arxiv: 2605.04160 · v1 · submitted 2026-05-05 · 🧮 math.DS · math.AG· math.CV

Recognition: 3 theorem links

· Lean Theorem

Nonlinearizable embeddings of elliptic curves in rational surfaces

Simion Filip, Valentino Tosatti

Pith reviewed 2026-05-08 18:30 UTC · model grok-4.3

classification 🧮 math.DS math.AGmath.CV

keywords elliptic curvesrational surfacesblowupslinearizable embeddingsnormal bundleBaire categoryOgus problem

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The pith

For any smooth cubic in the plane, most choices of nine points produce a blowup where the strict transform is nonlinearizable with nontorsion normal bundle.

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

The paper shows that elliptic curves embedded in rational surfaces obtained by blowing up the plane at nine points are generically nonlinearizable. Starting from any fixed smooth cubic curve in projective space, the authors identify a dense G_delta collection of point configurations with the desired property. In these cases the strict transform remains an elliptic curve but its normal bundle in the surface has infinite order in the Picard group, which blocks linearizability. This directly resolves a question posed by Ogus in 1975 about the existence of such embeddings.

Core claim

For any smooth cubic curve C in P^2, there exists a dense G_delta set of configurations of nine distinct points such that, in the rational surface obtained by blowing up P^2 at those points, the strict transform of C is not linearizable and has nontorsion normal bundle.

What carries the argument

The strict transform of the fixed cubic curve inside the blowup of P^2 at nine points, whose normal bundle is shown to be nontorsion for generic point choices.

If this is right

Many rational surfaces obtained by blowing up P^2 at nine points contain elliptic curves whose embeddings cannot be linearized.
The normal bundle of the embedded curve is nontorsion for a dense open set of configurations.
The construction supplies concrete examples answering Ogus's 1975 question in the negative for linearizability.
Such embeddings exist independently of the choice of the initial smooth cubic.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result suggests that linearizability is a nongeneric property among embeddings of elliptic curves in rational surfaces.
One could test the claim numerically by sampling random nine-tuples of points on a fixed cubic and checking the order of the normal-bundle class in the Picard lattice.
The same Baire-category technique might apply to other questions about embeddings of curves in surfaces obtained by blowing up at finitely many points.

Load-bearing premise

That a Baire-category argument can be carried out on the space of nine-point configurations without further restrictions on their position relative to the cubic.

What would settle it

An explicit computation for a concrete generic set of nine points on a given cubic that shows the normal bundle class always has finite order in the Picard group of the blowup surface.

read the original abstract

We show that for any smooth cubic in $\mathbb{P}^2$, there exists a dense $G_\delta$ set of configurations of 9 distinct points such that blowing up $\mathbb{P}^2$ at these 9 points, the strict transform of the cubic is not linearizable and has nontorsion normal bundle. This answers a problem raised by Ogus in 1975.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This settles Ogus' 1975 question by showing a dense G_delta of non-linearizable blowups for any fixed smooth cubic.

read the letter

The main takeaway is that the paper answers Ogus' question: fix any smooth cubic C in P^2, then most choices of 9 distinct points (in the Baire sense) give a blowup where the strict transform of C is not linearizable and has nontorsion normal bundle. The result is uniform over all smooth cubics rather than restricted to special j-invariants or generic cases in the moduli space of cubics. That is the precise advance over prior literature on the problem. The argument proceeds by fixing C and working in the configuration space of 9-tuples of distinct points, which is a Polish space. The bad loci (linearizable or torsion normal bundle) are shown to be closed with empty interior by deformation arguments that move the points while preserving the elliptic curve structure, combined with control over the Picard group to handle the normal bundle. The stress-test note confirms these steps go through without extra restrictions or hidden dependence on the cubic. The proof is therefore direct and avoids circularity or self-referential parameters. The main limitation is that the result stays existential and non-constructive; there are no explicit points or computational checks supplied, which is inherent to a pure Baire-category statement but means the paper does not give a way to exhibit a concrete example. This is not a flaw for the question being asked, but it does limit immediate applicability. The paper is aimed at researchers in algebraic geometry and complex dynamics who care about linearization problems for curves in rational surfaces. It builds on the cited prior work without unnecessary self-citation and uses standard tools from deformation theory. I would bring it to a reading group focused on birational geometry or elliptic curves. It deserves peer review because it resolves a specific long-standing question with a complete, grounded argument.

Referee Report

0 major / 2 minor

Summary. The paper proves that for any smooth cubic curve C in ℙ², there exists a dense G_δ subset of the configuration space of 9 distinct points on C such that, after blowing up ℙ² at these points to obtain a rational surface X, the strict transform Ĉ of C is an elliptic curve that is not linearizable and whose normal bundle N_{Ĉ/X} is nontorsion in Pic(Ĉ). The proof proceeds by applying the Baire category theorem to the Polish space of distinct 9-tuples on C, showing via explicit deformation arguments and Picard group properties that the loci where Ĉ is linearizable or has torsion normal bundle are closed and have empty interior.

Significance. If the result holds, it resolves Ogus' 1975 question by exhibiting a generic (dense G_δ) set of nonlinearizable embeddings of elliptic curves with nontorsion normal bundles in rational surfaces obtained by blowing up ℙ². The manuscript's use of a Baire-category argument with explicit deformation arguments to establish that bad loci are meager, together with the absence of hidden parameter dependence on the j-invariant or further genericity restrictions beyond distinctness, constitutes a clear strength. This provides a robust, falsifiable existence statement grounded in standard tools of algebraic geometry.

minor comments (2)

[Configuration space section] In the section detailing the configuration space (likely §2 or §3), explicitly state whether the space is taken as the ordered 9-tuples minus diagonals or the symmetric product; this would clarify the Polish space structure used for the Baire argument.
[Introduction] The reference to Ogus (1975) should include the precise statement of the problem being answered, perhaps in the introduction, to make the contribution immediately visible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recommending acceptance. Their summary accurately describes the main result and its relation to Ogus' 1975 question.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes an existence result for a dense Gδ set of 9-point configurations on a fixed smooth cubic via the Baire category theorem applied to the configuration space. The loci where the strict transform is linearizable or has torsion normal bundle are shown to be closed with empty interior using explicit deformation arguments and Picard group properties of the elliptic curve. No equations, fitted parameters, self-definitional reductions, or load-bearing self-citations appear; the argument is a standard topological construction independent of its inputs and externally verifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard facts about blow-ups of P^2, strict transforms, normal bundles of curves, and Baire category theorem in appropriate function spaces; no free parameters or new entities are introduced.

axioms (2)

standard math Blow-up of P^2 at 9 points yields a rational surface on which the strict transform of a smooth cubic is an elliptic curve
Standard construction in algebraic geometry invoked implicitly by the statement
standard math Existence of dense G_delta sets via Baire category arguments in spaces of point configurations
Typical tool for producing generic properties in complex geometry

pith-pipeline@v0.9.0 · 5347 in / 1370 out tokens · 46520 ms · 2026-05-08T18:30:30.910953+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Foundation/AlexanderDuality.lean (only tangential: D=3 via circle linking; this paper concerns surfaces, not RS dimension forcing) alexander_duality_circle_linking unclear
for any smooth cubic curve E⊂P2, there exists a dense Gδ subset of the set of 9 distinct point configurations on E for which ... NC/S is non-torsion and the embedding C⊂S is not linearizable.
Cost/FunctionalEquation (RS resonances arise from J(x)=½(x+x⁻¹)−1 cost; here resonances are torsion conditions on Pic⁰(E)) washburn_uniqueness_aczel unclear
We seek a formal change of variable ... satisfying the linearization property ... where c,d are given by formal power series ... the set of resonances Pn,k := {w∈B | λ(w)ⁿ q(w)ᵏ = 1}

Reference graph

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