REVIEW 6 minor 1 cited by
Retract rational varieties are uniformly retract rational
T0 review · 0 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read This paper proves that nonsingular retract rational algebraic varieties over any infinite field are uniformly retract rational, and that every rational, projective, nonsingular complex variety is therefore algebraically elliptic.
desk verdict Banecki proves that nonsingular retract rational implies uniformly retract rational over any infinite field; the result is significant, the proof is coherent, and the paper deserves peer review despite needing proofreading. 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
The machinery is Proposition 2.2, which produces a polynomial mapping $\sigma: X \times K^{n-m} \to K^n$ with three properties: it fixes $X$ on the slice $t=0$, its derivative at $(x_0,0)$ is an isomorphism, and for a given nonzero ideal $I \subset P(X)$ the induced local homomorphism onto $P(X)[t_1,\dots,t_{n-m}] / I(t_1,\dots,t_{n-m})$ is surjective at the point. The subspace $W$ defining the additive part of $\sigma$ is chosen generically (Lemma 2.3) so that it is transverse to $X$ at $x_0$ and meets the zero set of $I$ only at the origin. Proposition 3.1 then uses $\sigma$ to regularise a rational map $F: K^n \dashrightarrow Y$ that is regular on a subvariety $X$: one perturbs the coordinates of $F \circ \sigma$ modulo the denominator ideal, applies Lemma 2.4 to keep rational functions regular, and uses the derivative isomorphism (Observation 2.1) to transfer regularity back from $X \times K^{n-m}$ to $K^n$. The retraction claimed in Theorem 1.8 is this regularised germ.
What would settle it
A concrete way to test Theorem 1.8 would be to exhibit a nonsingular retract rational variety over an infinite field with a point $x_0$ for which no local retraction exists; equivalently, one could look for a nonzero ideal $I$ vanishing at $x_0$ such that every codimension-$m$ subspace transverse to $X$ at $x_0$ meets the zero set of $I$ away from the origin, which would invalidate Lemma 2.3 and the regularisation proposition built on it.
Extended reading notes
Core claim
The central claim is Theorem 1.8: every nonsingular retract rational algebraic variety over an infinite field $K$ is uniformly retract rational. Retract rational means there is a dense Zariski open subset $V \subset X$, an open subset $U$ of some affine space, and regular maps $V \to U \to V$ whose composition is the identity on $V$. Uniformly retract rational means the same data can be found for every point $x \in X$, with $V$ a Zariski open neighbourhood of $x$. The theorem promotes one global retraction datum into local retractions at every point, using nonsingularity in an essential way. The proof of Theorem 1.8 obtains, for each point $x_0$, a regular germ $G: (K^n, x_0) \to X$ that restricts to the identity on $X$ as a germ; the domain of $G$ then supplies the required neighbourhood $U'$ and retraction.
Load-bearing premise
The proof needs the geometric fact that after choosing a generic complementary subspace $W$, the zero set of the ideal $I$ meets $W$ only at the base point; if that choice were impossible, the regularisation lemma at the heart of the argument would fail.
Editorial extensions
If this is right
- Every nonsingular rational variety over an infinite field is uniformly retract rational, because rational varieties are retract rational.
- Every rational, projective, nonsingular complex variety is algebraically elliptic.
- Every nonsingular retract rational real variety is uniformly retract rational and hence malleable.
- For nonsingular varieties over infinite fields, retract rationality and uniform retract rationality coincide.
Reading between the lines
- The paper leaves open whether nonsingular retract rational varieties are actually uniformly rational; the result here does not produce biregular models of neighbourhoods, only retractions, so uniform rationality remains a separate and stronger question.
- The regularisation scheme of Proposition 3.1 may apply to other local-global problems: any rational map on affine space that is regular along a nonsingular subvariety can be made regular on a Zariski neighbourhood of a point if the denominator ideal is controlled by a generic transverse subspace.
- One could try to extend the theorem to singular retract rational varieties; the proof relies on the tangent-space isomorphism and regularity of local rings, so a singular counterexample would mark the boundary of the statement.
- In the real setting, the theorem simplifies the study of maps between real algebraic varieties: nonsingular retract rational varieties now carry the same local retraction data as uniformly retract rational ones, which may make the approximation results applicable to a wider class.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves Theorem 1.8: every nonsingular retract rational algebraic variety over an infinite field is uniformly retract rational. The proof reduces the main theorem to Proposition 3.1, a local extension statement for rational maps whose restriction to a nonsingular subvariety is regular, and Proposition 2.2, which constructs a polynomial map σ with a surjectivity property modulo a product ideal. The corollary that rational projective nonsingular complex varieties are algebraically elliptic follows from Observation 1.6. I checked the main line of reasoning; the potentially delicate dimension count in Lemma 2.3 is correct, and I found no circularity in the derivation of the main theorem.
Significance. If correct, the result settles the retract-rational analogue of Gromov's uniform rationality question and gives a new partial answer to Question 1.3 for rational complex projective varieties. The proof is short, mostly self-contained, and constructive in a useful way: Proposition 3.1 turns a rational retraction defined on one Zariski open set into regular local retractions around every point. The paper also records the real analogue through malleability, citing prior work [2] for Observation 1.6. No machine-checked proofs or code are supplied, but the algebraic steps are concrete and checkable; the main lemma is supported by an explicit dimension count. The use of the author's own previous paper is limited to a peripheral observation and does not affect the main theorem.
minor comments (6)
- [Section 2, Proposition 2.2, equation (2.1)] The indices in equation (2.1) are inconsistent with the definition of W. Since W is defined by x_{n-m+1} = ... = x_n = 0, the Noether normalization statement should read that P(X) is integral over K[\bar{x}_{n-m+1}, ..., \bar{x}_n], not over K[\bar{x}_1, ..., \bar{x}_m]. The later sentence 'for n-m < i ≤ n we have \bar{x}_i = x_i' confirms that the last m coordinates are intended.
- [Section 2, Proposition 2.2, proof] The sentence 'For 1 ≤ i ≤ k let t_i := x_i + I(W)' uses an undefined index k; it should be 1 ≤ i ≤ n-m. Correspondingly, the zero-set computation 'v ∈ Z_K(t_1,...,t_m) = {0}' should refer to the ideal (t_1,...,t_{n-m}), because W has dimension n-m.
- [Section 2, Proposition 2.2, final Nakayama step] The definition of the quotient module M := P(X×W)/(I(t_1,...,t_{n-m}) + P(K^n)) is ambiguous because P(K^n) is a subring, not an ideal. The displayed relation 'M ⊂ M m' appears to be a typographical corruption of M ⊂ \mathfrak{m}M or M_m = \mathfrak{m}M_m. Please rewrite this step, explicitly identifying the image of P(K^n) in P(X×W) and noting that R = n + A because the image of A maps onto the residue field R/n.
- [Section 2, Lemma 2.3] The proof is sound, but the dimension count could be stated more transparently: each fiber of Y over Z\{0} is Gr(n-m-1, n-1), of dimension (n-m-1)m, and dim Z ≤ m-1, giving dim Y ≤ (n-m)m - 1 < dim Gr(n-m, n). The reduction to the algebraically closed case is valid because K-points of the Grassmannian are Zariski dense for infinite K.
- [Section 2, Observation 2.1] The argument that divisibility passes from the completion back to the local ring is terse; since it uses faithful flatness of the completion over the local ring, adding one sentence with the standard reference or a brief explanation would improve readability.
- [Section 3, Proposition 3.1] The arrows in the displayed statement of Proposition 3.1 are garbled in the text (e.g., 'K^n /axisshort/axisshort/arrowaxisrightY'). This is a typesetting issue, but the intended statement should be written with a standard dashed arrow for the rational map and a solid arrow for the regular germ.
Circularity Check
No significant circularity: the main theorem is derived from independent local extension results; the only self-citation is non-central.
full rationale
I walked the claimed derivation chain. Theorem 1.8 is proved from Definition/Question 1.7 by applying Proposition 3.1 to the rational map F=r∘i0, obtaining a regular germ that restricts to the identity on X. Proposition 3.1 is established using Proposition 2.2 and Lemmas 2.3–2.4, none of which assumes retract rationality or uniform retract rationality. Lemma 2.3 is a dimension count on a Grassmannian, and Proposition 2.2 constructs σ(x,v)=x+v for a generic linear subspace W, verifying the required surjectivity by integrality and Nakayama's lemma; this construction does not encode the target conclusion. The only self-citation appears in the proof of Observation 1.6 for the real case, referring to the author's prior [2, Theorem 2.11], and in attributing Definition 1.5 to [2]. That citation is not load-bearing for Theorem 1.8 or Corollary 1.9, whose complex case is proved in the text using external results [8, Proposition 6.4.1] and [10, Theorem 1.1]. I found no equation that reduces to its own input, no fitted parameter renamed as a prediction, and no uniqueness claim imported from the authors' earlier work to force the conclusion. The proof is self-contained apart from standard external theorems; its textual typos concern exposition rather than circularity.
Assumptions & free parameters
assumptions (3)
- standard math Standard algebraic geometry results are assumed: Noether normalization lemma, Nakayama's lemma, properties of local rings and completions, Grassmannian dimension counting.
- domain assumption The ground field K is infinite.
- domain assumption For the corollary, X is a complex projective nonsingular rational variety, which is assumed to be irreducible and nonsingular.
Cite this review
Pith. "Pith review of Retract rational varieties are uniformly retract rational." pith.science (2026). https://pith.science/paper/PB7YA5LZ
@misc{pith2026241117892,
author = {Pith},
title = {Pith review of: Retract rational varieties are uniformly retract rational},
year = {2026},
howpublished = {\url{https://pith.science/paper/PB7YA5LZ}},
note = {Machine review of arXiv:2411.17892}
}
read the original abstract
We prove that nonsingular retract rational algebraic varieties over any infinite field are uniformly retract rational. As a consequence, every rational, projective, nonsingular complex variety is algebraically elliptic.
Forward citations
Cited by 1 Pith paper
-
Extension of $k$-regulous functions from varieties of arbitrary dimension
Banecki claims a general extension theorem for k-regulous functions, but a key step asserting the local extension fixes the variety is unjustified.
Reference graph
Works this paper leans on
-
[2]
J. Banecki. Relative Stone-Weierstrass theorem for map pings between varieties. arXiv preprint, 2408.09233, 2024
work page Pith review arXiv 2024
-
[1]
I. Arzhantsev, S. Kaliman, and M. Zaidenberg. Varieties covered by affine spaces, uniformly rational varieties and their cones. Advances in Mathematics , 437:109449, 2024
work page 2024
-
[3]
Approximation of maps from algebraic polyhedra to real algebraic varieties
M. Bilski and W. Kucharz. Approximation of maps from alge braic polyhedra to real algebraic varieties. arXiv preprint , 2410.23457, 2024
work page Pith review arXiv 2024
-
[4]
J. Bochnak, M. Coste, and M.-F. Roy. Real Algebraic Geometry. Springer Berlin Heidelberg, Berlin, Heidelberg, 1998
work page 1998
-
[5]
J. Bochnak and W. Kucharz. On approximation of maps into r eal algebraic homogeneous spaces (with an appendix by J´ anos Koll´ ar).Journal de Math´ ematiques Pures et Appliqu´ ees, 161:111–134, 2022
work page 2022
-
[6]
F. Bogomolov and C. B¨ ohning.On uniformly rational varieties , pages 33–48. American Math- ematical Society Translations: Series 2. American Mathema tical Society, 2014
work page 2014
-
[7]
C. H. Clemens and Ph. A. Griffiths. The intermediate jacobi an of the cubic threefold. Annals of Mathematics , 95(2):281–356, 1972
work page 1972
-
[8]
F. Forstneriˇ c. Stein Manifolds and Holomorphic Mappings: The Homotopy Pri nciple in Complex Analysis . Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folg e / A Series of Modern Surveys in Mathematics. Springer Berlin Heidelberg , 2011
work page 2011
Show all 14 references
-
[9]
M. Gromov. Oka’s principle for holomorphic sections of e lliptic bundles. Journal of the Amer- ican Mathematical Society , 2(4):851–897, 1989
1989
-
[10]
Kaliman and M
S. Kaliman and M. Zaidenberg. Gromov ellipticity and su bellipticity. Forum Mathematicum, 36(2):373–376, 2024
2024
-
[11]
Kaliman and M
S. Kaliman and M. Zaidenberg. Algebraic gromov’s ellip ticity of cubic hypersurfaces. arXiv preprint, 2402.04462, 2025
2025 arXiv
-
[12]
W. Kucharz. Spaces of maps between real algebraic varie ties. Bulletin of the London Math- ematical Society, 57(3):669–680, 2025
2025
-
[13]
D. J. Saltman. Noether’s problem over an algebraically closed field. Inventiones mathemati- cae, 77(1):71–84, Feb 1984
1984
-
[14]
Zaidenberg
M. Zaidenberg. Algebraic Gromov Ellipticity: A Brief S urvey. Taiwanese Journal of Mathe- matics, pages 1 – 25, 2024. F aculty of Mathematics and Computer Science, Jagiellonian University, ul. Lo- jasiewicza 6, 30-348 Krakow, Poland Email address : juliusz.banecki@student.uj.edu.pl
2024
Reviewed August 12, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.