{"id":"b052ba5c-be46-4875-87fb-dbfc16510b13","arxiv_id":"2411.17892","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Nonsingular retract rational varieties are uniformly retract rational, implying rational projective complex varieties are algebraically elliptic.","lead":"This paper proves that every nonsingular retract rational algebraic variety over any infinite field is uniformly retract rational. It then concludes that every rational, projective, nonsingular complex variety is algebraically elliptic, answering a partial question of Gromov.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the core argument of Theorem 1.8 is coherent and the potentially weak lemma is actually sound.","rationale":"The reader identified Lemma 2.3 as the weakest assumption, but the dimension count there is rigorous: for a nonzero ideal I with 0 in Z(I), the zero set has codimension at least 1, so the incidence variety Y has dimension strictly less than the Grassmannian. The reduction to the algebraically closed case is legitimate because K-points are Zariski dense for infinite K. The heavier machinery in Proposition 2.2 is obscured by typographical errors, but tracing the argument with corrected indexing shows the intended module-theoretic reasoning works: R is finite over A, the ideal mR+I·(t) is n-primary, and Nakayama gives the required surjectivity. The central proof of Theorem 1.8 then follows without circularity: Proposition 3.1 is a direct consequence, and the retract-rational hypothesis supplies the needed rational map. The typos are a serious readability problem but not a mathematical one. Since the reader's specific worry does not survive scrutiny, and no alternative load-bearing concern emerged, the conditional verdict need not change, but the paper would benefit from a careful rewrite of Section 2.","tokens_in":7120,"tokens_out":49970,"duration_ms":425616,"concrete_test":"Rewrite the proof of Proposition 2.2 with correct indices (W = {x_{n-m+1}=...=x_n=0}, Noether normalization over \\bar{x}_{n-m+1},...,\\bar{x}_n) and verify that the Nakayama step indeed yields M = m M, i.e., that A_{x0}+I·(t)+mR = R locally. Alternatively, explicitly compute the localization map for X={y=x^2}, W={y=x}, I=(x), checking that x and t are in the image of C[u,v]_0 in C[x,t]/(xt).","verdict_should_be":"UNCHANGED","load_bearing_attack":"After a careful pass through Proposition 2.2, Lemma 2.3, Lemma 2.4, and the proof of Theorem 1.8, I do not find a load-bearing flaw. Lemma 2.3 is correct: since I is nonzero, Z(I) has dimension at most m-1, and the fiber over any x in Z(I)\\{0} is Gr(n-m-1,n-1), giving dim Y <= (n-m-1)m + (m-1) = (n-m)m - 1 < dim Gr(n-m,n). The reduction to the algebraic closure is valid because the K-points of the Grassmannian are Zariski dense for infinite K. The proof of Proposition 2.2 contains several typos (notably the indices in equation (2.1) and the quotient module definition), but the intended argument is recoverable: choose W generically so it is both transverse and avoids Z(I); the addition map makes R=P(X x W) finite over A=P(K^n); the zero-set computation and Nakayama lemma then correctly show that the cokernel of A -> R/(I·(t)) is zero, giving condition (3). A concrete degree-2 example (X={y=x^2}, W={y=x}, I=(x)) confirms that the localization map is surjective, with preimages for x and t given by regular rational functions of the image coordinates. Thus the central claim appears robust.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":7319,"tokens_out":42166,"duration_ms":377105,"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.","major_comments":[],"minor_comments":[{"comment":"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":"Section 2, Proposition 2.2, equation (2.1)"},{"comment":"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":"Section 2, Proposition 2.2, proof"},{"comment":"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":"Section 2, Proposition 2.2, final Nakayama step"},{"comment":"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":"Section 2, Lemma 2.3"},{"comment":"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":"Section 2, Observation 2.1"},{"comment":"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.","section":"Section 3, Proposition 3.1"}],"recommendation":"minor_revision","confidential_remarks":"I recommend minor revision. The central theorem appears correct, and the proof strategy is convincing. The main issues are notational and typographical, concentrated in Proposition 2.2; once these are fixed, the paper should be suitable for publication. The author's self-citation [2] is appropriate and not load-bearing for the main argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: Banecki proves that nonsingular retract rational varieties over any infinite field are uniformly retract rational. That answers a question he posed in earlier work and closes the gap between the two properties for nonsingular varieties. The corollary that rational projective nonsingular complex varieties are algebraically elliptic is a nice payoff, giving a partial answer to Gromov's old question.\n\nWhat's new is the main theorem itself, plus Proposition 3.1, a technical tool about extending a rational map through a germ. The proof is short and mostly self-contained, using Noether normalization, Nakayama's lemma, and a generic Grassmannian argument. I checked the potentially shaky piece, Lemma 2.3, and it holds: for a nonzero ideal I, Z(I) has dimension at most m-1, so the incidence variety has dimension strictly smaller than the Grassmannian. Proposition 2.2 is dense and has several typos (indices in (2.1), the quotient module definition), but the intended argument is recoverable, and the second-pass stress-test example confirms the local surjectivity claim. The main proof, with Observation 2.1 and Lemma 2.4, is coherent.\n\nSoft spots: the manuscript is not polished. There are LaTeX artifacts, minor notation slips, and parts of Proposition 2.2 are more terse than a referee would like. These are cosmetic, not load-bearing. The reduction in Lemma 2.3 to the algebraic closure is fine because K-points are Zariski dense for infinite K. The self-citation to [2] for the real case of Observation 1.6 is appropriate and not central. The author could add more context on examples, but that is optional.\n\nWho is this for: algebraic geometers working on rationality properties, Gromov ellipticity, or real algebraic approximation. They will want this paper. The result deserves a serious referee, and I would send it out rather than desk-reject, even though the presentation needs revision. I would cite it if I worked in the area. My verdict: accept after minor revision, with a specialist checking the algebra in Proposition 2.2.\n\nHappy to talk more over coffee.","headline":"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.","tokens_in":7900,"tokens_out":2019,"would_cite":true,"duration_ms":18943,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E08","14M20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["retract rational variety","uniformly retract rational","algebraic ellipticity","uniformly rational variety","rational complex projective variety","real algebraic variety"],"falsifier":"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.","tokens_in":6846,"feed_emoji":"🔁","tokens_out":10440,"duration_ms":82579,"temperature":0.7,"pith_summary":"The paper proves a local-to-global statement for retract rational varieties: over any infinite field, if a nonsingular algebraic variety admits a retraction of a Zariski open subset of affine space onto a dense open subset, then it admits such retractions around every point. This answers the retract rational analogue of the question whether all nonsingular rational varieties are uniformly rational. The result matters because uniformly retract rational varieties support the vector-bundle constructions that define algebraic ellipticity and, in the real case, malleability. A direct corollary is that every rational, projective, nonsingular complex variety is algebraically elliptic, a partial answer to a question left open in the classical theory of algebraic ellipticity.","feed_headline":"Retract rational varieties are uniformly retract rational","feed_subtitle":"Uniform retractions at every point make rational complex projective varieties algebraically elliptic.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines algebraic ellipticity and poses the question that the corollary partly answers.","marker":"[9]"},{"why":"Supplies the theorem that local dominating sprays at every point imply algebraic ellipticity, used in Observation 1.6.","marker":"[10]"},{"why":"Supplies the result that a Zariski open subset of affine space with large complement is algebraically elliptic, used to extend retractions.","marker":"[8]"},{"why":"Introduces uniformly retract rational varieties and establishes the real analogue used in the malleability statement.","marker":"[2]"}],"fun_headline_variants":["Retract rational varieties retract uniformly","Uniform retractions for every point on nonsingular varieties","One retraction spawns retractions at every point","Rational complex projective varieties are algebraically elliptic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Retract rational varieties retract uniformly","Uniform retractions for every point on nonsingular varieties","One retraction spawns retractions at every point","Rational complex projective varieties are algebraically elliptic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000201,"raw_usage":{"total_tokens":1281,"prompt_tokens":749,"completion_tokens":532,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":365,"completion_tokens_details":{"reasoning_tokens":473}},"tokens_in":365,"tokens_out":532,"duration_ms":5170,"temperature":1.0,"reasoning_tokens":473,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:41:43.808386+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines algebraic ellipticity and poses the question that the corollary partly answers."},{"cited_title":"Kaliman and M","cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that local dominating sprays at every point imply algebraic ellipticity, used in Observation 1.6."},{"cited_title":"Forstneriˇ c","cited_arxiv_id":null,"evidence_quote":"Supplies the result that a Zariski open subset of affine space with large complement is algebraically elliptic, used to extend retractions."},{"cited_title":"Relative Stone-Weierstrass theorem for mappings between varieties","cited_arxiv_id":"2408.09233","evidence_quote":"Introduces uniformly retract rational varieties and establishes the real analogue used in the malleability statement."}],"review_version":1}