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pith:MB3OC7QI

pith:2026:MB3OC7QIJ3XV6PVNOYEXVUHPIW

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Categorical Torelli theorems for higher Picard rank Fano double covers

Augustinas Jacovskis, Reinder Meinsma

An equivalence of Kuznetsov components implies an isomorphism for four families of Fano double covers with Picard rank greater than 1.

arxiv:2604.12453 v2 · 2026-04-14 · math.AG

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Claims

C1strongest claim

We prove categorical Torelli theorems for four families of Fano double covers with Picard rank greater than 1. Among these is the family of Verra fourfolds. ... For the three families of threefolds, our proof is based on reducing equivalences between the Kuznetsov components ... to derived equivalences of their respective K3 branch divisors, and deducing that the resulting isomorphism of branch divisors gives rise to an isomorphism of the Fano threefolds for each family. For Verra fourfolds, we show that an equivalence of their Kuznetsov components induces an isomorphism of the branch divisors using the theory of 2-torsion Brauer classes on K3 surfaces.

C2weakest assumption

The reduction from equivalences of Kuznetsov components of the Fano threefolds to derived equivalences of the K3 branch divisors holds within each deformation family, and that an isomorphism of the branch divisors lifts to an isomorphism of the ambient Fano threefolds (or, for Verra fourfolds, that the 2-torsion Brauer class data suffices to recover the isomorphism). This premise is invoked in the abstract's description of the proof strategy for the three families and the Brauer class argument for Verra fourfolds.

C3one line summary

Categorical Torelli theorems are proved for Verra fourfolds and three families of Fano threefolds by reducing Kuznetsov component equivalences to derived equivalences of K3 branch divisors or using 2-torsion Brauer classes.

References

5 extracted · 5 resolved · 0 Pith anchors

[1] Cyclic covers: Hodge theory and categorical Torelli theorems.Int 2025

[2] Lian, Keiji Oguiso, and Shing-Tung Yau 2003

[3] Semiorthogonal decompositions in algebraic geometry 2014

[4] Derived Equivalence for Elliptic K3 Surfaces and Jacobians.International Mathematics Research Notices, 2024(13):10139–10164, 04 2024

[5] [Muk87] S. Mukai. On the moduli space of bundles on K3 surfaces. I. InVector bundles on algebraic varieties (Bombay, 1984), volume 11 ofTata Inst. Fund. Res. Stud. Math., pages 341–413. Tata Inst. Fun 1984

Formal links

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Receipt and verification

First computed	2026-05-20T00:00:37.898027Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

6076e17e084eef5f3ead76097ad0ef45bc402f4dfb8c14bf69c75b9ccf7f2fab

Aliases

arxiv: 2604.12453 · arxiv_version: 2604.12453v2 · doi: 10.48550/arxiv.2604.12453 · pith_short_12: MB3OC7QIJ3XV · pith_short_16: MB3OC7QIJ3XV6PVN · pith_short_8: MB3OC7QI

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/MB3OC7QIJ3XV6PVNOYEXVUHPIW \
  | 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: 6076e17e084eef5f3ead76097ad0ef45bc402f4dfb8c14bf69c75b9ccf7f2fab

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "57c42faecd6a2a1350a10258b6f73144ad20bf01a76d5c86af96ae9b1e4d1a13",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.AG",
    "submitted_at": "2026-04-14T08:40:46Z",
    "title_canon_sha256": "d32ae22b823524423591ae69b538c97ceb6290e471f7e67b6f9f7b97edab7fd2"
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  "schema_version": "1.0",
  "source": {
    "id": "2604.12453",
    "kind": "arxiv",
    "version": 2
  }
}