When do Ten Points Lie on a Quadric Surface?

Will Traves

REVIEW 4 major objections 5 minor 17 references

When do Ten Points Lie on a Quadric Surface?

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Ten points in 3-space lie on a quadric exactly when four constructed points are coplanar.

desk verdict A genuine new construction for the Bruxelles Problem's generic case, but Section 4's reduction is not verifiable as written due to broken cross-references and unshipped Macaulay2 code. read the letter →

arxiv 2412.05678 v1 pith:NV5VBJCO submitted 2024-12-07 math.AG math.AC

Will Traves This is my paper · ORCID

classification math.AGmath.AC MSC 14N0551A0514N15

keywords BruxellesProblemquadricsurfacesyntheticconstructionGrassmann-CayleyalgebrabracketpolynomialAreaPrinciplecrossratioprojective3-space

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims a complete synthetic solution to the Bruxelles Problem, the 1825 question of deciding whether ten points in projective 3-space lie on a quadric surface using only incidence constructions. The proposed criterion builds four new points from the original ten by lines, planes, and intersections; the original ten points lie on a quadric exactly when the four new points are coplanar. The paper further claims that every configuration, including degenerate ones, is quadric-decidable: either a special-case rule answers the question directly or a relabeling puts the points in the generic position where the four-point test applies. If correct, the result converts a determinant computation in invariant theory into a geometric construction that can, in principle, be drawn.

What carries the argument

The load-bearing object is the 4x4 matrix M whose rows are labelled by the last four points 6, 7, 8, 9 and whose columns list the values of four reducible quadrics in the basis S at those points; its columns are the four test points, so det(M) = 0 is coplanarity. The construction relies on three pieces: the Grassmann-Cayley meet and join operations that turn bracket polynomials into incidence constructions; an extension of the Area Principle to P3 (Lemma 3.2) showing that the point de ∧ abc has local parameter -[abcd]/[abce] on the line de; and von Staudt's projective multiplication, which realizes products of ratios as points on a line. The basis S itself is discovered with computer algebra and is valid after a relabeling that makes Q nonzero, where Q is a bracket binomial equivalent to a Ceva-like concurrence of three lines.

What would settle it

A concrete check: compute the bracket identity det(M) = Q det(N) on random rational point configurations; any mismatch between coplanarity of the constructed test points and the vanishing of the Veronese determinant would refute the generic case. For the exhaustion claim, enumerate all possible positions of the intersections a = 01 ∩ π, b = 23 ∩ π, c = 45 ∩ π relative to the six lines in the plane π through 6, 7, 8, 9 and verify that each position is covered by Lemma 4.1, 4.2, or 4.3.

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Extended reading notes

Core claim

The central claim is a solution to the Bruxelles Problem: for any ten points in projective 3-space, there is a synthetic construction—using only lines through two points, planes through three points, and their intersections—of four new points Q1, Q2, Q3, Q4 such that the original ten points lie on a quadric surface if and only if the four new points are coplanar. In the generic case, the paper chooses six of the points (0, 1, 2, 3, 4, 5) so that the lines 01, 23, and 45 are mutually skew and the remaining four points are in general position, then forms a basis S of the 3-dimensional space of quadrics through the first six from four reducible quadrics [015X][234X], [012X][345X], [024X][135X], and [045X][123X]. Substituting X = 6, 7, 8, 9 gives a 4x4 matrix M whose columns are test points; det(M) = 0 is the condition that a quadric through the first six also contains the last four. The paper proves det(M) = Q det(N), where N is the 10x10 Veronese constraint matrix, so the determinant condition is exactly the classical criterion. It then shows how to construct the test points (or their images under a projective automorphism) using cross-ratio coordinates and von Staudt's product construction, and it handles all degenerate configurations by a case analysis that either decides the question immediately or relabels the points into the generic situation. The conclusion is that all ten-point configurations are quadric-decidable.

Load-bearing premise

The load-bearing premise is that the Section 4 case analysis is complete and valid: every non-generic configuration is either directly quadric-decidable by Lemma 4.1 or can be relabeled into the generic case using Lemmas 4.2 and 4.3, including the subcases the text cites as 'Lemma 5.1' and 'Lemma 5.2'.

Editorial extensions

If this is right

The 1825 Bruxelles Problem is settled: quadric membership of any ten points in P3 can be decided by a finite sequence of incidence constructions, with no determinant or coordinate arithmetic required.
Every configuration of ten points is quadric-decidable, including configurations with repeated points, four collinear points, or the last four points lying in a plane.
In the generic case, the classical Veronese determinant criterion det(N) = 0 is equivalent, through det(M) = Q det(N), to the coplanarity of four constructed test points, connecting the invariant-theoretic and synthetic formulations.
The construction gives the first synthetic analog for quadrics in P3 of the Pascal-Braikenridge-Maclaurin test that six coplanar points lie on a conic.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension: the determinant factorization det(M) = Q det(N) may be the shadow of a general construction in which a 'test-point matrix' for any Veronese hypersurface-membership problem is built from a computer-found basis of reducible hypersurfaces through a first subset of points.
The paper's use of Macaulay2 to discover the reducible-quadric basis suggests a general recipe: compute a simple basis symbolically, then use Cayley factorization to translate the determinant into a straightedge construction.
An empirical stress test is straightforward: implement the Section 3 construction over a finite field and compare the coplanarity of the constructed points with det(N) on random configurations; any mismatch would pinpoint a missing case.
The references in the text to 'Lemma 5.1' and 'Lemma 5.2' do not match the lemmas as printed; verifying that the intended statements cover the exceptional configurations is the natural way to test the exhaustion claim.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The Bruxelles Problem gets a serious, novel synthetic solution in the generic case, but the paper as submitted is not fully checkable: Section 4's reduction cites lemmas that don't exist, and the Macaulay2 results are asserted without code. I'd send it to a good referee, but they need to be asked to fix those before I'd take the global claim at face value.

What's new: the explicit construction of four test points from the ten original points via meet/join operations and local coordinates is genuinely new. Prior work (Turnbull–Young, White, Sturmfels–Whiteley) established bracket polynomials and a Cayley factorization existence theorem, but not this compact geometric test with a four-point coplanarity condition. The generic case in Section 3 is the heart of the paper and it looks sound in outline: when the basis S is valid (Q ≠ 0), the matrix M has determinant zero exactly when a nonzero quadric vanishes on all ten points, and the columns of M are the test points. The use of von Staudt's product construction to build the Q_i is clever, and Lemma 3.1's interpretation of Q as a Ceva-type incidence is a nice touch. The paper also does its homework on the history and on the relevant literature.

The soft spots are proportionately real. The main one is Section 4. The reduction from general position to the generic case is load-bearing: the central theorem claims every configuration is quadric-decidable, and Section 3 only applies after relabeling to satisfy the four genericity conditions. But that reduction is written in a way I can't certify. It repeatedly cites "Lemma 5.1" and "Lemma 5.2"; those lemmas are not in Section 5. They're almost certainly Lemma 4.1 and Lemma 4.3, but the paper doesn't say so, and the exhaustiveness claim—"the three cases (A), (B), (C) exhaust all possibilities"—is asserted rather than systematically demonstrated. The skew swap construction (Lemma 4.2) is applied under conditions that are checked case by case, but with the broken references, a conscientious reader cannot verify that every subcase is covered. This is repairable, and I suspect it is correct, but it is not proven in the manuscript as it stands.

Second, the Macaulay2 computations in Section 3 are not shipped. The claim that the ideal of 4×4 minors cuts out exactly Q=0, and the identity det(M)=Q det(N), are both asserted on the basis of computer algebra. I believe those claims are checkable, but the code or certificates need to be included for the proof to be complete.

Who is this for: synthetic geometers, invariant theorists, and anyone who likes Grassmann-Cayley algebra. It deserves a serious referee, not a desk reject, but the referee should be given a clear instruction to verify Section 4 with the correct lemma references, and to ask for the Macaulay2 data. After major revision, I'd accept it.

Referee Report

4 major / 5 minor

Summary. The paper claims to solve the Bruxelles Problem: to give a synthetic construction, using only lines, planes, and intersections, that decides whether ten given points in projective 3-space lie on a quadric surface. The proposed solution constructs four new test points that are coplanar exactly when the original ten points lie on a quadric. The main body develops a generic-case construction (Section 3) under four genericity assumptions, based on a basis of reducible quadrics through six of the points and on a 4x4 matrix M whose determinant vanishes exactly in the desired case. Section 4 attempts to show that every configuration either is already decidable or can be relabeled to satisfy the genericity assumptions. The paper uses Grassmann-Cayley algebra, bracket polynomials, cross-ratios, and computations in Macaulay2.

Significance. If the proof were complete, this would be a notable result: it would solve a classical problem posed in 1825, and it would demonstrate a fully synthetic, incidence-based solution to a nontrivial algebraic decision problem. The generic construction is elegant and the use of Grassmann-Cayley algebra is appropriate. The paper also contains a useful algebraic verification in Lemma 3.1, and it explicitly connects the determinant condition to the coplanarity of test points. However, the manuscript as written has several load-bearing gaps: the reduction in Section 4 cites lemmas that do not exist in the cited section, the case analysis is not fully justified, and the synthetic construction in Section 3.2 is only described for a convenient coordinate chart. These issues prevent the central claim from being verified from the manuscript alone, although they appear repairable.

major comments (4)

[Section 4, paragraphs after Lemma 4.3] The reduction argument repeatedly refers to 'Lemma 5.1' and 'Lemma 5.2', but Section 5 is 'Open Problems and Further Reading' and contains no such lemmas. For example, 'In case (A) all ten points lie on quadric by Lemma 5.1 (a)' and 'use Lemma 5.2 to replace the skew lines 23 and 45' have no valid referent. These references appear to be intended as Lemma 4.1 and Lemma 4.3, but as written the proof of the reduction cannot be followed. This is load-bearing because the conclusion that every configuration is either quadric-decidable or reducible to the generic case depends on these lemmas.
[Section 4, cases (B) and (C)] The case analysis that is supposed to exhaust all possible positions of the points a, b, c relative to the lines determined by 6, 7, 8, 9 is asserted rather than proved. The text says 'after relabeling, we can assume that one of the following cases holds' for case (B) and similarly for case (C), but it does not justify why these finite lists of subcases cover every configuration. In particular, the applications of the skew swap construction (Lemma 4.2) are stated with conditions such as 'a ∉ 67' and 'b ∉ 89' that are not derived from the case hypotheses in a systematically verified way. Since the generic construction in Section 3 applies only after this reduction, any omitted configuration would falsify the paper's central claim. A formal proof of exhaustiveness of the subcase enumeration is needed.
[Section 3.2] The construction of the four test points Q1, Q2, Q3, Q4 is only described under the assumption that the corresponding point Pi lies in the coordinate chart U0. The text says, 'for illustrative purposes, we suppose that P1 lies in U0,' and then states that the same construction applies to the other columns of M. No synthetic method is provided to determine a chart containing each Pi, nor to handle the case where a Pi lies on a coordinate plane (so that some of the bracket ratios in the affine formula have zero denominators). This is a gap in the generic-case construction, which is supposed to be a fully synthetic algorithm.
[Section 3.1] The claim that the space of quadrics through points 0,...,5 is spanned by the set S rests on two Macaulay2 computations: the primary decomposition of the ideal of 2x2 minors of a 2x6 matrix, and the assertion that the ideal generated by the 4x4 minors of the coefficient matrix has minimal prime generated by Q. These computations are not reproduced, and no code or ancillary file is provided. Because the spanning property of S is used to set up the matrix M and hence to define the test points, this is a load-bearing computational assertion that a reader cannot independently check from the manuscript. The author should provide the Macaulay2 code or replace these computations with explicit proofs.

minor comments (5)

[Section 2.4] In the sentence 'then we their join is a j + k-extensor', the word 'we' appears to be a typo; it should read 'then their join is'.
[Section 4.1] The text 'We start by finding find three skew lines' contains a duplicated word; it should read 'We start by finding three skew lines.'
[Section 3.2 / Section 1] The numbering of the test points is inconsistent: Section 1 speaks of 'four test points P0, P1, P2, and P3', while Section 3 uses P1, P2, P3, P4 and Q1, Q2, Q3, Q4. This should be harmonized.
[Lemma 3.2 proof] The phrase 'recovered using the cross product' should likely be 'cross ratio', since the lemma concerns local parameters on a projective line.
[Section 3.1, after the definition of M] The identity det(M) = Q det(N) is stated without proof. If it is intended only as an observation, it should be labeled as such; if it is used, a proof or reference is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the four-point coplanarity criterion is an independently derived incidence construction, not an input-equivalent restatement; the only self-citations occur in Open Problems and are not load-bearing.

full rationale

The derivation chain is self-contained. The target condition is encoded by det(N)=0 (the Veronese matrix), and the paper proves that under the Section 3 genericity assumptions the same condition is equivalent to det(M)=0, where M is the 4x4 matrix of basis-quadric evaluations at points 6,7,8,9; this is ordinary linear algebra (M v=0 has a nonzero solution iff det M=0) and is not a fitted relation. The columns of M are then identified with four points P1..P4, and the synthetic Q_i are their images under an explicit projective isomorphism tau; coplanarity is preserved by tau, so the constructed coplanarity test is equivalent to det(M)=0 by a proved incidence construction (Lemma 3.2, Section 3.2), not by definition. The spanning of the basis S of quadrics through points 0..5 is verified by Macaulay2 ideal computations and Lemma 3.1, with a relabeling argument ensuring Q != 0; no parameter is fitted to the ten-point data and no prediction is extracted from a subset of the data. The reductions in Section 4 are geometric case analyses (Grassmann-Cayley meets, skew swap, Grassmann-Plucker relations) that do not invoke the desired conclusion. The only self-citations, [14] and [15], appear in Section 5 open problems and are not load-bearing for the Bruxelles theorem. The broken cross-references to Lemma 5.1 and Lemma 5.2 are a correctness/completeness concern about the Section 4 case analysis, not a circularity; they do not make the central claim equivalent to its inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; the construction is algebraic and parameter-free. The central claim rests on standard projective geometry and Grassmann-Cayley algebra, and on two unverified computational assertions from Macaulay2 (the primary decomposition and the radical computation in Section 3.1). No new entities are postulated; the test points are constructed from the given points.

assumptions (4)

standard math Grassmann-Cayley meet and join operations correspond to intersections and spans in projective space.
Invoked throughout Sections 2.4 and 3.2 to translate bracket expressions into incidence constructions.
standard math Bracket algebra and Grassmann-Pluecker relations are valid; determinants transform correctly under projective transformations.
Used in Section 2.1 cross-ratio computations and in the proof of Lemma 3.1.
standard math von Staudt's geometric product construction computes products of local coordinates using only projections and intersections.
Section 2.2 is the basis for the product construction used in Section 3.2 to build the coordinates of the test points.
ad hoc to paper The Macaulay2 computations are correct: the primary decomposition of the ideal of minors and the claim that the ideal generated by the 4x4 minors has radical generated by Q.
Section 3.1 states 'We used Macaulay2 to find the primary decomposition...' and 'Using Macaulay2, we find that the ideal generated by these minors is a primary ideal and its minimal prime ideal is generated by Q...' No code or output is provided, so correctness is assumed.

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Cite this review

Pith. "Pith review of When do Ten Points Lie on a Quadric Surface?." pith.science (2026). https://pith.science/paper/NV5VBJCO

@misc{pith2026241205678,
  author       = {Pith},
  title        = {Pith review of: When do Ten Points Lie on a Quadric Surface?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NV5VBJCO}},
  note         = {Machine review of arXiv:2412.05678}
}

read the original abstract

A solution is provided to the Bruxelles Problem, a geometric decision problem originally posed in 1825, that asks for a synthetic construction to determine when ten points in 3-space lie on a quadric surface, a surface given by the vanishing of a degree-2 polynomial. The solution constructs four new points that are coplanar precisely when the ten original points lie on a quadric surface. The solution uses only lines constructed through two known points, planes constructed through three known points, and intersections of these objects. The tools involved include an extension of the Area Principle to three-dimensional space, bracket polynomials and the Grassmann-Cayley algebra, and von Staudt's results on geometric arithmetic. Many special cases are treated directly, leading to the generic case, where three pairs of the points generate skew lines and the remaining four points are in general position. A key step in the generic case involves finding a nice basis for the quadrics that pass through six of the ten points, which uses insights derived from Macaulay2, a computational algebra package not available in the nineteenth century.

Figures

Figures reproduced from arXiv: 2412.05678 by the authors.

**Figure 2.** The product construction. Reversing the order of the construction in an appropriate way gives a construction of p1/x from px. First we draw a new line L ′ through p0. Then we pick a point b off L ∪ L ′ and draw lines p∞b, pxb and p1b. We draw the line (pxb ∩ L ′ )p1 and let a denote the intersection of this line and p∞b. Then we draw (p1b ∩ L ′ )a, which intersects L at the point p1/x. This construction works fine i… view at source ↗

**Figure 3.** 2 3 1 p = 23 ∩ 015 q = 13 ∩ 024 r = 12 ∩ 345 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Instances of cases (A) (left), (B) (middle), and (C) (rig [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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