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Point-line geometries with a generating set that depends on the underlying field

Geometry & topology · math.AG · math.SG · posed by R.J. Blok, A. Pasini · open

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Statement

Computations in some cases with n = 3 suggest that if S is a generating set for Γ0, then F1 = F0 and ⟨S⟩Γ is precisely the collection of points L of Γ for which there exists an i-element E of Δ(F0) such that L ⊂ E^⟂ (in Δ(F)) and L ∩ E ≠ ∅.

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  • Point-line geometries with a generating set that depends on the underlying field
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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: Reconstructed statement: for a classical polar space Δ(F)\Delta(F) over a field FF, with subgeometry Δ(F0)\Delta(F_0) over a subfield F0F_0, let Γ\Gamma be the ii-Grassmann point-line geometry and Γ0\Gamma_0 its F0F_0-subgeometry. If SS generates Γ0\Gamma_0, then

    ⟨S⟩Γ={L∈Γ:∃E∈Δi(F0) with L⊆E⊥, L∩E≠∅},\langle S\rangle_\Gamma=\{L\in\Gamma:\exists E\in\Delta_i(F_0)\text{ with }L\subseteq E^\perp,\ L\cap E\ne\varnothing\},

    and F1=F0F_1=F_0. The wording is ambiguous about the polar type; the standard context of the paper includes line-Grassmannians of symplectic polar spaces. The statement is false already for the rank 33 symplectic line-Grassmannian.

    Result: Let F0=F2F_0=\mathbb F_2, F=F4=F2(ω)F=\mathbb F_4=\mathbb F_2(\omega), ω2+ω+1=0\omega^2+\omega+1=0. Let V=F6V=F^6 with symplectic form

    B(x,y)=x1y4+x2y5+x3y6+x4y1+x5y2+x6y3.B(x,y)=x_1y_4+x_2y_5+x_3y_6+x_4y_1+x_5y_2+x_6y_3.

    Let Γ\Gamma be the line-Grassmannian of W(5,F)W(5,F); its points are totally isotropic 22-spaces, and its lines are pencils [p,Π][p,\Pi], where pp is a point and Π\Pi a totally isotropic plane. Let RR denote the conjectured right-hand side.

    A generated set ⟨S⟩Γ\langle S\rangle_\Gamma is always a subspace of Γ\Gamma. Thus, if the conjecture were true, RR would have to be a subspace.

    Set

    u=(1,ω+1,0,0,0,0),v=(0,0,0,1,ω,0),z=(0,0,1,0,0,ω+1),u=(1,\omega+1,0,0,0,0),\quad v=(0,0,0,1,\omega,0),\quad z=(0,0,1,0,0,\omega+1), p=u+v,Π=⟨u,v,z⟩F.p=u+v,\qquad \Pi=\langle u,v,z\rangle_F.

    One checks B(u,v)=B(u,z)=B(v,z)=0B(u,v)=B(u,z)=B(v,z)=0, so Π\Pi is totally isotropic. Define three lines through pp in Π\Pi:

    L=⟨u,v⟩F,L=\langle u,v\rangle_F, A=⟨p,z+v⟩F,M=⟨p,z+ωv⟩F.A=\langle p,z+v\rangle_F,\qquad M=\langle p,z+\omega v\rangle_F.

    Thus A,M,LA,M,L are points of one line of Γ\Gamma.

    Now A,M∈RA,M\in R. Indeed,

    EA=⟨(1,0,0,1,0,1),(0,1,1,1,1,1)⟩FE_A=\langle(1,0,0,1,0,1),(0,1,1,1,1,1)\rangle_F

    is an F0F_0-rational singular line, A∩EA≠0A\cap E_A\ne0, and direct calculation gives A⊆EA⊥A\subseteq E_A^\perp. Similarly,

    EM=⟨(1,0,1,1,0,1),(0,1,1,1,1,0)⟩FE_M=\langle(1,0,1,1,0,1),(0,1,1,1,1,0)\rangle_F

    is F0F_0-rational singular, M∩EM≠0M\cap E_M\ne0, and M⊆EM⊥M\subseteq E_M^\perp.

    But L∉RL\notin R. The five points of LL are represented by u+tvu+tv, t∈Ft\in F, and vv. For each such point, the unique F0F_0-line containing it is either not singular, or is one of

    ⟨e1,e2⟩,⟨f1,f2⟩,⟨(1,0,0,0,1,0),(0,1,0,1,0,0)⟩,\langle e_1,e_2\rangle,\quad \langle f_1,f_2\rangle,\quad \langle(1,0,0,0,1,0),(0,1,0,1,0,0)\rangle,

    and in each of these three cases LL is not contained in the perp of that line. Hence no F0F_0-rational singular line EE satisfies L⊆E⊥L\subseteq E^\perp and L∩E≠0L\cap E\ne0.

    Therefore RR contains two points A,MA,M of a line of Γ\Gamma but omits another point LL on that same line. Hence RR is not a subspace, while every generated closure ⟨S⟩Γ\langle S\rangle_\Gamma is a subspace. Taking SS to be the full point set of Γ0\Gamma_0, which certainly generates Γ0\Gamma_0, the conjectured equality fails.

    Citation: No external resolution is used. Source of the reconstructed problem: R. J. Blok and A. Pasini, “Point-Line Geometries with a Generating Set that Depends on the Underlying Field,” in Finite Geometries, Developments in Mathematics 3, Kluwer/Springer, 2001, DOI: 10.1007/978-1-4613-0283-4_1.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The counterexample attacks the stated conjectural equality for the rank-3 symplectic line-Grassmannian with F0=F2⊂F=F4F_0=\mathbb F_2\subset F=\mathbb F_4. The coordinate checks are finite and consistent: A,MA,M lie in the proposed right-hand set via the displayed F0F_0-rational singular lines, while LL does not, since the only F0F_0-rational singular lines meeting LL are the three listed and none has LL in its perp. Thus the proposed right-hand set is not a subspace, whereas any generated closure ⟨S⟩Γ\langle S\rangle_\Gamma is a subspace; taking SS to be the full point set of Γ0\Gamma_0 disproves the conjectured equality. I found no prior literature resolution superseding this counterexample.

    Novelty assessment

    TYPE1

    Classification rationale: The accepted resolution is an explicit small finite counterexample to a computationally suggested statement, not a broad structural theorem. It is useful as a correction/remark to the Blok–Pasini problem, but by itself is likely too narrow and coordinate-specific to support a standalone standard-journal paper.

    Literature check: I found no prior publication, note, survey, or repository entry giving this counterexample or a stronger disproof. I checked exact-title and exact-phrase searches, author/publication records, OpenAlex metadata/citation data, arXiv/ORCID records, and related searches for generating sets, polar/line Grassmannians, F1=F0F_1=F_0, and subfield geometries. The relevant later accessible Pasini material concerns general generating ranks/chains, not this proposed equality.

    Citation: No prior-resolution citation found. Original source: R. J. Blok and A. Pasini, “Point-Line Geometries with a Generating Set that Depends on the Underlying Field,” in Finite Geometries, Developments in Mathematics 3, Springer/Kluwer, 2001, pp. 1–25, DOI: 10.1007/978-1-4613-0283-4_1.

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