pith. sign in

arxiv: 2606.22587 · v1 · pith:TK4U6M7Pnew · submitted 2026-06-21 · 🧮 math.AG

Jacobian algebras and variation of hyperplane sections

Giovanna Ilardi , Abbas Nasrollah Nejad , Saeed Tafazolian This is my paper

Pith reviewed 2026-06-26 09:37 UTC · model grok-4.3

classification 🧮 math.AG
keywords Jacobian algebraMilnor algebrahyperplane sectionsmoduli of hypersurfacesLefschetz mapisolated singularitiesGIT quotientvariation in moduli
0
0 comments X

The pith

The Milnor algebra supplies an infinitesimal quotient criterion that decides when the hyperplane-section map of a hypersurface with isolated singularities is generically finite onto its image in moduli.

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

The paper establishes that the Milnor algebra M(f) of a hypersurface equation f yields a concrete algebraic test for whether the map sending hyperplanes to the moduli class of their sections is generically finite. The test is first formulated at the level of infinitesimal quotients and then lifted to the coarse moduli space by a local GIT slice argument. In the smooth case the test reproduces the classical Lefschetz criterion and proves generic finiteness once the degree is large enough relative to dimension; when isolated singularities are allowed, an extra linear Jacobian syzygy must be absent before the test reduces to injectivity of a single map on the Milnor algebra. Readers care because the criterion converts a geometric question about families of sections into a finite-dimensional linear-algebra computation inside the Jacobian quotient ring.

Core claim

Using the Milnor algebra M(f), the authors give an infinitesimal quotient criterion for the hyperplane-section map Φ(f) : (P^n)* ⇢ M(d,n-1) to be generically finite onto its image. The passage from the infinitesimal quotient to the coarse moduli space is justified by a local GIT slice argument. In the smooth case the criterion recovers the Lefschetz criterion and, with recent weak Lefschetz theorems, yields generic finiteness for n ≥ 3 and d ≥ n+2. In the singular case a linear Jacobian syzygy appears as a new obstruction; once it is excluded, injectivity of the critical Lefschetz map ℓ : M(f)_{d-1} → M(f)_d governs maximal infinitesimal variation. The criterion is applied to plane curves, s

What carries the argument

The Milnor algebra M(f), the quotient of the polynomial ring by the Jacobian ideal of f, which carries both the infinitesimal quotient condition and the critical Lefschetz map ℓ.

If this is right

  • In the smooth case the new criterion recovers the classical Lefschetz criterion for generic finiteness.
  • For smooth hypersurfaces of degree d ≥ n+2 in P^n with n ≥ 3 the hyperplane-section map is generically finite.
  • A linear Jacobian syzygy (equivalently, positive-dimensional projective automorphism group for non-cones) is the new obstruction that appears once isolated singularities are allowed.
  • After the syzygy obstruction is removed, injectivity of the critical Lefschetz map on the Milnor algebra is necessary and sufficient for maximal infinitesimal variation.
  • The criterion produces new explicit conditions for generic finiteness when hyperplane sections are required to be nodal.

Where Pith is reading between the lines

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

  • The same Milnor-algebra test may classify hypersurfaces whose hyperplane sections realize the full expected dimension in the moduli space for small values of n and d.
  • Computational verification of the critical Lefschetz map could be used to decide whether a given singular hypersurface has maximal variation without constructing the full moduli map.
  • The obstruction coming from linear Jacobian syzygies suggests analogous algebraic conditions may control variation of sections for other classes of varieties with isolated singularities.

Load-bearing premise

The hypersurface has at most isolated singularities and the local GIT slice argument correctly lifts the infinitesimal quotient condition to the coarse moduli space.

What would settle it

An explicit hypersurface with isolated singularities, no linear Jacobian syzygy, and injective critical Lefschetz map, yet whose hyperplane-section map fails to be generically finite onto its image in the moduli space.

read the original abstract

We study the variation in moduli of hyperplane sections of a hypersurface $V(f)\subseteq \mathbf P^n$ with at most isolated singularities. Using the Milnor algebra $M(f)$, we give an infinitesimal quotient criterion for the hyperplane-section map $\Phi(f):(\mathbf P^n)^*\dashrightarrow M(d,n-1)$ to be generically finite onto its image. The passage from the infinitesimal quotient to the coarse moduli space is justified by a local GIT slice argument. Our approach gives a Jacobian-algebraic extension of the Beauville--Patel--Riedl--Tseng theory from smooth hypersurfaces to hypersurfaces with isolated singularities. In the smooth case it recovers the Lefschetz criterion and, using recent weak Lefschetz results, gives generic finiteness for $n\geq 3$ in the range $d\geq n+2$. In the singular case a new obstruction appears: a linear Jacobian syzygy, equivalently, for non-cones, a positive-dimensional projective automorphism group. After this obstruction is excluded, maximal infinitesimal variation is governed by the injectivity of the critical Lefschetz map $\ell:M(f)_{d-1}\to M(f)_d$. We apply the criterion to plane curves, surfaces in $\mathbf P^3$, and hypersurfaces admitting singular hyperplane sections, obtaining new criteria involving nodal sections and an application to the Schoen quintic threefold.

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

1 major / 2 minor

Summary. The paper studies variation of hyperplane sections for hypersurfaces V(f) in P^n with at most isolated singularities. It uses the Milnor algebra M(f) to formulate an infinitesimal quotient criterion for generic finiteness of the hyperplane-section map Φ(f):(P^n)* ⇢ M(d,n-1). The lift from the infinitesimal condition (injectivity of the critical Lefschetz map ℓ:M(f)_{d-1}→M(f)_d after excluding linear Jacobian syzygies) to the coarse moduli space is justified by a local GIT slice argument. The work extends the Beauville–Patel–Riedl–Tseng theory to the singular setting, recovers the classical Lefschetz criterion when f is smooth, identifies linear Jacobian syzygies as a new obstruction (equivalent to positive-dimensional automorphism groups for non-cones), and applies the criterion to plane curves, surfaces in P^3, hypersurfaces with singular hyperplane sections, and the Schoen quintic threefold, yielding new finiteness criteria involving nodal sections.

Significance. If the local GIT slice construction is valid, the paper supplies a Jacobian-algebraic criterion that extends existing Lefschetz-type results to isolated-singularity hypersurfaces and produces concrete applications (including to the Schoen quintic). The approach is parameter-free once the Milnor algebra is fixed and recovers known smooth-case statements via weak Lefschetz theorems; the identification of the linear syzygy obstruction is a clear conceptual advance.

major comments (1)
  1. [local GIT slice argument] The section justifying the local GIT slice argument (the bridge from injectivity of the critical Lefschetz map on M(f) to generic finiteness of Φ(f) on the coarse moduli space M(d,n-1)): the manuscript states that a local GIT slice lifts the infinitesimal quotient condition, but supplies no explicit construction of the slice, no verification that the slice is étale at the point corresponding to f, and no analysis of how isolated singularities affect orbit closures or stabilizer dimensions. This step is load-bearing for the central claim that the infinitesimal criterion implies the stated finiteness on the coarse space.
minor comments (2)
  1. [Introduction] The abstract and introduction use the notation M(d,n-1) without an explicit definition of the target moduli space; a sentence clarifying whether this is the GIT quotient of the space of degree-d hypersurfaces in P^{n-1} or a different coarse space would improve readability.
  2. [Applications] In the applications to the Schoen quintic, the manuscript invokes the criterion after excluding linear Jacobian syzygies; a brief remark on whether the quintic satisfies this exclusion (or how it is checked) would make the application self-contained.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the single major comment below and will revise the manuscript to strengthen the exposition of the local GIT slice argument.

read point-by-point responses

  1. Referee: The section justifying the local GIT slice argument (the bridge from injectivity of the critical Lefschetz map on M(f) to generic finiteness of Φ(f) on the coarse moduli space M(d,n-1)): the manuscript states that a local GIT slice lifts the infinitesimal quotient condition, but supplies no explicit construction of the slice, no verification that the slice is étale at the point corresponding to f, and no analysis of how isolated singularities affect orbit closures or stabilizer dimensions. This step is load-bearing for the central claim that the infinitesimal criterion implies the stated finiteness on the coarse space.

    Authors: We agree that the current manuscript provides only a brief reference to the local GIT slice without an explicit construction or detailed verification. In the revised version we will expand the relevant section to supply an explicit local GIT slice at the point corresponding to f, verify that the slice is étale there, and analyze the effect of isolated singularities on orbit closures and stabilizer dimensions, using standard results from GIT for hypersurface moduli spaces adapted to the isolated-singularity setting. This will make the passage from the infinitesimal condition to generic finiteness on the coarse space fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses independent Milnor algebra and external GIT slice

full rationale

The paper presents an infinitesimal criterion via the Milnor algebra M(f) for generic finiteness of the hyperplane-section map, extending the Beauville-Patel-Riedl-Tseng theory while recovering the Lefschetz criterion in the smooth case. The local GIT slice argument is cited to lift the infinitesimal condition to the coarse moduli space, but the provided text contains no self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claim to its inputs by construction. The derivation remains self-contained against the stated algebraic and geometric inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption of at most isolated singularities and on the validity of the local GIT slice lifting; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption V(f) has at most isolated singularities
    Explicitly stated as the setting for the variation study.
  • domain assumption Local GIT slice argument lifts infinitesimal quotients to the coarse moduli space
    Invoked to justify passage from infinitesimal to global statement.

pith-pipeline@v0.9.1-grok · 5789 in / 1184 out tokens · 15551 ms · 2026-06-26T09:37:16.638830+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

19 extracted references · 1 canonical work pages

  1. [1]

    Beauville, A.: Hyperplane sections of cubic threefolds. Proc. Amer. Math. Soc. 153, 5167–5170 (2025) 1, 3, 3, 6.7

    2025

  2. [2]

    Preprint

    Beauville, A.: Maximal variation of linear systems. Preprint. arXiv:2511.22329 (2025) 1, 3, 3, 5, 5.3

    arXiv 2025

  3. [3]

    Applications

    Beorchia, V., Mir´ o-Roig, R.-M.: Weak Lefschetz property of equigenerated complete intersections. Applications. Preprint. arXiv:2503.17991 (2025) 1, 3, 3, 5.3

    arXiv 2025

  4. [4]

    M., Nagel, U.: The weak Lefschetz property for height four complete intersections

    Boij, M., Migliore, J., Mir´ o-Roig, R. M., Nagel, U.: The weak Lefschetz property for height four complete intersections. Trans. Amer. Math. Soc. Ser. B 10, 1254–1286 (2023) 5, 5.3

    2023

  5. [5]

    S., Parkes, L.: An exactly soluble superconformal theory from a mirror pair of Calabi-Yau manifolds

    Candelas, P., de la Ossa, X., Green, P. S., Parkes, L.: An exactly soluble superconformal theory from a mirror pair of Calabi-Yau manifolds. Phys. Lett. B 258, 118–126 (1991) 6.8

    1991

  6. [6]

    http://www.singular.uni-kl.de (2014) 6.8

    Decker, W., Greuel, G.-M., Pfister, G., Sch¨ onemann, H.:Singular4-0-1 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2014) 6.8

    2014

  7. [7]

    An introduction

    Dimca, A.:Topics on real and complex singularities. An introduction. Advanced Lectures in Mathe- matics. Friedr. Vieweg & Sohn, Braunschweig (1987) 2.2

    1987

  8. [8]

    Dimca, A., Ilardi, G.: On the duals of smooth projective complex hypersurfaces. Publ. Mat. 68, 431–438 (2024) 6

    2024

  9. [9]

    Dimca, A., Ilardi, G.: Lefschetz properties of Jacobian algebras and Jacobian modules. Ann. Sc. Norm. Super. Pisa Cl. Sci. https://doi.org/10.2422/2036-2145.202301-014 6.1

    work page doi:10.2422/2036-2145.202301-014 2036

  10. [10]

    Dimca, A., Popescu, D.: Hilbert series and Lefschetz properties of dimension one almost complete intersections. Comm. Algebra 44, 4467–4482 (2016) 1, 4.1

    2016

  11. [11]

    A., Wall, C

    du Plessis, A. A., Wall, C. T. C.: Curves inP 2(C) with one-dimensional symmetry. Rev. Mat. Complut. 12, 117–132 (1999) 2.9

    1999

  12. [12]

    A., Wall, C

    du Plessis, A. A., Wall, C. T. C.: Discriminants, vector fields and singular hypersurfaces. In:New developments in singularity theory(Cambridge, 2000). NATO Sci. Ser. II Math. Phys. Chem., vol. 21, pp. 351–377. Kluwer Acad. Publ., Dordrecht (2001) 2.11

    2000

  13. [13]

    Graduate Texts in Mathematics, vol

    Hartshorne, R.:Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977) 1

    1977

  14. [14]

    Ilardi, G.: Jacobian ideals, arrangements and Lefschetz properties. J. Algebra 508, 418–430 (2018) 5

    2018

  15. [15]

    Jardim, M., Nasrollah Nejad, A., Simis, A.: The Bourbaki degree of a plane projective curve. Trans. Amer. Math. Soc. 377, 7633–7655 (2024) 4.3

    2024

  16. [16]

    Luna, D.: Slices ´ etales. Bull. Soc. Math. France, M´ emoire 33, 81–105 (1973) 2.5, 2.5

    1973

  17. [17]

    Mumford, D., Fogarty, J., Kirwan, F.:Geometric Invariant Theory. 3rd edn. Ergebnisse der Mathe- matik und ihrer Grenzgebiete (2), vol. 34. Springer, Berlin (1994) 2.5

    1994

  18. [18]

    Algebra Number Theory 18 (2024) 1, 3, 3

    Patel, A., Riedl, E., Tseng, D.: Moduli of linear slices of high degree smooth hypersurfaces. Algebra Number Theory 18 (2024) 1, 3, 3

    2024

  19. [19]

    R. Caccioppoli

    Schoen, C.: On the geometry of a special determinantal hypersurface associated to the Mumford- Horrocks vector bundle. J. Reine Angew. Math. 364, 85–111 (1986) 6.8 Dipartimento Matematica e Applicazioni “R. Caccioppoli”, Universit`a degli Studi Di Napoli “Federico II” Via Cintia - Complesso Universitario Di Monte S. Angelo 80126 - Napoli - Italia Email ad...

    1986