embedding dimension five.
- For r=6 to 15 the corresponding loci cannot be elementary components.
- The finite-field differential rank certificate supplies an explicit computational check of dominance for each r from 6 to 14.
- The r=15 case reduces to a flat degeneration of 21 general reduced points onto the fat point defined by m^3.
Reading between the lines
- The same dominance technique might extend the classification to embedding dimension six once an analogous map is constructed.
- If the smoothable component is irreducible in these degrees, the result would imply that the one-step loci are dense in it for r≠3,5.
- The flat degeneration for r=15 suggests that similar degenerations could handle boundary cases in higher embedding dimensions.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to resolve the one-step Shafarevich gap in embedding dimension five. For a codimension-r subspace Q of S_2 = Sym^2 k^5, it defines I_Q = (Q) + m^3 with Hilbert function (1,5,r). It proves that the translated one-step locus (closure of GL(5)-translates of such I_Q) lies in the smoothable component of the Hilbert scheme for every r in {6,...,15}. The proof uses a finite-field differential-rank certificate establishing dominance of the Erman-Velasco map GL(5) x (A^5)^r dashrightarrow Gr(r, Sym^2 k^5) for r=6 to 14, together with a flat degeneration of 21 general reduced points to the fat point defined by m^3 for r=15. Combined with known small cases and the elementary components at r=3,5, this yields the complete classification: the loci are smoothable for all r ≠ 3,5.
Significance. If the containment statements hold, the result completes the one-step classification in embedding dimension five, distinguishing the smoothable cases from the two known elementary components. The differential-rank certificate technique for proving dominance over finite fields is a potentially reusable tool for similar dominance questions in Grassmannians of quadratic forms.
major comments (2)
- [argument following the differential rank certificate for r=6 to 14] The abstract and the description of the argument for r=6 to 14 treat dominance of the Erman-Velasco map as immediately implying that the translated one-step locus lies inside the smoothable component. The precise geometric reduction step—whether dominance produces a dense set of smoothable points whose closure fills the locus, or whether an auxiliary flat family or identification of the image with a known smoothable family is required—is not visible. This step is load-bearing for the central claims in that range.
- [the separate treatment of the endpoint r=15] For the r=15 case, the flat degeneration of 21 general reduced points to the fat point defined by m^3 is invoked to place the locus in the smoothable component. The manuscript should explicitly verify that this degeneration preserves the one-step property and lands in the smoothable component without additional assumptions on the quadratic span.
minor comments (1)
- The abstract contains several LaTeX rendering artifacts (e.g., GL*5, sum*{i=1}^5) that should be cleaned for the published version.
Simulated Author's Rebuttal
We thank the referee for their thorough review and valuable suggestions. We address each major comment below and plan to revise the manuscript to improve the clarity of the geometric arguments.
read point-by-point responses
-
Referee: [argument following the differential rank certificate for r=6 to 14] The abstract and the description of the argument for r=6 to 14 treat dominance of the Erman-Velasco map as immediately implying that the translated one-step locus lies inside the smoothable component. The precise geometric reduction step—whether dominance produces a dense set of smoothable points whose closure fills the locus, or whether an auxiliary flat family or identification of the image with a known smoothable family is required—is not visible. This step is load-bearing for the central claims in that range.
Authors: We agree with the referee that the link between dominance and containment in the smoothable component should be spelled out more clearly. Dominance of the Erman-Velasco map means that the general point in Gr(r, Sym^2 k^5) is in the image, so the corresponding general I_Q is a GL(5)-translate of an ideal I constructed from r vectors a^{(1)},...,a^{(r)} in A^5 via the quadratic forms q(a^{(i)}). The construction via the a^{(i)} ensures that these ideals are smoothable, as they arise in a flat family degenerating to r distinct reduced points. Therefore, the general points in the translated one-step locus are smoothable, and its closure lies in the smoothable component. We will insert an explicit explanation of this reduction in the revised version. revision: yes
-
Referee: [the separate treatment of the endpoint r=15] For the r=15 case, the flat degeneration of 21 general reduced points to the fat point defined by m^3 is invoked to place the locus in the smoothable component. The manuscript should explicitly verify that this degeneration preserves the one-step property and lands in the smoothable component without additional assumptions on the quadratic span.
Authors: We will add the requested verification. The flat degeneration is constructed so that the general member is the ideal of 21 general reduced points in affine 5-space, which necessarily have Hilbert function (1,5,15) and thus quadratic span of codimension 15 (i.e., the full Sym^2), satisfying the one-step condition with no further assumptions. The special fiber is m^3, which has the same Hilbert function. Since the family is flat and the general fiber consists of smoothable schemes, the special fiber m^3 is in the smoothable component. The GL(5)-translates of m^3 remain in the component as well. This will be detailed in the revised manuscript. revision: yes
Circularity Check
No circularity: new dominance certificate and degeneration supply independent content
full rationale
The derivation introduces a finite-field differential-rank certificate establishing dominance of the Erman-Velasco map for r=6..14 and a separate flat degeneration for r=15; these steps are constructed explicitly in the paper and do not reduce by the paper's own equations to fitted parameters, self-definitions, or prior self-citations. Containment in the smoothable component is asserted to follow from the image of this map together with external known cases for small r; no load-bearing step equates a claimed prediction to its input by construction. The combination with known elementary components is standard citation practice and does not create a self-referential chain.
Assumptions & free parameters
assumptions (1)
- domain assumption k is an algebraically closed field of characteristic zero
Cite this review
Pith. "Pith review of The one-step Shafarevich gap in embedding dimension five." pith.science (2026). https://pith.science/paper/2YWLXKFV
@misc{pith2026260622368,
author = {Pith},
title = {Pith review of: The one-step Shafarevich gap in embedding dimension five},
year = {2026},
howpublished = {\url{https://pith.science/paper/2YWLXKFV}},
note = {Machine review of arXiv:2606.22368}
}
abstract
Let $k$ be an algebraically closed field of characteristic zero and let $S=k[x_1,\ldots,x_5]$ with maximal ideal $\mathfrak m=(x_1,\ldots,x_5)$. For a codimension-$r$ subspace $Q\subset S_2$, set $I_Q=(Q)+\mathfrak m^3$. Then $S/I_Q$ has Hilbert function $(1,5,r)$. We prove that the translated one-step locus defined by these ideals is contained in the smoothable component for every $r\in{6,7,\ldots,15}$. We introduce a finite field differential rank certificate proving dominance, for $6\le r\le 14$, of the Erman--Velasco map $\operatorname{GL}*5\times (\mathbb A^5)^r\dashrightarrow \operatorname{Gr}(r,\operatorname{Sym}^2 k^5)$, $(g,a^{(1)},\ldots,a^{(r)})\mapsto g\cdot\langle q(a^{(1)}),\ldots,q(a^{(r)})\rangle$, where $q(a)=\sum*{i=1}^5 a_i y_i^2-\left(\sum_{i=1}^5 a_i y_i\right)^2$. The endpoint $r=15$ is handled separately by a flat degeneration of $21$ general reduced points to the fat point defined by $\mathfrak m^3$. Combined with the known small cases and with the known elementary components for $r=3$ and $r=5$, this gives the complete one-step classification in embedding dimension five: the one-step loci with Hilbert function $(1,5,r)$ are smoothable for all $r\neq 3,5$, and the cases $r=3,5$ are precisely the generically reduced elementary component cases. In this sense the one-step Shafarevich gap in embedding dimension five is completely resolved.
Reference graph
Works this paper leans on
-
[1]
Cartwright, Daniel Erman, Mauricio Velasco, and Bianca Viray,Hilbert schemes of 8 points, Algebra Number Theory3(2009), no
Dustin A. Cartwright, Daniel Erman, Mauricio Velasco, and Bianca Viray,Hilbert schemes of 8 points, Algebra Number Theory3(2009), no. 7, 763–795
2009
-
[2]
Math.224(2010), no
Daniel Erman and Mauricio Velasco,A syzygetic approach to the smoothability of zero-dimensional schemes, Adv. Math.224(2010), no. 3, 1143–1166
2010
- [3]
- [4]
-
[5]
Anthony Iarrobino,Punctual Hilbert schemes, Mem. Amer. Math. Soc.10(1977), no. 188, viii+112
1977
-
[6]
Joachim Jelisiejew,Elementary components of Hilbert schemes of points, J. Lond. Math. Soc. (2)100(2019), no. 1, 249–272
2019
-
[7]
Math.805, Amer
Joachim Jelisiejew,Open problems in deformations of Artinian algebras, Hilbert schemes and around, in Deformation of Artinian algebras and Jordan type, Contemp. Math.805, Amer. Math. Soc., 2024, pp. 3–25
2024
-
[8]
Grayson and Michael E
Daniel R. Grayson and Michael E. Stillman,Macaulay2, a software system for research in algebraic geometry, available athttps://macaulay2.com/
Show all 9 references
-
[9]
Shafarevich,Deformations of commutative algebras of class 2, Algebra i Analiz2(1990), no
Igor R. Shafarevich,Deformations of commutative algebras of class 2, Algebra i Analiz2(1990), no. 6, 178–196; English transl., Leningrad Math. J.2(1991), no. 6, 1335–1351. Department of Mathematics, Imperial College London, United Kingdom Email address:cz2922@ic.ac.uk
1990
Reviewed June 26, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.