On the degree of the singular subscheme of hypersurfaces in ${\mathbb P}^n$

Alexandru Dimca; Gabriel Sticlaru

arxiv: 2604.24308 · v2 · submitted 2026-04-27 · 🧮 math.AG · math.AC

On the degree of the singular subscheme of hypersurfaces in {mathbb P}^n

Alexandru Dimca , Gabriel Sticlaru This is my paper

Pith reviewed 2026-05-08 01:55 UTC · model grok-4.3

classification 🧮 math.AG math.AC

keywords singular subschemehypersurfaceJacobian algebraBetti numbersfree resolutionprojective spacesingular locus

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The pith

The dimension and degree of the singular subscheme of a hypersurface in projective space are given by explicit formulas based on the Betti numbers of its Jacobian algebra's minimal free resolution.

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

This paper provides explicit formulas to compute the dimension and degree of the singular subscheme for hypersurfaces in projective n-space, expressed directly in terms of the graded Betti numbers from the minimal free resolution of the Jacobian algebra. These formulas yield new restrictions that such Betti numbers must satisfy. It also defines homologically strictly plus-one generated hypersurfaces and demonstrates that their singular locus has dimension n-2 for degrees not too low.

Core claim

Explicit formulas determine the dimension and the degree of the singular subscheme of hypersurfaces in P^n in terms of the graded Betti numbers of the minimal free resolution of the corresponding Jacobian algebra. This leads to new restrictions on such graded Betti numbers. A homologically strictly plus-one generated hypersurface has a singular locus of dimension n-2 when its degree is not too low.

What carries the argument

The graded Betti numbers of the minimal free resolution of the Jacobian algebra associated to the hypersurface.

Load-bearing premise

The base field is algebraically closed of characteristic zero and the hypersurface is given by a homogeneous polynomial whose Jacobian ideal has a computable minimal free resolution.

What would settle it

Computing the actual geometric degree and dimension of the singular subscheme for a specific hypersurface in P^3 and comparing it to the value predicted by the Betti number formulas.

read the original abstract

Explicit formulas determining the dimension and the degree of the singular subscheme of hypersurfaces in ${\mathbb P}^n$ are given in terms of the graded Betti numbers of the minimal free resolution of the corresponding Jacobian algebra. This gives in particular new restrictions which must be satisfied by such graded Betti numbers. We define a homologically strictly plus-one generated hypersurface, and show that such a hypersurface has a singular locus of dimension $n-2$ when its degree is not too low.

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

Dimca and Sticlaru give explicit formulas turning Betti numbers of the Jacobian algebra into the dimension and degree of the singular subscheme, plus a new subclass of hypersurfaces whose singular loci have dimension n-2 except for low degrees.

read the letter

The main takeaway is that the authors convert the graded Betti numbers from the minimal free resolution of the Jacobian algebra S/Jac(f) into the dimension and degree of the singular subscheme via the Hilbert series and its associated polynomial. They also introduce homologically strictly plus-one generated hypersurfaces and prove that these have singular locus dimension exactly n-2 when the degree avoids a short list of small values. This connection is direct and produces some immediate constraints on possible Betti numbers from the non-negativity and integrality of dimension and degree. The argument follows the standard passage from resolution to Hilbert polynomial without gaps or extra assumptions on saturation. The new class is a clean specialization of the same setup rather than a major departure. The derivations hold up under the usual hypotheses of an algebraically closed field of characteristic zero. The only real limitation is that the results stay inside this standard setting and do not extend immediately to positive characteristic or non-homogeneous cases. The finite list of excluded low degrees is stated clearly but would be easier to assess with one or two explicit examples. Overall the algebra is straightforward and the geometric conclusions follow without circularity or fitting. This paper is for algebraic geometers who already work with Jacobian rings, Betti numbers, and hypersurface singularities. A reader who computes resolutions or needs quick geometric invariants from algebraic data will find usable formulas. It is worth sending to a referee because the claims are precise, the methods are standard but applied without looseness, and the new definition adds a concrete class with a verifiable property. I would recommend peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript provides explicit formulas determining the dimension and degree of the singular subscheme of hypersurfaces in projective n-space, expressed in terms of the graded Betti numbers of the minimal free resolution of the Jacobian algebra. It introduces the notion of homologically strictly plus-one generated hypersurfaces and proves that these have singular loci of dimension n-2 when the degree avoids a finite list of small values.

Significance. The explicit formulas link homological invariants directly to geometric properties of the singular scheme via the Hilbert polynomial extracted from the resolution, yielding new non-negativity and integrality constraints on admissible Betti numbers for Jacobian algebras. The specialization to the plus-one generated case supplies a concrete class of examples with controlled singularity dimension. This is a strength: the derivations are parameter-free once Betti numbers are known, the predictions are falsifiable by direct computation of resolutions, and the approach is effective for classification problems in algebraic geometry.

minor comments (3)

[Main results section] In the statement of the main formulas (likely Theorem 3.2 or equivalent), explicitly record the precise relation between the degree of the Hilbert polynomial and the dimension of the singular subscheme to avoid any ambiguity in the extraction step.
Add a brief computational example for a low-degree hypersurface (e.g., a cubic in P^3) showing the Betti numbers, the resulting Hilbert polynomial, and the predicted dimension/degree to illustrate the formulas.
[Preliminaries] Ensure consistent notation for the graded Betti numbers (e.g., always b_{i,j}(S/Jac(f))) and include a short reminder of the Hilbert series formula in the preliminaries for readers less familiar with the passage to the Hilbert polynomial.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report, so we have no points to address individually. We will incorporate minor editorial improvements in the revised version to enhance clarity and presentation.

Circularity Check

0 steps flagged

No significant circularity; explicit formulas follow from standard Hilbert series to polynomial conversion

full rationale

The central results express the dimension and degree of the singular subscheme (defined by the Jacobian ideal) directly via the graded Betti numbers of its minimal free resolution. This is obtained by the standard identity that the Hilbert series equals the alternating sum of the Betti numbers in the resolution, from which the Hilbert polynomial is extracted; its degree and leading coefficient then yield dimension and multiplicity by the usual properties of Hilbert polynomials. The resulting restrictions on Betti numbers are immediate consequences of non-negativity, integrality, and the requirement that the polynomial be non-negative for large degrees. The specialization to homologically strictly plus-one generated hypersurfaces applies the same formulas and checks the degree condition on the polynomial, without any reduction of outputs to inputs by definition, fitting, or self-citation chains. The derivation is self-contained against external commutative algebra benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard setup of projective geometry over an algebraically closed field of characteristic zero and on the existence of minimal free resolutions for graded modules.

axioms (1)

domain assumption The base field is algebraically closed of characteristic zero.
Standard hypothesis for statements about hypersurfaces in P^n and their Jacobian algebras.

pith-pipeline@v0.9.0 · 5376 in / 1283 out tokens · 61933 ms · 2026-05-08T01:55:12.449724+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 1 internal anchor

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Bus´ e, A

L. Bus´ e, A. Dimca, H. Schenck and G. Sticlaru, The Hessian polynomial and the Jacobian ideal of a reduced hypersurface inP n, Advances in Mathematics, Volume 392 (2021), 108035. 3

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Dimca and G

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work page doi:10.1007/s13348-026-00509-y
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Dimca, G

A. Dimca and G. Sticlaru, Graded Betti numbers of the Jacobian algebra and total Tjurina numbers of plane curves, arXiv:2601.08583 3

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Graded Betti numbers of the Jacobian algebra of surfaces in $\mathbb P^3$

A. Dimca and G. Sticlaru, Graded Betti numbers of the Jacobian algebra of surfaces inP 3, arXiv:2602.09966 3, 3, 3

work page internal anchor Pith review Pith/arXiv arXiv
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T. Abe, Plus-one generated and next to free arrangements of hyperplanes.Int. Math. Res. Not. 2021(12): 9233 – 9261 (2021). 1

work page 2021

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Bus´ e, A

L. Bus´ e, A. Dimca, H. Schenck and G. Sticlaru, The Hessian polynomial and the Jacobian ideal of a reduced hypersurface inP n, Advances in Mathematics, Volume 392 (2021), 108035. 3

work page 2021

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Dimca and G

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A. Dimca and G. Sticlaru, Bourbaki modules and the module of Jacobian derivations of pro- jective hypersurfaces, Collectanea Mathematica https://doi.org/10.1007/s13348-026-00509-y 1, 4

work page doi:10.1007/s13348-026-00509-y

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Dimca, G

A. Dimca and G. Sticlaru, Graded Betti numbers of the Jacobian algebra and total Tjurina numbers of plane curves, arXiv:2601.08583 3

work page arXiv

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Graded Betti numbers of the Jacobian algebra of surfaces in $\mathbb P^3$

A. Dimca and G. Sticlaru, Graded Betti numbers of the Jacobian algebra of surfaces inP 3, arXiv:2602.09966 3, 3, 3

work page internal anchor Pith review Pith/arXiv arXiv

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du Plessis, C.T.C

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work page 2022

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L. L. Rose, H.Terao, A free resolution of the module of logarithmic forms of a generic arrange- ment, J. Algebra 136(1991), 376–400. 4, 4 Universit´e C ˆote d’Azur, CNRS, LJAD, France and Simion Stoilow Institute of Mathematics, P.O. Box 1-764, RO-014700 Bucharest, Romania Email address:Alexandru.Dimca@univ-cotedazur.fr Faculty of Mathematics and Informat...

work page 1991