Pith Number

pith:SDBNX4VM

pith:2026:SDBNX4VMNCDRCF5BVIWN7UAEWH

not attested not anchored not stored refs pending

On the Jacobian algebras of Ziegler pairs of plane arrangements

Alexandru Dimca, Piotr Pokora

Ziegler pairs of plane arrangements in projective 3-space can have isomorphic intersection lattices yet different Betti numbers for the minimal resolutions of their Jacobian algebras.

arxiv:2604.25637 v2 · 2026-04-28 · math.AG

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\pithnumber{SDBNX4VMNCDRCF5BVIWN7UAEWH}

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We consider a Ziegler pair of plane arrangements A:f=0 and A':f'=0 in P^3 such that L(A) ≅ L(A') but the Betti numbers of the minimal resolutions of their Jacobian algebras are not the same.

C2weakest assumption

That the observed difference in Betti numbers is intrinsic and not an artifact of the choice of defining equations or of some undetected isomorphism between the algebras.

C3one line summary

Ziegler pairs of plane arrangements in P^3 have isomorphic intersection lattices but different Betti numbers for Jacobian algebra resolutions and relate to cones over Ziegler pairs of line arrangements in P^2.

Cited by

1 paper in Pith

On Ziegler pairs of line arrangements: from non-existence to abundance

Receipt and verification

First computed	2026-06-24T01:15:03.088304Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

90c2dbf2ac68871117a1aa2cdfd004b1e0b74a905f85ed78db1320130e7af56f

Aliases

arxiv: 2604.25637 · arxiv_version: 2604.25637v2 · doi: 10.48550/arxiv.2604.25637 · pith_short_12: SDBNX4VMNCDR · pith_short_16: SDBNX4VMNCDRCF5B · pith_short_8: SDBNX4VM

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/SDBNX4VMNCDRCF5BVIWN7UAEWH \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 90c2dbf2ac68871117a1aa2cdfd004b1e0b74a905f85ed78db1320130e7af56f

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "8f1f0aa45ba9151c2eeec9c066fba572bd5184148dba99648fbf92dee4eafecc",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AG",
    "submitted_at": "2026-04-28T13:37:24Z",
    "title_canon_sha256": "607cdeb96860eda505894f457d541e7bd2b0f7fe7903dccdbb53173f08b691f6"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.25637",
    "kind": "arxiv",
    "version": 2
  }
}