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arxiv: 2510.16417 · v2 · submitted 2025-10-18 · 🧮 math.AG

The Hesse Pencil Variety

Elisabetta Rocchi This is my paper

Pith reviewed 2026-05-18 06:27 UTC · model grok-4.3

classification 🧮 math.AG
keywords Hesse pencilplane cubicHessianGrassmannianSL(3) orbitSchur moduleskew-invariantsingular locus
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The pith

The Hesse pencil variety H_8 is the 8-dimensional Zariski closure of pencils from a smooth plane cubic and its Hessian, realized as the SL(3)-orbit closure of a specific pencil inside the Grassmannian.

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

The paper defines and studies the Hesse pencil variety H_8 as the closure in the Grassmannian G(1,9) of all pencils generated by a smooth plane cubic curve together with its Hessian. It establishes that this variety has dimension 8 and equals the intersection of G(1,9) with ten hyperplanes coming from the Schur module S_{(5,1)} C^3. The variety is identified with the closure of the SL(3)-orbit of the pencil spanned by x^3 + y^3 + z^3 and xyz, and it contains eight further orbits whose points are also pencils of this type. A cubic skew-invariant R whose vanishing selects precisely these pencils supplies explicit equations, while a count of six Hesse configurations through four general points in the plane is used to compute the multidegree and confirm the identification.

Core claim

H_8 is the Zariski closure in G(1,9) of the set of pencils generated by a smooth plane cubic and its Hessian. It has dimension 8, coincides with the SL(3)-orbit closure of the pencil <x^3 + y^3 + z^3, xyz>, contains eight additional orbits, and is singular precisely along the union of the two orbits O(<x^3, x^2 y>) and O(<x^2 y, x^2 z>). It is cut out by the vanishing of the cubic skew-invariant R in ∧^3(Sym^3 C^3) given by R(l^3, m^3, n^3) = (l ∧ m ∧ n)^3, and this description is verified by showing that the multidegree matches the one obtained from the geometric count that exactly six Hesse configurations pass through four general points of P^2.

What carries the argument

The cubic skew-invariant R in ∧^3(Sym^3 C^3) defined by R(l^3, m^3, n^3) = (l ∧ m ∧ n)^3, whose vanishing characterizes pencils generated by a cubic and its Hessian.

If this is right

  • H_8 is realized as the intersection of G(1,9) with ten hyperplanes corresponding to the Schur module S_{(5,1)} C^3.
  • The singular locus of H_8 is exactly the union of the two orbits O(<x^3, x^2 y>) and O(<x^2 y, x^2 z>).
  • H_8 contains eight orbits in addition to the closure of the orbit of <x^3 + y^3 + z^3, xyz>.
  • The multidegree of H_8 equals that of the zero locus of the invariant R, computed via the six Hesse configurations through four general points.
  • Explicit equations for H_8 are furnished by the components of the invariant R.

Where Pith is reading between the lines

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

  • The explicit equations from R may allow computation of the Chow ring or cohomology of H_8 via representation-theoretic methods.
  • The same counting technique for Hesse configurations could be applied to compute degrees of related orbit closures in other Grassmannians of lines.
  • The controlled singularities along two orbits suggest that the generic point of H_8 corresponds to a smooth pencil in a natural sense.
  • The construction supplies a model for studying other classical configurations of cubics and their associated invariants in the Grassmannian.

Load-bearing premise

Through four general points of the projective plane there pass exactly six Hesse configurations.

What would settle it

A direct computation or geometric check showing that the orbit closure of the pencil <x^3 + y^3 + z^3, xyz> under SL(3) has dimension different from 8, or that some pencil generated by a smooth cubic and its Hessian fails to satisfy R = 0.

read the original abstract

We introduce and study the Hesse pencil variety $H_8$, obtained as the Zariski closure in the Grassmannian $G(1,9)$ of the set of pencils generated by a smooth plane cubic and its Hessian. We prove that $H_8$ has dimension $8$ and can be realized as the intersection of $G(1,9)$ with ten hyperplanes corresponding to the Schur module $\mathbb{S}_{(5,1)}\mathbb{C}^3$. Moreover, $H_8$ coincides with the closure of the $SL(3)$-orbit of the pencil $\langle x^3+y^3+z^3,\ xyz\rangle$ and contains eight additional orbits. The variety is singular, and its singular locus is precisely the union of two orbits, $O(\langle x^3,x^2y\rangle)$ and $O(\langle x^2y,x^2z\rangle)$. A key ingredient in our study is a cubic skew-invariant $R\in \bigwedge^3(\mathrm{Sym}^3\mathbb{C}^3)$ defined by $R(l^3,m^3,n^3)=(l\wedge m\wedge n)^3$, whose vanishing characterizes pencils generated by a cubic and its Hessian. This invariant allows us to write explicit equations defining $H_8$. A crucial geometric step in our argument is the fact that through four general points of $\mathbb{P}^2$ there pass exactly six Hesse configurations, which enables us to compute the multidegree of $H_8$ and conclude that it coincides with the variety defined by the invariant $R$.

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 introduces the Hesse pencil variety H_8 as the Zariski closure in G(1,9) of pencils spanned by a smooth plane cubic and its Hessian. It proves that H_8 has dimension 8, is cut out by ten hyperplanes corresponding to the Schur module S_{(5,1)}C^3, coincides with the SL(3)-orbit closure of the pencil <x^3+y^3+z^3, xyz>, contains eight additional orbits, and has singular locus equal to the union of the orbits O(<x^3, x^2 y>) and O(<x^2 y, x^2 z>). A cubic skew-invariant R in ∧^3(Sym^3 C^3) is defined by R(l^3,m^3,n^3)=(l∧m∧n)^3 whose vanishing characterizes Hesse pencils; the identification of H_8 with the zero set of R is obtained by matching multidegrees, using the enumerative fact that exactly six Hesse configurations pass through four general points of P^2.

Significance. If the results hold, the paper gives a concrete description of a classical orbit closure in the Grassmannian of lines in P^9, linking the geometry of plane cubics and their Hessians to Schur-module equations and an explicit invariant R. The explicit equations, orbit decomposition, and singularity analysis provide a model case for studying special varieties arising from invariant theory of ternary cubics.

major comments (1)
  1. [Abstract / multidegree argument] Abstract and the multidegree computation section: the identification of H_8 with the zero locus of R rests on equating their multidegrees in G(1,9), which in turn relies on the enumerative statement that exactly six Hesse configurations pass through four general points of P^2. This count is described as crucial but is not supplied with a reference to a standard result in the literature on Hesse pencils or with a self-contained parameter count or proof; an error in the number or in the generality assumption would change the computed multidegree and block the equality of the two varieties.
minor comments (2)
  1. [Section on Schur-module equations] The notation for the Grassmannian G(1,9) and the embedding into projective space via Plücker coordinates could be recalled explicitly when the hyperplane equations from S_{(5,1)}C^3 are introduced.
  2. [Introduction] A brief sentence recalling the classical definition of the Hessian of a plane cubic would help readers who are not specialists in ternary forms.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need to strengthen the justification of the enumerative count in the multidegree argument. We address the major comment below and will revise the paper to improve clarity and completeness.

read point-by-point responses

  1. Referee: Abstract and the multidegree computation section: the identification of H_8 with the zero locus of R rests on equating their multidegrees in G(1,9), which in turn relies on the enumerative statement that exactly six Hesse configurations pass through four general points of P^2. This count is described as crucial but is not supplied with a reference to a standard result in the literature on Hesse pencils or with a self-contained parameter count or proof; an error in the number or in the generality assumption would change the computed multidegree and block the equality of the two varieties.

    Authors: We agree that the current presentation relies on this enumerative fact without sufficient justification in the text. The count of exactly six Hesse configurations through four general points of P^2 is used to determine the multidegree of H_8 and thereby conclude that it coincides with the zero locus of the cubic invariant R. While this is a classical enumerative statement in the geometry of plane cubics, we acknowledge that the manuscript should either cite a standard reference or include a brief self-contained argument. In the revised version we will add a short parameter count (or an appropriate citation from the literature on Hesse pencils and ternary cubic invariants) to make the multidegree computation fully rigorous and independent of external results. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation relies on external enumerative fact

full rationale

The paper defines H_8 geometrically as the Zariski closure of pencils generated by a smooth cubic and its Hessian in G(1,9). It introduces the skew-invariant R with R(l^3,m^3,n^3)=(l∧m∧n)^3 whose zero set is claimed to characterize such pencils. Identification of H_8 with the intersection by ten hyperplanes from S_{(5,1)}C^3, with the SL(3)-orbit closure of <x^3+y^3+z^3,xyz>, and with V(R) proceeds via dimension count, orbit decomposition, and multidegree comparison. The multidegree of H_8 is obtained from the stated fact that exactly six Hesse configurations pass through four general points of P^2. This enumerative input is presented as external and independent; it is not derived from the definition of H_8, from R, or from any fitted parameter within the paper. No self-citation is load-bearing, no ansatz is smuggled, and no step equates a derived quantity to its own input by construction. The argument is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on standard facts from algebraic geometry and representation theory plus one key geometric count; no free parameters are fitted and no new physical entities are postulated.

axioms (2)
  • standard math Standard properties of Grassmannians G(1,9), Zariski closures, and Schur modules S_{(5,1)}C^3 hold in characteristic zero.
    Invoked to realize H_8 as an intersection and to define the ambient space and equations.
  • standard math The SL(3)-action on the space of cubics and on the Grassmannian is well-defined and the orbit closures behave as stated.
    Used to identify H_8 with the closure of the orbit of <x^3+y^3+z^3, xyz> and to classify the additional orbits.
invented entities (1)
  • Cubic skew-invariant R in ∧^3(Sym^3 C^3) no independent evidence
    purpose: To characterize pencils generated by a cubic and its Hessian by the vanishing of R(l^3,m^3,n^3)=(l∧m∧n)^3.
    Explicitly constructed in the paper; its vanishing is shown to select the desired pencils and to cut out H_8.

pith-pipeline@v0.9.0 · 5814 in / 1879 out tokens · 58378 ms · 2026-05-18T06:27:04.704822+00:00 · methodology

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