Computations and ML for surjective rational maps

Ilya Karzhemanov

REVIEW 3 major objections 4 minor 19 references

Computations and ML for surjective rational maps

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read For generic cubic rational maps P^2 ⇢ P^2, surjectivity is exactly the condition that the indeterminacy locus has at most six points; seven points generically give non-surjective maps.

desk verdict The two explicit cubic surjective maps are proven and look correct, but the generic-surjectivity theorem has a load-bearing genericity gap and the abstract contradicts the body. read the letter →

arxiv 2510.08093 v2 pith:4XFBFGAM submitted 2025-10-09 math.AG cs.LG

Ilya Karzhemanov This is my paper · ORCID

classification math.AGcs.LG MSC 14E0514D0568-04

keywords surjectiverationalmapendomorphismcubicindeterminacylocusdelPezzosurfaceunrulypencilmachinelearningprojectiveplane

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 asks when a cubic rational endomorphism of the complex projective plane — a map defined by three cubic polynomials but undefined at its indeterminacy locus I_f — nevertheless hits every point of the target plane. The main claim, proved in the body, is that for a generic cubic map with a fixed generic set of base points, surjectivity is determined by one number: the map is surjective exactly when #I_f ≤ 6, equivalently when the blow-up of I_f is a del Pezzo surface of degree at least 3, while the generic map with seven indeterminate points is not surjective. The argument blows up I_f, views f as a linear projection of a del Pezzo surface, and shows that every fiber of that projection escapes the exceptional divisors. The paper also develops a finite-field enumeration coupled with a neural-network predictor, producing two new explicit surjective cubic maps. A reader should care because the result reduces a qualitative question about rational maps to a sharp cardinality threshold, and shows that machine-generated candidates can be certified by direct geometry.

What carries the argument

The central device is the del Pezzo surface X = Bl_{I_f} P^2 of degree δ = 9 − #I_f. For δ ≥ 3 the anticanonical linear system |−K_X| embeds X as a degree-δ surface in P^δ = Λ^*, and the cubic endomorphism becomes the linear projection ψ from a codimension-3 subspace Σ onto the target plane Π^*; surjectivity is governed by the condition that no fiber ψ^{-1}(o) lies inside an exceptional divisor E_i. The key local identity is the hyperplane section X∩H = E_i + R, where R is a smooth residual curve with R·E_i = 2; checking that ψ|_R is unramified at the two intersection points R∩E_i rules out the obstruction. Equivalently, in the dual plane, f fails to be surjective exactly when Π contains an

What would settle it

Run the finite-field enumeration from Section 3 over a prime p ≥ 5 for a fixed generic set of six base points; any plane Π ⊂ Λ containing a pencil ℓ with base locus Bs(ℓ) = I_f gives a cubic endomorphism with #I_f = 6 that is not surjective, contradicting Theorem 2.6. Equivalently, in the del Pezzo model, look for a hyperplane H with X∩H = E_i + R where R is singular or meets E_i non-transversely; showing such H occur for a non-empty family of generic centers would break the proof's central step.

Watch

Extended reading notes

Core claim

For a generic cubic rational endomorphism f: P^2 ⇢ P^2 with indeterminacy locus I_f = {P_1,...,P_{9−δ}} in general position, the paper proves that surjectivity is equivalent to δ ≥ 3, i.e. #I_f ≤ 6. For δ = 2 (#I_f = 7), Proposition 2.2 constructs an 'unruly pencil' ℓ ⊂ Π whose base locus equals I_f, so the generic f is not surjective (Corollary 2.3). For δ ≥ 3, the blow-up X of I_f is a del Pezzo surface of degree δ, and f becomes a regular projection ψ: X → Π^*; Theorem 2.6 shows ψ^{-1}(o) ⊄ E_i for every point o and exceptional divisor E_i, hence f is onto. Its proof uses the hyperplane section X∩H = E_i + R with R smooth and R·E_i = 2, and the fact that ψ|_R is unramified at R∩E_i. Propo

Load-bearing premise

The proof of Theorem 2.6 depends on an unverified generality assertion: for a generic map and center Σ, the residual curve R in X∩H = E_i + R is smooth and the projection ψ is unramified at the two points of R∩E_i — if a non-empty family of generic projections had R tangent to E_i at an intersection point, the argument would not settle surjectivity.

Editorial extensions

If this is right

For any fixed generic set of at most six points in P^2, every sufficiently general cubic map with exactly those indeterminacy points covers the whole target plane.
For seven points in general position, generic cubic maps through them are not surjective; the cutoff is sharp for δ = 2.
Surjectivity of a cubic rational map has an explicit geometric certificate: for every exceptional divisor E_i and every point o, the fiber ψ^{-1}(o) must not be contained in E_i; equivalently, Π contains no unruly pencil.
The finite-field search yields two new explicit surjective cubic endomorphisms; one of them is found even though every plane tested over F_2 was labelled non-surjective, showing that the finite-field heuristic can be wrong while the complex map is still surjective.

Reading between the lines

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

The abstract's formulation ('cardinality at least 3') is inconsistent with the body's proved condition δ ≥ 3, which is #I_f ≤ 6; if the body is correct, the intended threshold is 'at most six points,' and the theorem also covers maps with one or two base points, whereas the abstract's wording would exclude them.
The neural network's output is a real-valued 'measure of surjectivity' rather than a certificate; a natural strengthening would be to train on a graded invariant — for example, the minimal degree of a pencil whose base locus equals I_f — so that predictions come with error estimates instead of binary labels.
The connection drawn to rational elliptic surfaces and Painlevé families suggests a testable extension: when the projection ψ is viewed as an elliptic fibration, surjectivity may coincide with the existence of a section, and one could look for such a section in the two explicit examples.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the two explicit cubic surjective maps in Propositions 3.2 and 3.6 are genuinely checked and are new; the ML part is a heuristic that doesn't do much; the main theorem (generic surjectivity iff delta >= 3) is plausible but not proved as written because a genericity step is asserted rather than argued.

What the paper does well: it frames the cubic case as the natural next step after the quadratic classification of Kulikov–Zhdanovskiy, and the del Pezzo interpretation (blow up the base points, f becomes a linear projection) is the right framework. Propositions 3.2 and 3.6 are elementary but careful: the authors reduce surjectivity to roots of a quartic and check the edge cases. Those arguments appear correct, and the examples give usable test cases.

The soft spots are real. In Theorem 2.6, after defining R via X ∩ H = E_i + R, the proof needs R ∩ E_i to consist of two distinct points and needs ψ|_R to be unramified at them. The text says this is immediate by generality of f and Σ. But R·E_i = 2 only gives a zero-cycle of length 2; tangency would give one point of multiplicity 2, and ramification would break the local isomorphism argument. No open condition is specified, so as written the theorem is conditional on an unproven Bertini-type statement. This is load-bearing, not cosmetic: the criterion from [15, Prop. 3] requires ψ^{-1}(o) not subset E_i. A referee should ask for a precise genericity statement.

There is also a plain inconsistency between the abstract ('cardinality at least 3') and Corollary 2.3, since #I_f = 7 satisfies 'at least 3' but the body says those maps are not surjective. The correct statement is #I_f ≤ 6. That needs fixing.

The experimental part is the weakest. The code and the output.txt file are not shipped reproducibly, there are no error bars, and the claim that openness fails over F2 rests on a small orthonormal sample and a neural-network prediction. The authors acknowledge this is heuristic, so it doesn't damage the geometry, but it's not something a referee can verify. The overlap with the author's previous del Pezzo paper [12] is also not clarified, so the novelty, while real, is hard to calibrate.

Who should read it: people working on rational maps, del Pezzo surfaces, or algebraic statistics. The explicit examples are worth having even if the theorem remains conditional. My recommendation: send it to peer review, but the referee should require a rigorous genericity argument for Theorem 2.6 and a corrected abstract.

Referee Report

3 major / 4 minor

Summary. The paper studies surjective rational endomorphisms f: P^2 ⇢ P^2 given by cubics with nonempty indeterminacy locus I_f. For a general set I_f of 9−δ points in general position, the main theorem (Theorem 2.6) claims that the induced map is surjective exactly when δ ≥ 3, i.e. #I_f ≤ 6; Corollary 2.3 handles δ = 2 (#I_f = 7) by constructing an unruly pencil. The proof uses the blow-up of the indeterminacy points to realize f as a projection of a del Pezzo surface Xδ ⊂ P^δ. The paper also presents two explicit cubic surjective maps (Propositions 3.2 and 3.6) found by finite-field experiments, together with a neural-network-based heuristic 'measure of surjectivity'. The abstract states the criterion as '#I_f has cardinality at least 3', which contradicts the body.

Significance. If the genericity step in Theorem 2.6 can be rigorously justified, the paper would provide a clean classification for generic cubic plane rational endomorphisms with prescribed general indeterminacy locus, extending the quadratic case of Kulikov–Zhdanovskiy. The two explicit surjective cubic maps are nontrivial and correctly proven modulo small gaps; they are genuine contributions and illustrate that experimental finite-field search can identify interesting examples. The ML/NN part is anecdotal: it gives no reproducible data, error bars, or a precise predictive claim, so it should be viewed as motivational rather than as a load-bearing component of the paper.

major comments (3)

[§2.4, proof of Theorem 2.6] The proof asserts that R·E_i = 2 implies R∩E_i = {o_1,o_2} with distinct points, and then that ψ|_R is unramified at o_1,o_2, saying 'this is immediate — again by generality of f (and Σ)'. This is not an explicit open condition. Intersection number 2 does not exclude tangency, and Lemma 2.7 only proves smoothness of R. If R is tangent to E_i or if ψ|_R ramifies at the intersection points, the criterion ψ^{-1}(o) ⊄ E_i from [15, Prop. 3] is not verified. The proof needs a concrete genericity statement (e.g. an open condition on the plane Π in the Grassmannian, or on (f,Σ)) guaranteeing that R∩E_i is reduced and that ψ|_R is unramified there. As written, Theorem 2.6 — the central classification — is conditional on this missing argument.
[Abstract; Corollary 2.3] The abstract states that a general non-regular cubic endomorphism is surjective iff I_f has cardinality at least 3. This is inconsistent with the body: Corollary 2.3 says the generic map is not surjective when #I_f = 7, while Theorem 2.6 says it is surjective when #I_f ≤ 6. Since #I_f = 7 satisfies 'at least 3', the abstract condition is false. The correct statement is '#I_f ≤ 6' (equivalently δ ≥ 3), or the abstract must be rephrased to match Corollary 2.3 and Theorem 2.6.
[§3.2, proof of Proposition 3.2] In the line L = (y=0) part, the proof claims that for [1:0:a] ∈ L\{P,Q}, f([0:1:a−2]) = [1:0:a]. For a = 2, the proposed preimage is [0:1:0], which lies in I_f, so the map is not defined there. Thus the given argument does not cover the point [1:0:2]. This can be repaired separately (for instance [1:2:0] maps to [1:0:2]), so the proposition may still be true, but the proof as written has a gap. Please fix this and check whether any other value of a is affected.

minor comments (4)

[§3.4, ML experiment] The experimental claims lack quantitative support: no dataset size, no number of test samples, no variance/error bars, and the value 0.0737 is reported as a single average with no standard deviation. The file output.txt is not included, and the provided code has formatting and truncation issues. Since these experiments are not used to prove the main theorems, this does not block acceptance, but the claims should be marked as anecdotal or supplemented with reproducible details.
[§1.5, §3.4] The statement that 'the openness property ... does not hold over F2' is too strong for the evidence given; an average prediction value near 0 does not establish that no Zariski-open surjectivity set exists over F2. Please soften the claim or provide a precise finite-field statement.
[§3.6, proof of Proposition 3.6] After the transformation (a,b) ↦ (a−b,b), the displayed map is written as a map in (x,y), but the right-hand side depends on x and z (e.g. x^2 + z − xz over x − z). This is presumably a typo for (x,z). Please clarify the variable names to avoid confusion.
[General] The relation to the author's previous paper [12] on surjective rational maps and del Pezzo surfaces should be clarified: the current geometric setup overlaps with [12], and the reader is not told which statements are new and which are recalled from there.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorem is proved by independent projective geometry, examples are verified directly, and the ML heuristic is not load-bearing.

full rationale

The paper's central claim (Theorem 2.6 and Corollary 2.3) is a geometric statement about generic cubic rational endomorphisms. Its proof uses a projection criterion from the external reference [15, Proposition 3], not a fitted parameter or a prior result by the same authors. The finite-field table and the neural-network function Ψ / Ylearn are explicitly experimental: they motivate examples but are not used as evidence for Theorem 2.6. Propositions 3.2 and 3.6 are proved by explicit preimage calculations over C, independent of the ML output. The only self-citation, [12], appears in a contextual sentence ('have already interacted with each other') and is not load-bearing for any derivation. The proof of Theorem 2.6 does contain an unproven genericity assertion ('this is immediate — again by generality of f (and Σ)'), but a missing or terse justification is a gap in argument, not circularity. Similarly, the abstract's phrasing ('cardinality at least 3') is inconsistent with Corollary 2.3's #I_f ≥ 7, but this is a correctness/consistency issue, not a reduction of the result to its inputs. No step in the paper equates a prediction with a fitted input or derives a conclusion from a self-citation chain. Hence the appropriate circularity score is 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The theorem depends on standard del Pezzo/Bertini material and, critically, on an unproved genericity assertion in Theorem 2.6. The ML component introduces fitted NN weights but is explicitly heuristic and does not support the theorem. No physical entities are introduced.

free parameters (1)

Neural-network hyperparameters/weights for Ylearn = not fully given; test_loss 0.078, average Ylearn 0.0737
Section 3.4 fits a Keras net (Conv2D 256, Dense 256, 150 epochs, batch 32) to the F2 output.txt table to define Psi. This is a fitted model for heuristic claims only; it is not used to prove Theorem 2.6 or Propositions 3.2/3.6.

assumptions (5)

standard math Anticanonical embedding of del Pezzo surfaces of degree delta >= 3 into P^delta, with exceptional divisors E_i as lines.
Used in sections 1.3 and 2.4 to identify f with a linear projection of X subset P^delta.
standard math Bertini's theorem applies to the residual linear system |R| once Bs(|R|) is empty, giving R smooth for generic H.
Lemma 2.7.
ad hoc to paper For generic f, H is generic and psi|_R is unramified at R cap E_i; asserted without proof.
Theorem 2.6 proof, last paragraph; the load-bearing genericity step.
domain assumption The fixed indeterminacy set consists of 9-delta points in general position and the parameter count gives 2 <= delta <= 8.
Section 1.3; the 'generic' in the theorem is with respect to planes Pi inside Lambda with this base locus.
domain assumption Existence of an inflection cubic C in Pi and a pencil sharing a tangent at P1 in the delta=2 case.
Proposition 2.2; justified by a parameter count, but the count is only sketched.

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Pith. "Pith review of Computations and ML for surjective rational maps." pith.science (2026). https://pith.science/paper/4XFBFGAM

@misc{pith2026251008093,
  author       = {Pith},
  title        = {Pith review of: Computations and ML for surjective rational maps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4XFBFGAM}},
  note         = {Machine review of arXiv:2510.08093}
}

abstract

The present note studies \emph{surjective rational endomorphisms} $f: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2$ with \emph{cubic} terms and the indeterminacy locus $I_f \ne \emptyset$. We develop an experimental approach, based on some Python programming and Machine Learning, towards the classification of such maps; a couple of new explicit $f$ is constructed in this way. We also prove (via pure projective geometry) that a general non-regular cubic endomorphism $f$ of $\mathbb{P}^2$ is surjective if and only if the set $I_f$ has cardinality at least $3$.

Figures

Figures reproduced from arXiv: 2510.08093 by the authors.

**Figure 2.** Projection onto ψ(Ei) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 2 linked inside Pith

[12]

Karzhemanov and A

I. Karzhemanov and A. Lekontseva, Surjective rational maps and del Pezzo surfaces , 2023, arXiv:2311.01835

arXiv 2023
[1]

V. I. Arnol’d, Russian Math. Surveys 59 (2004), no. 6, 1029–1046; translated from Uspekhi Mat. Nauk 59 (2004), no. 6(360), 23–40

2004
[2]

G. E. Carlsson, Topology and data , Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 2, 255–308

2009
[3]

T. - C. Dinh and N. Sibony, Dynamics in several complex variables: endomorphisms of pr ojective spaces and polynomial-like mappings, in Holomorphic dynamical systems , 165–294, Lecture Notes in Math., 1998, Springer, Berlin

1998
[4]

I. V. Dolgachev, Classical algebraic geometry , Cambridge University Press, Cambridge, 2012

2012
[5]

Gukov, J

S. Gukov, J. Halverson, F. Ruehle, Rigor with machine learning from ﬁeld theory to the Poincaré conjecture, Nat. Rev. Phys. 6 (2024), 310–319

2024
[6]

N. J. Hitchin, A lecture on the octahedron , Bull. London Math. Soc. 35 (2003), no. 5, 577–600

2003
[7]

Kajiwara, M

K. Kajiwara, M. Noumi and Y. Yamada, Geometric aspects of Painlevé equations , J. Phys. A 50 (2017), no. 7, 073001, 164 pp

2017

Show all 19 references

[8]

Karzhemanov, Gradient property of quadratic maps , Lobachevskii J

I. Karzhemanov, Gradient property of quadratic maps , Lobachevskii J. Math. 42 (2021), no. 10, 2333–2336

2021
[9]

Karzhemanov, Maximum Likelihood Degree of Surjective Rational Maps , Arnold Math

I. Karzhemanov, Maximum Likelihood Degree of Surjective Rational Maps , Arnold Math. J. 8 (2022), no. 3 - 4, 513–516

2022
[10]

Karzhemanov, One instance of wild automorphisms , Lobachevskii J

I. Karzhemanov, One instance of wild automorphisms , Lobachevskii J. Math. 46 (2025), no. 6, 2560–2565

2025
[11]

Karzhemanov, Projective - Geometric Aspects of Quantization , Lobachevskii J

I. Karzhemanov, Projective - Geometric Aspects of Quantization , Lobachevskii J. Math. 43 (2022), no. 7, 1651–1654

2022
[13]

Karzhemanov and I

I. Karzhemanov and I. Zhdanovskiy, Some properties of surjective rational maps , Eur. J. Math. 4 (2018), no. 1, 326–329

2018
[14]

King, Mysterious duality and helical line bundles on del Pezzo sur faces, 2025, arXiv:2507.10169

A. King, Mysterious duality and helical line bundles on del Pezzo sur faces, 2025, arXiv:2507.10169

2025 arXiv
[15]

Kulikov and I

A. Kulikov and I. Zhdanovskiy, On Surjective Quadratic Maps of P2, Lobachevskii J. Math. 44 (2023), no. 6, 2072–2078

2023
[16]

Z. Liu, Y. W ang, S. Vaidya, F. Ruehle, J. Halverson, M. So ljacic, T. Y. Hou, M. Tegmark, KAN: Kolmogorov–Arnold Networks , International Conference on Learning Representations (IC LR), 2025, https:openreview.net/forum?id=Ozo7qJ5vZi

2025
[17]

Saito and H

M.-H. Saito and H. Umemura, Painlevé equations and deformations of rational surfaces w ith rational double points , in Physics and combinatorics 1999 (Nagoya) , 320–365, W orld Sci. Publ., River Edge, NJ

1999
[18]

V. V. Serganova and A. N. Skorobogatov, On the equations for universal torsors over del Pezzo surfac es, J. Inst. Math. Jussieu 9 (2010), no. 1, 203–223

2010
[19]

Usnich, Symplectic automorphisms of CP2 and the Thompson group T , 2006, arXiv:0611604

A. Usnich, Symplectic automorphisms of CP2 and the Thompson group T , 2006, arXiv:0611604. Laboratory of AGHA, Moscow Institute of Physics and Technology , 9 Institutskiy per., Dolgo- prudny, Moscow Region, 141701, Russia E-mail address : karzhemanov.iv@mipt.ru 15

2006

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [12]

Karzhemanov and A

I. Karzhemanov and A. Lekontseva, Surjective rational maps and del Pezzo surfaces , 2023, arXiv:2311.01835

arXiv 2023

[2] [1]

V. I. Arnol’d, Russian Math. Surveys 59 (2004), no. 6, 1029–1046; translated from Uspekhi Mat. Nauk 59 (2004), no. 6(360), 23–40

2004

[3] [2]

G. E. Carlsson, Topology and data , Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 2, 255–308

2009

[4] [3]

T. - C. Dinh and N. Sibony, Dynamics in several complex variables: endomorphisms of pr ojective spaces and polynomial-like mappings, in Holomorphic dynamical systems , 165–294, Lecture Notes in Math., 1998, Springer, Berlin

1998

[5] [4]

I. V. Dolgachev, Classical algebraic geometry , Cambridge University Press, Cambridge, 2012

2012

[6] [5]

Gukov, J

S. Gukov, J. Halverson, F. Ruehle, Rigor with machine learning from ﬁeld theory to the Poincaré conjecture, Nat. Rev. Phys. 6 (2024), 310–319

2024

[7] [6]

N. J. Hitchin, A lecture on the octahedron , Bull. London Math. Soc. 35 (2003), no. 5, 577–600

2003

[8] [7]

Kajiwara, M

K. Kajiwara, M. Noumi and Y. Yamada, Geometric aspects of Painlevé equations , J. Phys. A 50 (2017), no. 7, 073001, 164 pp

2017