Question & scope
For Δᵣ=sdʳ(Δ), where Δ is the six-vertex projective plane, how do characteristic-two Betti differences behave in the square and cube of IΔᵣ?
Let Δ₀=Δ and Δᵣ₊₁=sd(Δᵣ). Set Jᵣ=IΔᵣ in one variable for each vertex. Study βᵢ(Jᵣʰ), the resolution of the ideal itself, not of its quotient. The first subdivision has 31 vertices, so complete resolutions require a feasibility check.
First calculation: separate subdivision from augmentation
Construct the exact facets and minimal nonfaces of sd(Δ), then generate J₁² and J₁³ without duplicates. Use polarization or the lcm lattice to isolate candidate multidegrees before requesting a complete resolution. Keep the r=0 calculation as a literature baseline.
Known boundaries
Bolognini et al. (2022), Proposition 3.1: for fixed i and prime p, characteristic dependence is eventually constant as a yes/no condition for high powers. Their §3.1 reports no dependence for the original six-vertex example at h=2,…,10, a finite computation. Theorem 4.3 propagates dependence after adding a monomial on new disjoint variables. Corollary 4.8 also uses subdivisions in a construction followed by such an addition. Those results must be separated from the unaugmented Jᵣ considered here. For r≥1, barycentric subdivision makes Δᵣ flag. Javadekar (2026 preprint), Corollary 1.2, adds field independence of β₇(S/Jᵣ) to the previously known indices 1,…,6. Equivalently, β₀,…,β₆(Jᵣ) are field independent at h=1. This statement does not cover Jᵣ² or Jᵣ³.
Powers need not be squarefree. Hochster’s formula for the original complex cannot simply be reused for every power. The proposed family’s novelty remains to be checked.
One boundary matrix, two coefficient fields
The characteristic-two phenomenon is classical. The saved boundary-matrix and Smith-form calculations are assistant-prepared groundwork, not an independently completed research result. In the displayed difference sequence, β refers to the resolution of the ideal Iʰ itself.
Supporting study notes
Classical background and worked examples for this proposal.
Work to undertake
- Construct J₀ and J₁ from a fixed facet list and record their generators.
- Compare h=2,3 in selected certified multidegrees over Q and F₂, checking resource feasibility.
- Compare the resulting mechanism with Proposition 3.1, Theorem 4.3 and Corollary 4.8 before formulating any propagation claim.
- Integral homology and Smith normal form
- Hochster’s formula
- Multigraded resolutions
- Polarization and lcm lattices