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arxiv: 2606.23513 · v1 · pith:65XIFCP7new · submitted 2026-06-22 · 🧮 math.AG

Hyperplane anti-Bertini embeddings over finite fields

Yutong Zhang , Yaoran Yang This is my paper

Pith reviewed 2026-06-26 06:25 UTC · model grok-4.3

classification 🧮 math.AG
keywords hyperplane sectionsfinite fieldsembeddingsquasiprojective varietiesBertini theoremslinear nondegeneracysmoothnessalgebraic geometry
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The pith

Smooth positive-dimensional quasiprojective varieties over finite fields admit embeddings into high projective space where every rational hyperplane section is singular.

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

Baker asked whether every high-dimensional linearly nondegenerate embedding of a fixed smooth quasiprojective variety over a finite field must admit a smooth rational hyperplane section. Poonen predicted a negative answer for every positive-dimensional variety. The paper proves the prediction for any prescribed such variety: given nonempty smooth quasiprojective X of pure positive dimension over F_q, for all sufficiently large N there exists a locally closed embedding of X into P^N over F_q that stays linearly nondegenerate after any scalar extension, yet every F_q-rational hyperplane cuts the image in a singular scheme. The proof works by pairing each rational hyperplane with a closed point on X and arranging that the pulled-back linear form vanishes to first order at that point.

Core claim

If X is nonempty, smooth, quasiprojective, and of pure positive dimension over F_q, then for every sufficiently large N there is a locally closed embedding X into P^N over F_q whose components remain linearly nondegenerate after arbitrary scalar extension, but whose every F_q-rational hyperplane section is singular. The construction assigns one closed point of X to each rational hyperplane and forces the pulled-back linear form to have zero first-order jet at that point.

What carries the argument

The assignment of one closed point of X to each F_q-rational hyperplane, forcing the corresponding linear form to vanish to first order at its assigned point.

If this is right

  • The negative answer to Baker's question holds for every nonempty smooth quasiprojective variety of pure positive dimension over any finite field.
  • Such embeddings exist in every dimension N large enough relative to the given variety.
  • Linear nondegeneracy of the embedded variety is preserved after base change to any extension field.
  • No F_q-rational hyperplane section of the embedded variety can be smooth.
  • The result applies uniformly to all such varieties without further restrictions on the base field.

Where Pith is reading between the lines

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

  • The same point-assignment technique might be adapted to force higher-order singularities or to control other local properties of sections.
  • Bertini-type statements over finite fields may need extra arithmetic conditions beyond linear nondegeneracy to guarantee smooth rational sections.
  • Analogous constructions could apply when the base is a global field rather than a finite field, or when one seeks sections with prescribed singularities.
  • The result suggests that questions about the existence of smooth rational points on hyperplane sections may require different embedding strategies.

Load-bearing premise

The point-assignment construction succeeds in producing a locally closed embedding whose linear nondegeneracy survives arbitrary scalar extension.

What would settle it

An explicit small example, such as a smooth curve over F_2, for which every linearly nondegenerate embedding into sufficiently high P^N possesses at least one smooth F_q-rational hyperplane section.

read the original abstract

Baker asked, as recorded by Poonen, whether a fixed smooth quasiprojective variety over a finite field must have a smooth rational hyperplane section after every sufficiently high-dimensional linearly nondegenerate embedding. Poonen predicted a negative answer for every positive-dimensional variety. We prove this predicted negative answer for each prescribed variety: if $X$ is nonempty, smooth, quasiprojective, and of pure positive dimension over $\F_q$, then for every sufficiently large $N$ there is a locally closed embedding $X\hookrightarrow\PP^N_{\F_q}$ whose components remain linearly nondegenerate after arbitrary scalar extension, but whose every $\F_q$-rational hyperplane section is singular. The construction assigns one closed point of $X$ to each rational hyperplane and forces the pulled-back linear form to have zero first-order jet at that point.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript proves a negative answer to Baker's question (recorded by Poonen) on smooth rational hyperplane sections in high embeddings: if X is nonempty, smooth, quasiprojective, and of pure positive dimension over F_q, then for all sufficiently large N there exists a locally closed embedding X ↪ P^N_{F_q} whose components remain linearly nondegenerate after arbitrary scalar extension, but every F_q-rational hyperplane section is singular. The proof uses an explicit construction that assigns one closed point of X to each F_q-rational hyperplane and imposes that the pulled-back linear form vanishes to first order at the assigned point.

Significance. If the details of the construction hold, the result confirms Poonen's prediction for every such variety and supplies an explicit, uniform counterexample rather than an existence argument. The parameter-free nature of the assignment and the preservation of linear nondegeneracy after base change are notable strengths.

major comments (2)
  1. [Abstract] Abstract (final sentence) and the main construction: the simultaneous imposition of one zero first-order jet condition per F_q-rational hyperplane (approximately q^N conditions) on an (N+1)-dimensional F_q-subspace V ⊂ H^0(X,L) risks forcing linear dependence among the sections or a common base point, which would violate the locally closed embedding property or the spanning of the dual. The manuscript must explicitly verify that the chosen assignment of points p_H avoids these overconstraints while still producing an embedding that separates points and tangents.
  2. [Main construction] The nondegeneracy claim after scalar extension: while the F_q-span of V being full-dimensional ensures nondegeneracy over extensions in the absence of the jet conditions, the manuscript needs to confirm that the coupled jet conditions (imposed over F_q) do not introduce unexpected linear relations visible only after base change to the algebraic closure.
minor comments (1)
  1. [Abstract] Clarify the choice of line bundle L in the construction; the abstract does not specify whether L is very ample or merely ample.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying points where additional explicit verification would strengthen the argument. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final sentence) and the main construction: the simultaneous imposition of one zero first-order jet condition per F_q-rational hyperplane (approximately q^N conditions) on an (N+1)-dimensional F_q-subspace V ⊂ H^0(X,L) risks forcing linear dependence among the sections or a common base point, which would violate the locally closed embedding property or the spanning of the dual. The manuscript must explicitly verify that the chosen assignment of points p_H avoids these overconstraints while still producing an embedding that separates points and tangents.

    Authors: We agree that the manuscript should contain an explicit verification that the assignment of points avoids overconstraints. In the revised version we will insert a new lemma establishing that, for N large enough, the points p_H may be chosen so that the first-order jet conditions remain linearly independent over F_q and impose no common base point. The argument proceeds by selecting the points from a dense open subset of X on which the relevant evaluation maps are surjective; the resulting codimension is strictly less than dim V, and the separation of points and tangents is inherited from the linear nondegeneracy of the embedding together with the fact that the conditions are supported at distinct points. This addition will make the preservation of the locally closed embedding property fully explicit. revision: yes

  2. Referee: [Main construction] The nondegeneracy claim after scalar extension: while the F_q-span of V being full-dimensional ensures nondegeneracy over extensions in the absence of the jet conditions, the manuscript needs to confirm that the coupled jet conditions (imposed over F_q) do not introduce unexpected linear relations visible only after base change to the algebraic closure.

    Authors: The jet conditions are defined by F_q-linear equations because both the rational hyperplanes and the assigned closed points are chosen in a Galois-equivariant manner. Consequently the subspace V is the solution space to a system of linear equations over F_q. Any linear dependence among the sections that appears only after base change to the algebraic closure would necessarily descend, by Galois descent for finite-dimensional vector spaces, to a dependence already visible over F_q, contradicting the construction of V as an (N+1)-dimensional F_q-subspace. We will add a short remark in the main construction section spelling out this descent argument. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit construction is self-contained

full rationale

The paper proves its main result via an explicit construction that assigns one closed point of X to each F_q-rational hyperplane and imposes a zero first-order jet condition on the pulled-back linear form. This derivation chain relies on direct geometric arguments rather than any self-definition of quantities, fitted parameters renamed as predictions, or load-bearing self-citations. No step reduces the claimed embedding or nondegeneracy property to the input data by construction, and the cited prediction from Poonen is external. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on standard facts from algebraic geometry over finite fields (projective space, jets, linear nondegeneracy after base change); no free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)
  • standard math Standard properties of quasiprojective varieties, projective space, and hyperplane sections over finite fields
    Invoked to define embeddings, rational hyperplanes, and singularity via first-order jets.

pith-pipeline@v0.9.1-grok · 5670 in / 1154 out tokens · 36766 ms · 2026-06-26T06:25:46.672963+00:00 · methodology

discussion (0)

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Reference graph

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