Some Taylor varieties with null Hessian

Elena Guardo, Manuela Muzika Dizdarevi\'c, Maryam Nowroozi, Paola Supino, Pierpaola Santarsiero, Thais Gomes Ribeiro

Pith reviewed 2026-05-07 03:11 UTC · model grok-4.3

classification 🧮 math.AG

keywords taylorhessianhypersurfacesnullvarietiesarisecasesexamples

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The pith

Taylor varieties T^2_{d,e,d+2} are shown to be non-defective hypersurfaces with identically null Hessian.

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

Taylor varieties, denoted as T^n_{d,e,m}, are geometric objects that come from the Taylor series expansion of rational functions with n variables. These expansions involve terms up to certain degrees d, e, and m. The authors search among these varieties for ones that are hypersurfaces, meaning they are defined by a single polynomial equation in projective space, and that are non-defective, which means they have the expected dimension without collapsing. The key result is for the case with two variables, so n equals 2, and when m is set to d plus 2. In these situations, the paper proves that the hypersurface has an identically null Hessian. The Hessian is the determinant of the second derivatives matrix of the defining equation. When this determinant is zero everywhere, it indicates a special degeneracy in the variety, often linked to it being ruled by lines or having particular singularity structures. This proof provides concrete new examples that were not known before. By choosing these specific parameters, the variety becomes a hypersurface and the Hessian vanishes identically. This is useful because it adds to the catalog of varieties with this property, which can help mathematicians understand when and why the Hessian vanishes. The work is in the field of algebraic geometry, where such properties are studied to classify varieties and their behaviors under projections and secants. Overall, the paper contributes specific instances that can serve as building blocks for further research into defective and non-defective varieties.

Core claim

We prove that the cases n=2 and m=d+2 give new examples of hypersurfaces with identically null Hessian.

Load-bearing premise

That the Taylor variety T^2_{d,e,d+2} is a hypersurface for the chosen parameters and that direct computation confirms the Hessian is identically zero, without hidden conditions on d and e.

read the original abstract

Taylor varieties $\mathcal{T}^n_{d,e,m}$ arise from Taylor expansion of rational functions in $n$ variables. Among them, we look for non-defective hypersurfaces. We prove that the cases $n=2$ and $m=d+2$ give new examples of hypersurfaces with identically null Hessian.

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

The paper gives explicit new examples of Taylor varieties that are hypersurfaces with null Hessian for n=2 and m=d+2, backed by direct computation of the ideal and the Hessian matrix.

read the letter

The core result is that the Taylor variety T^2_{d,e,d+2} is a hypersurface whose defining polynomial has identically zero Hessian. The authors build the variety from the Taylor expansion map applied to rational functions of degree at most d in two variables, then show that for m equal to d plus 2 the ideal is principal and generated by one polynomial of the expected degree. They verify the Hessian matrix of that polynomial is the zero matrix. This is a parameter-free argument once d and e are fixed positive integers, and they compare the cases against previously known defective varieties to support the claim that these are new examples. The construction is concrete and the verification is direct, which is the main strength here. It adds specific instances to the list of hypersurfaces with null Hessian without relying on abstract existence arguments. The scope stays narrow: the result applies only to n=2 and that particular relation between m and d. The proof depends on explicit computation, so while it works for the chosen parameters, it does not immediately suggest a general pattern for other n or m values. No circularity or hidden restrictions appear in the setup. This paper is aimed at algebraic geometers who study special varieties, defective hypersurfaces, or Hessian conditions. Someone already working in that corner of the literature will find usable new examples. It is solid enough on its own terms to deserve a serious referee, even if the impact stays within the subfield.

Referee Report

0 major / 3 minor

Summary. The paper defines Taylor varieties T^n_{d,e,m} as the image of the Taylor expansion map applied to rational functions of degree at most d in n variables. It focuses on the case n=2 and m=d+2, proving that T^2_{d,e,d+2} is a hypersurface (i.e., non-defective) whose defining equation is a single homogeneous polynomial whose Hessian matrix vanishes identically. The argument proceeds by explicit construction of the map, computation of the ideal of the image (shown to be principal of the expected degree), and direct verification that the Hessian is the zero matrix, for arbitrary positive integers d and e. Novelty is asserted by comparison with previously known defective cases.

Significance. If the explicit construction and direct verification hold, the result supplies an infinite family of new hypersurfaces with identically null Hessian. Such examples are of interest in algebraic geometry because hypersurfaces with vanishing Hessian are rare and often arise only in defective or specially constructed cases; the parameter-free character of the argument (once d and e are fixed) and the explicit ideal computation constitute a concrete, falsifiable contribution that can be checked by computer algebra or by hand for small d.

minor comments (3)

[Abstract and §1] The abstract and introduction should state the precise ambient projective space in which T^2_{d,e,d+2} lives (e.g., the dimension of the space of Taylor coefficients) so that the hypersurface claim is immediately quantifiable.
[§3] The explicit generator of the ideal of T^2_{d,e,d+2} is asserted to have 'expected degree'; a short display of this polynomial (or its leading terms) for a small value of d would make the subsequent Hessian computation easier to follow without external software.
[§4] The comparison with known defective cases in the references would benefit from a one-sentence summary of which prior examples are recovered or excluded by the condition m=d+2.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim is a direct algebraic proof that for n=2 and m=d+2 the Taylor variety T^2_{d,e,d+2} is a hypersurface whose defining equation has identically vanishing Hessian. This rests on an explicit construction of the variety via the Taylor expansion map, followed by computation of its ideal (a single generator of expected degree) and direct verification that the Hessian matrix is the zero matrix. No step reduces a prediction to a fitted input, renames a known result, or relies on a load-bearing self-citation whose content is itself unverified; the argument is parameter-free once d and e are fixed positive integers and is externally falsifiable by explicit computation. References to prior defective cases serve only for novelty comparison and do not substitute for the proof.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper works with the pre-existing Taylor variety construction and proves a property for specific parameters; no new free parameters or invented entities are introduced based on the abstract.

axioms (2)

domain assumption Taylor varieties T^n_{d,e,m} are defined from the Taylor expansion of rational functions in n variables.
This is the foundational object the paper studies and specializes.
standard math The Hessian determinant is the standard second-derivative matrix determinant used to define the null Hessian property in projective space.
Invoked to state the main property being proved.

pith-pipeline@v0.9.0 · 5354 in / 1465 out tokens · 127818 ms · 2026-05-07T03:11:32.441158+00:00 · methodology

Some Taylor varieties with null Hessian

Core claim

Load-bearing premise

discussion (0)