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arxiv: 2606.25060 · v1 · pith:4PH7BO4Mnew · submitted 2026-06-23 · 🧮 math.AG · math.AC

Weddle schemes

Luca Chiantini , {\L}ucja Farnik , Giuseppe Favacchio , Brian Harbourne , Juan Migliore , Tomasz Szemberg , Justyna Szpond This is my paper

Pith reviewed 2026-06-25 22:06 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords Weddle schemesMacaulay dualitynon-Lefschetz lociunexpected conesinterpolation matricesprojective geometryArtinian algebras
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The pith

For a general set of binom(d+n,n) points in P^n the d-Weddle scheme is a hypersurface whose degree is computed explicitly.

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

The paper generalizes the classical Weddle surface to d-Weddle schemes for a finite point set Z in P^n, defined as the locus of centers from which Z projects onto a degree-d hypersurface. Macaulay duality supplies a multiplication map in the associated Artinian algebra whose non-Lefschetz locus is identified with this geometric condition. For a general Z of exactly binom(d+n,n) points the scheme is shown to be a hypersurface and its degree is determined. The work also treats nearby cardinalities, where the scheme has higher codimension, and special six-point sets in which the scheme becomes reducible or nonreduced.

Core claim

For a general set Z subset P^n of binom(d+n,n) points, the d-Weddle scheme is a hypersurface and we compute its degree.

What carries the argument

Macaulay duality applied to powers of linear forms, yielding a multiplication map in an Artinian algebra whose non-Lefschetz locus coincides with the d-Weddle scheme defined by the projection condition.

If this is right

  • When the cardinality equals binom(d+n,n) the scheme has codimension one.
  • For cardinalities near binom(d+n,n) the scheme has higher codimension.
  • For six points not in linear general position the appropriate Weddle scheme can be reducible or nonreduced.

Where Pith is reading between the lines

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

  • The equivalence between the projection definition and the non-Lefschetz locus supplies a geometric test for when multiplication maps fail to be Lefschetz.
  • The explicit degree can be checked directly in low-dimensional cases such as the classical Weddle surface.
  • The same duality technique may apply to other classical loci defined by projection conditions.

Load-bearing premise

The Macaulay duality construction produces a multiplication map whose non-Lefschetz locus exactly coincides with the geometric projection condition defining the d-Weddle scheme.

What would settle it

For d=2 and n=3 with six general points, compute the 2-Weddle scheme explicitly and check whether it is a surface whose degree and support match the classical Weddle surface.

Figures

Figures reproduced from arXiv: 2606.25060 by Brian Harbourne, Giuseppe Favacchio, Juan Migliore, Justyna Szpond, Luca Chiantini, {\L}ucja Farnik, Tomasz Szemberg.

Figure 1
Figure 1. Figure 1: A Weddle surface of six points in P 3 (5 of 6 nodes visible) are nodes; see Remark 4.9). In Remark 4.1 we give a new proof that this surface has degree 4. In the next section we endow the d-Weddle locus with a natural scheme structure. 2. The d-Weddle scheme and two approaches to finding it In this section we present two different approaches to compute the Weddle loci, and we show that they induce the same… view at source ↗
Figure 2
Figure 2. Figure 2: Three sets of harmonic points in Proposition 4.4 on the rulings of a quadric. Bk such that {i, j, k} = {1, 2, 3}. In particular, L contains Aj and Bk, so rj = L = r ′ k , and hence r ′ j = rk. If k < j, then Aj ∈ r ′ k , contrary to assumption. If j < k, then Ak ∈ r ′ j , contrary to assumption. (2) We now check that m1, m2 ⊂ W(Z). Let O be a general point of m1. Project￾ing from O, the images of the point… view at source ↗
read the original abstract

The classical Weddle surface is the locus of vertices of quadric cones through six points in $\mathbb{P}^3$ in linear general position. Equivalently, it is the closure of the locus of centers of projection from which those six points map to six points on a plane conic. Motivated by this 1850 construction of T. Weddle, we introduce $d$-Weddle schemes for finite point sets $Z\subset \mathbb{P}^n$, defined by an analogous projection-to-degree-$d$ condition. Our main tool is Macaulay duality, which yields a natural multiplication map in an Artinian algebra defined by powers of linear forms. This viewpoint connects $d$-Weddle schemes to unexpected cones and interprets them as non-Lefschetz loci for these multiplication maps. Parallel to this, we give an analysis from the point of view of interpolation matrices, and we explain the connections between these approaches. For a general set $Z\subset \mathbb{P}^n$ of $\binom{d+n}{n}$ points, we show that the $d$-Weddle scheme is a hypersurface and we compute its degree. We also study general sets whose cardinalities are "near" such a binomial coefficient, where the Weddle scheme has higher codimension. Returning to sets of six points (not always in linear general position), we discuss special configurations in which the appropriate Weddle scheme is reducible, or even nonreduced.

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 d-Weddle schemes for a finite set Z of points in P^n, defined geometrically as the locus of centers of projection such that the images of Z lie on a degree-d hypersurface. Using Macaulay duality applied to the ideal of powers of linear forms vanishing on Z, the authors interpret these schemes as non-Lefschetz loci of a natural multiplication map on the associated Artinian algebra. The central result states that when |Z| equals binom(d+n,n) and Z is general, the d-Weddle scheme is a hypersurface whose degree is computed explicitly; the paper also treats nearby cardinalities (higher codimension) and special six-point configurations in P^3 where the scheme may be reducible or nonreduced.

Significance. If the identification between the geometric projection locus and the algebraic non-Lefschetz locus holds with matching scheme structure, the work supplies an explicit degree formula and a new algebraic framework for classical Weddle-type constructions, extending them uniformly to higher d and n. The Macaulay-duality approach and the interpolation-matrix analysis provide two independent viewpoints whose agreement would strengthen the result; the treatment of near-binomial cardinalities and reducible cases adds concrete examples.

major comments (1)
  1. [Main theorem on hypersurface property (likely §3 or the statement following the Macaulay-duality construction)] The equality of the geometric projection locus with the non-Lefschetz locus of the multiplication map (including scheme structure and multiplicities) is invoked to obtain both the hypersurface property and the degree formula. The abstract states that the algebraic viewpoint 'interprets' the geometric condition, but the manuscript must contain an explicit proposition or lemma establishing that a point lies in one locus if and only if it lies in the other, with identical ideal sheaves or at least the same support and multiplicity. Without this, the degree computation applies to a possibly different scheme.
minor comments (2)
  1. [Macaulay duality subsection] Notation for the Artinian algebra and the multiplication map should be fixed once at the beginning of the Macaulay-duality section and used consistently; currently the same symbol appears to be reused for the dual space and the quotient ring.
  2. [Degree computation paragraph] The degree formula is stated for general Z; an explicit example with small d and n (e.g., d=2, n=3, six points) would help verify that the computed degree matches the classical Weddle surface degree.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the precise observation regarding the identification of loci. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The equality of the geometric projection locus with the non-Lefschetz locus of the multiplication map (including scheme structure and multiplicities) is invoked to obtain both the hypersurface property and the degree formula. The abstract states that the algebraic viewpoint 'interprets' the geometric condition, but the manuscript must contain an explicit proposition or lemma establishing that a point lies in one locus if and only if it lies in the other, with identical ideal sheaves or at least the same support and multiplicity. Without this, the degree computation applies to a possibly different scheme.

    Authors: We agree that an explicit statement is needed. The current manuscript derives the identification from the Macaulay duality construction and the definition of the projection condition (see the paragraph following the definition of the Artinian algebra and the subsequent discussion of the multiplication map), but does not isolate it as a standalone proposition with a full proof of scheme-theoretic equality. We will add a new lemma (likely in §3) that proves the geometric projection locus coincides with the non-Lefschetz locus of the multiplication map as schemes, including matching support and multiplicities. The proof will use the explicit correspondence between the vanishing of the relevant minors in the interpolation matrix and the kernel dimension of the multiplication map, together with the fact that both loci are defined by the same Fitting ideals. This will make the invocation of the identification fully rigorous and justify the degree formula. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation applies standard Macaulay duality to independent geometric definition

full rationale

The paper defines d-Weddle schemes directly via the geometric projection-to-degree-d condition on point sets Z. It then invokes Macaulay duality as an external tool to produce a multiplication map whose non-Lefschetz locus is shown to interpret the same scheme. The hypersurface claim and degree computation for |Z|=binom(d+n,n) follow from this identification plus interpolation-matrix analysis, without any quoted step that reduces the output to a fitted parameter, self-citation chain, or definitional tautology. No uniqueness theorems, ansatzes, or renamings are imported from the authors' prior work in a load-bearing way. The derivation remains self-contained against standard commutative-algebra benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard facts from projective geometry and Macaulay duality; no free parameters or invented entities are visible from the abstract.

axioms (1)
  • domain assumption Macaulay duality yields a natural multiplication map in the Artinian algebra defined by powers of linear forms whose non-surjectivity locus coincides with the geometric d-Weddle condition.
    Invoked when the abstract states that the viewpoint connects d-Weddle schemes to non-Lefschetz loci.

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