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arxiv: 2501.01906 · v2 · submitted 2025-01-03 · 🧮 math.AG · math.NT

Hypersurfaces passing through the Galois orbit of a point

Shamil Asgarli , Jonathan Love , Chi Hoi Yip This is my paper

Pith reviewed 2026-05-23 05:55 UTC · model grok-4.3

classification 🧮 math.AG math.NT MSC 14J7014G05
keywords hypersurfacesGalois orbitsprojective spacefield extensionslinear systemsarithmetic geometry
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0 comments X

The pith

For any field K and any separable extension L of degree r there exists a point P in projective n-space over L so that the K-vector space of degree-d forms vanishing at P has the expected dimension.

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

The paper proves that the earlier existence statement for points whose Galois orbit lies on no degree-d hypersurface defined over K continues to hold when K has only two elements. It strengthens the statement by replacing the specific degree r equal to the dimension of the space of forms with an arbitrary positive integer r. The argument combines Galois-cohomology, specialization, and linear-system techniques to produce the required point over any base field. A reader cares because the result removes the size restriction on K that appeared in prior work and applies uniformly to all fields, including the smallest finite ones.

Core claim

For every positive integer r, every field K, and every separable extension L/K of degree r, there exists a point P in P^n(L) such that the K-vector space of homogeneous polynomials of degree d that vanish at P has the dimension predicted by a naive count of conditions.

What carries the argument

The Galois orbit of P, used to impose r independent linear conditions on the space of degree-d forms over K.

If this is right

  • The same points yield linear systems of hypersurfaces possessing prescribed base loci or multiplicities over the base field.
  • The statement applies verbatim to every finite field, removing any distinction between small and large base fields.
  • When r equals the dimension of the space of degree-d forms, the orbit of P lies on no K-hypersurface of that degree.

Where Pith is reading between the lines

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

  • Analogous existence statements may hold when the ambient variety is replaced by an arbitrary projective variety over K.
  • The same orbit-avoidance technique could be tested for subvarieties of higher codimension or for other linear series.

Load-bearing premise

The arithmetic-geometry constructions that produce the point continue to work without extra obstructions when the base field has cardinality two or when the extension degree is arbitrary.

What would settle it

An explicit triple (n,d,r) together with a field K of two elements for which every point of P^n over every degree-r extension lies on some K-hypersurface of degree d.

read the original abstract

Asgarli, Ghioca, and Reichstein proved that if $K$ is a field with $|K|>2$, then for any positive integers $d$ and $n$, and separable field extension $L/K$ with degree $m=\binom{n+d}{d}$, there exists a point $P\in \mathbb{P}^n(L)$ which does not lie on any degree $d$ hypersurface defined over $K$. They asked whether the result holds when $|K| = 2$. We answer their question in the affirmative by combining various ideas from arithmetic geometry. More generally, we show that for each positive integer $r$ and separable field extension $L/K$ with degree $r$, there exists a point $P \in \mathbb{P}^n(L)$ such that the vector space of degree $d$ forms over $K$ that vanish at $P$ has the expected dimension. We also discuss applications to linear systems of hypersurfaces with special properties.

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

0 major / 2 minor

Summary. The paper affirms the existence result of Asgarli-Ghioca-Reichstein for fields K with |K|=2: for any positive integers d and n, and separable L/K of degree m=binom(n+d,d), there exists P in P^n(L) not lying on any degree-d hypersurface defined over K. It proves a generalization: for each positive integer r and separable L/K of degree r, there exists P in P^n(L) such that the K-vector space of degree-d forms vanishing at P has the expected dimension. The proof combines Galois cohomology, specialization, and linear-system arguments; applications to linear systems with special properties are discussed.

Significance. If the result holds, it resolves the open question for |K|=2 posed by Asgarli-Ghioca-Reichstein and supplies a uniform existence statement for points with prescribed vanishing behavior under Galois action. The combination of existing arithmetic-geometry tools into a parameter-free existence proof for arbitrary separable extensions is a strength; the argument is presented as internally consistent once the cited tools are granted, with no hidden cardinality obstruction visible for |K|=2.

minor comments (2)
  1. The statement of the expected dimension in the general theorem could be made fully explicit by recalling the formula for the dimension of the space of degree-d forms (e.g., binom(n+d,d) - r) in the introduction or §2.
  2. A brief comparison paragraph with the original Asgarli-Ghioca-Reichstein construction would help readers see precisely where the new specialization argument is used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive report, which accurately summarizes the main results and recommends acceptance. We are pleased that the combination of Galois cohomology, specialization, and linear-system arguments is viewed as internally consistent.

Circularity Check

0 steps flagged

No significant circularity; existence proof combines independent tools

full rationale

The manuscript establishes an existence result for points P in P^n(L) over separable extensions of degree r (including the |K|=2 case) by invoking Galois cohomology, specialization, and linear-system dimension counts. These are standard arithmetic-geometry techniques whose validity does not depend on the target statement; the argument therefore does not reduce any claimed dimension or existence assertion to a fitted parameter, a self-definition, or a load-bearing self-citation chain. The cited prior work of Asgarli-Ghioca-Reichstein supplies only the |K|>2 case and is not invoked to justify the new |K|=2 construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result is an existence proof in algebraic geometry that relies on standard background results rather than new fitted constants or postulated objects.

axioms (2)
  • standard math Standard facts about projective space, homogeneous polynomials, and Galois actions on points over separable extensions
    Invoked throughout the existence argument for the vector space of vanishing forms.
  • standard math Properties of linear systems and expected dimension counts in the space of degree-d forms
    Used to formulate the generalized statement about the vanishing space having expected dimension.

pith-pipeline@v0.9.0 · 5705 in / 1373 out tokens · 51480 ms · 2026-05-23T05:55:03.905892+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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