Sylvester line quadrics: an exact rank-gap characterization and its syzygetic boundary

Status

This note records one exact reformulation of the Sylvester condition, an explicit counterexample to the resulting geometric shortcut, and a sharper reduction of the remaining attack. It neither proves nor refutes Conjecture 4.1 of Fouli, Grate, Kara, LaClair, McCullough, Nguyen, Sobieska, and Udanshive.

The source conjecture says that if the graded Möbius algebra of a simple matroid is quadratic, then its defining ideal is linearly presented exactly when the matroid is Sylvester. The source proves the implication from linear presentation to Sylvester. The reverse implication remains open.

1. The quadratic line ideal

Let $M$ be a simple matroid on $E$, let $V$ have basis $(x_e)_{e\in E}$, and write $S=\operatorname{Sym}(V)$. For every rank-two flat $L$, all squarefree pair monomials $x_ex_f$ with $e,f\in L$ have the same image in the graded Möbius algebra. Consequently its quadratic relation space is

$$R_M=(I_M)_2 =\left\langle x_e^2:e\in E\right\rangle +\left\langle x_ex_f-x_gx_h:\operatorname{cl}(e,f)=\operatorname{cl}(g,h) \right\rangle .$$

Call $J_M=(R_M)$ the quadratic line ideal. It depends only on the rank-two flats of $M$. If the graded Möbius algebra is quadratic, then

$$I_M=J_M.$$

Thus the open reverse implication is reduced exactly to the first syzygies of this line ideal under the additional assumption that it already presents the full graded Möbius algebra.

2. The inverse quadratic system detects Sylvester exactly

Pair monomials belonging to different rank-two flats have disjoint supports. Under the coefficient pairing on $\operatorname{Sym}^2(V)$, the orthogonal space of $R_M$ is therefore

$$W_M=R_M^\perp =\operatorname{span}\{Q_L:L\in L_2(M)\}, \qquad Q_L=\sum_{\{e,f\}\subset L}z_ez_f.$$

Here the variables $z_e$ are dual to the $x_e$. The following statement is valid over every characteristic-zero field, after extending scalars when testing matrix rank.

Proposition 2.1

For a simple matroid $M$, the following are equivalent:

1. $M$ is Sylvester: every pair of elements belongs to a 3-circuit;
2. every nonzero quadric in $W_M$ has rank at least three;
3. $\mathbb P(W_M)$ is disjoint from the rank-at-most-two symmetric determinantal locus.

Proof

If $M$ is not Sylvester, some pair $\{e,f\}$ has a two-point closure $L$. Then $Q_L=z_ez_f$ has rank two.

Conversely, assume $M$ is Sylvester and take $0\ne Q=\sum_L a_LQ_L$. Choose a line $L$ with $a_L\ne0$. It contains three distinct points $e,f,g$. Because every pair belongs to a unique rank-two flat, the principal matrix of $Q$ on these three coordinates is

$$a_L\begin{pmatrix} 0&1&1\\ 1&0&1\\ 1&1&0 \end{pmatrix}.$$

Its determinant is $2a_L^3\ne0$. Hence $\operatorname{rank}(Q)\ge3$. $\square$

This packages the Sylvester axiom as a canonical rank gap. The next section shows that the rank gap, by itself, does not control the first syzygies.

3. A four-variable countermodel to the secant shortcut

Order the six off-diagonal monomials as

$$12,13,14,23,24,34.$$

Let $W\subset\operatorname{Sym}^2(\mathbb Q^4)^*$ be the three-dimensional space whose coefficient map is

$$C= \begin{pmatrix} -1&-1&-1\\ 1&-1&-1\\ 0&1&-1\\ 0&0&-1\\ -1&0&1\\ 0&1&1 \end{pmatrix}.$$

Thus a parameter vector $u\in\mathbb Q^3$ gives the off-diagonal coefficients $Cu$ of a quadric in $W$, and every quadric in $W$ has zero diagonal.

Proposition 3.1

$\mathbb P(W)$ contains no quadric of rank at most two.

Proof

Over an algebraic closure of a characteristic-zero field, a nonzero zero-diagonal symmetric matrix of rank two has its support across a bipartition. Indeed it can be written $uv^t+vu^t$; the diagonal equations $u_iv_i=0$ partition the nonzero coordinates according to which factor vanishes. Rank one is impossible unless the matrix is zero.

It is therefore enough to check the seven bipartitions of four coordinates, up to exchanging their sides. For the four cuts of type $1+3$, restriction to the three within-side edges gives $3\times3$ determinants

$$1,\quad 2,\quad -3,\quad -2.$$

Thus no nonzero parameter vector can be supported on such a cut. For the three cuts of type $2+2$, the two within-side equations have respective kernel vectors

$$(0,1,-1),\qquad(-1,0,-1),\qquad(-1,0,0).$$

At those vectors, the determinants of the remaining $2\times2$ cross blocks are

$$-2,\quad -3,\quad 1.$$

The resulting matrices have rank four, not two. This exhausts the possible bipartitions. $\square$

Now set $I=(W^\perp)\subset S=\mathbb Q[x_1,x_2,x_3,x_4]$. The ideal is generated by the four squares and the three quadrics

$$\begin{aligned} r_1&=-x_1x_2-x_1x_3-2x_1x_4+4x_2x_3,\\ r_2&=-x_1x_2+3x_1x_3+2x_1x_4+4x_2x_4,\\ r_3&= x_1x_2+x_1x_3+2x_3x_4. \end{aligned}$$

The quotient $A=S/I$ is quadratic and Artinian. Exact row reduction gives

$$\operatorname{HF}_A=(1,4,3),$$

because the products $x_ir_j$ span all four squarefree cubic monomials. In internal degree four, therefore,

$$K_{2,4}(A)=A_2\otimes\bigwedge^2\mathbb Q^4$$

is an 18-dimensional cycle space. The preceding Koszul differential

$$d_3:A_1\otimes\bigwedge^3\mathbb Q^4 \longrightarrow A_2\otimes\bigwedge^2\mathbb Q^4$$

has exact rank 15. Hence

$$\boxed{\beta^S_{2,4}(A)=18-15=3.}$$

So $I$ is not linearly presented even though its inverse quadratic system avoids the full rank-at-most-two locus. Secant avoidance is not a sufficient syzygetic criterion, even for a quadratic Artinian ideal containing all variable squares.

4. Why this does not refute the matroid conjecture

The countermodel is not a graded Möbius algebra. A matroid inverse system has the stronger form

$$W_M=\bigoplus_{L\in L_2(M)}\mathbb k Q_L,$$

where the edge supports of the $Q_L$ partition all unordered pairs. The countermodel lacks this disjoint line-support structure. Proposition 3.1 therefore stops the generic Veronese/secant route but leaves the matroid-specific incidence route open.

There is a useful exact positive control. The rank-three truncation $T(PG(3,2))$ and the full projective geometry $PG(3,2)$ have the same points and rank-two flats, hence the same quadratic line ideal. The truncation has extra essential cubic relations and the earlier audit found $\dim H_{2,4}=209$. For the full projective geometry, the line ideal is the entire defining ideal. Exact rational Koszul ranks give

$$\beta_{2,4}=\beta_{2,5}=\beta_{2,6}=0.$$

Thus the full quadratic coherence removes the truncation’s entire nonlinear second-homology block. This is a control, not a proof by examples.

5. Next decisive question

The surviving structural target is now sharper:

Is the quadratic line ideal of every finite linear space whose lines all have at least three points linearly presented? If not, what additional incidence identity is forced when that line ideal is the defining ideal of a quadratic graded Möbius algebra?

A proof should construct the presentation from the line-partition tensor, not from chosen third points or a matroid census. A failure should be an intrinsic obstruction in that tensor. The generic secant geometry supplies no further leverage.

6. Reproducibility and credit

The dependency-free script verify-secant-gap-boundary.py checks every cut determinant, the orthogonal relation space, the cubic span, the exact Koszul rank, and the full $PG(3,2)$ positive control over $\mathbb Q$.

The conjecture, its proved direction, and the $T(PG(3,2))$ source example are due to Fouli, Grate, Kara, LaClair, McCullough, Nguyen, Sobieska, and Udanshive (arXiv:2609.03827v1, 3 September 2026). The presentation of graded Möbius algebras and the quadraticity framework used here are due to LaClair, Mastroeni, McCullough, and Peeva (arXiv:2412.18499v1). The rank-gap reformulation, countermodel, and exact audits in this note have not received external review.

References

• L. Fouli, S. Grate, S. Kara, A. LaClair, J. McCullough, V. Nguyen, A. Sobieska, and P. Udanshive, Graded Betti numbers of graded Möbius algebras of uniform matroids, arXiv:2609.03827v1 (2026), https://arxiv.org/html/2609.03827v1.
• A. LaClair, M. Mastroeni, J. McCullough, and I. Peeva, Koszul graded Möbius algebras and strongly chordal graphs, arXiv:2412.18499v1 (2024), https://arxiv.org/html/2412.18499v1.