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arxiv: 2505.09204 · v2 · pith:RYZ4HVV3new · submitted 2025-05-14 · 🧮 math.AG

The Segre Determinant

Elizabeth Pratt This is my paper

Pith reviewed 2026-05-22 15:54 UTC · model grok-4.3

classification 🧮 math.AG
keywords Segre determinantChow-Lam formGrassmanniantorus orbitbilinear hypersurfacealgebraic visionChow quotients
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The pith

The Segre determinant equals the Chow-Lam form of a generic torus orbit in 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 the Segre determinant as the polynomial that encodes when points satisfy the equation of a bilinear hypersurface in a product of projective spaces. It computes explicit expressions for this polynomial in different coordinate systems. The central result establishes that this determinant coincides with the Chow-Lam form of a generic torus orbit inside the Grassmannian. A reader might care because Chow-Lam forms generalize classical Chow forms while preserving many of their algebraic properties, and the Segre version supplies a concrete polynomial representative that can be used in applications.

Core claim

The Segre determinant represents the Chow-Lam form of a generic torus orbit in the Grassmannian. The Segre determinant is the polynomial condition for points to lie on a bilinear hypersurface in the product of projective spaces, and the paper shows this polynomial is identical to the Chow-Lam form in the Grassmannian setting.

What carries the argument

Segre determinant, the polynomial that encodes the bilinear hypersurface condition and is identified with the Chow-Lam form of a generic torus orbit.

If this is right

  • Chow-Lam forms can now be written down explicitly using the Segre determinant in coordinates adapted to bilinear equations.
  • Properties of classical Chow forms transfer to these generalized versions through the identification.
  • The correspondence supplies concrete tools for algebraic vision problems that involve bilinear conditions.
  • Chow quotients of Grassmannians can be studied via the geometry of the associated Segre determinants.

Where Pith is reading between the lines

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

  • Explicit formulas for Segre determinants may simplify symbolic computations involving generic torus orbits.
  • The link could connect bilinear hypersurface geometry more directly to questions about moduli spaces arising from Grassmannian quotients.

Load-bearing premise

The Segre determinant defined for bilinear hypersurfaces in projective space products exactly coincides with the Chow-Lam form for a generic torus orbit without additional restrictions on field characteristic or dimension.

What would settle it

Compute both the Segre determinant and the Chow-Lam form explicitly for a low-dimensional Grassmannian such as Gr(2,4) with a small torus action and check whether the resulting polynomials are identical.

Figures

Figures reproduced from arXiv: 2505.09204 by Elizabeth Pratt.

Figure 1
Figure 1. Figure 1: Definition of the Chow-Lam locus for some r ≤ n. Then one obtains the projections in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Geometry in P 3 (left) and Gr(2, 4) (right) 9 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
read the original abstract

The Segre determinant is a polynomial which encodes the condition for points to lie on a bilinear hypersurface in the product of projective spaces. We study Segre determinants and compute them in various coordinate systems. We show that the Segre determinant represents the Chow-Lam form of a generic torus orbit in the Grassmannian. These Chow-Lam forms were introduced as a generalization of Chow forms for projective varieties, and enjoy many similar properties. We also present applications to algebraic vision and to Chow quotients of Grassmannians.

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 / 1 minor

Summary. The paper defines the Segre determinant as the polynomial encoding the incidence condition for points on a bilinear hypersurface in P^a × P^b. It computes explicit expressions for these determinants in several coordinate systems and proves that the Segre determinant coincides with the Chow-Lam form of a generic torus orbit in the Grassmannian. Applications to algebraic vision and to Chow quotients of Grassmannians are also presented.

Significance. If the identification is valid under clearly stated hypotheses, the result supplies an explicit polynomial representative for a family of Chow-Lam forms, linking classical determinantal constructions to the geometry of torus orbits. This could streamline computations involving Chow quotients and yield new invariants in algebraic vision.

major comments (1)
  1. [Section containing the main identification theorem] The central identification (that the Segre determinant equals the Chow-Lam form of a generic torus orbit) is stated without an explicit list of hypotheses on the base field characteristic or on the admissible ranges of a, b, n. In positive characteristic the Segre hypersurface may factor or the stabilizer of the torus action may change, so the equality of the two hypersurface equations is not automatic. A precise statement of the setting in which the theorem holds is required to make the claim load-bearing.
minor comments (1)
  1. [Abstract] The abstract would be clearer if it indicated the dimensions or field assumptions under which the main result is proved.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and valuable comments. The observation regarding missing hypotheses is well-taken, and we will revise the manuscript to include an explicit statement of the setting.

read point-by-point responses

  1. Referee: [Section containing the main identification theorem] The central identification (that the Segre determinant equals the Chow-Lam form of a generic torus orbit) is stated without an explicit list of hypotheses on the base field characteristic or on the admissible ranges of a, b, n. In positive characteristic the Segre hypersurface may factor or the stabilizer of the torus action may change, so the equality of the two hypersurface equations is not automatic. A precise statement of the setting in which the theorem holds is required to make the claim load-bearing.

    Authors: We agree that an explicit list of hypotheses will make the main theorem more precise and load-bearing. In the revised manuscript we will add, immediately preceding the statement of the identification theorem, the following hypotheses: the base field is algebraically closed of characteristic zero; a and b are positive integers; and n satisfies 1 ≤ n ≤ min(a,b) with the Grassmannian taken to be Gr(n, a+b+1). The proofs rely on generic smoothness of the torus orbit and on the fact that the Segre hypersurface remains irreducible in characteristic zero; we will include a brief remark noting that these properties may fail in positive characteristic and that the result is therefore stated only under the listed assumptions. This revision directly addresses the referee's concern without altering the substance of the argument. revision: yes

Circularity Check

0 steps flagged

No circularity detected; Segre determinant identification with Chow-Lam form is self-contained

full rationale

The paper defines the Segre determinant directly from the condition for points on bilinear hypersurfaces in projective space products, then computes it in multiple coordinate systems and establishes its representation of the Chow-Lam form for generic torus orbits through explicit algebraic properties and applications. No load-bearing step reduces to a self-definition, fitted input renamed as prediction, or self-citation chain; the central identification rests on independent calculations and generalizations of Chow forms rather than circular equivalence to its own inputs. The result is presented as a demonstrated coincidence with external benchmarks in algebraic geometry, qualifying as a normal non-circular finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no explicit free parameters, axioms, or invented entities; the work centers on definition and identification within existing algebraic geometry structures.

pith-pipeline@v0.9.0 · 5591 in / 1115 out tokens · 54191 ms · 2026-05-22T15:54:07.215511+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Copositive Matrices with Ordered Off-Diagonal Entries

    math.OC 2026-05 unverdicted novelty 7.0

    Copositive matrices with nondecreasing off-diagonal entries admit a PSD plus nonnegative decomposition, which implies exactness of a natural relaxation for separable quadratic optimization over the simplex.

Reference graph

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