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Non-commutative Fano varieties and three-dimensional minifolds

  • Konstantin Loginov
  • Dmitri Orlov
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Abstract

We introduce notions of non-commutative Fano varieties and non-commutative minifolds, and study the latter in dimension three. We prove that the Euler matrices of full exceptional collections generating geometric helices on these minifolds form five mutation classes, distinguished by their Euler degrees 22,40,54,64,72. The first four classes are realized by (commutative) Fano threefolds X22,V5,Q3,ℙ3, respectively. The fifth is realized by the non-commutative family M72 and cannot be realized by a full exceptional collection on a smooth projective threefold.

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Cite

BibTeXRIS
BibTeX
@misc{loginov2026noncommutative,
  title = {{Non-commutative Fano varieties and three-dimensional minifolds}},
  author = {Konstantin Loginov and Dmitri Orlov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39766},
  eprint = {2609.39766},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39766},
}
RIS
TY  - UNPB
TI  - Non-commutative Fano varieties and three-dimensional minifolds
AU  - Konstantin Loginov
AU  - Dmitri Orlov
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39766
UR  - https://arxiv.org/abs/2609.39766
AB  - We introduce notions of non-commutative Fano varieties and non-commutative minifolds, and study the latter in dimension three. We prove that the Euler matrices of full exceptional collections generating geometric helices on these minifolds form five mutation classes, distinguished by their Euler degrees 22,40,54,64,72. The first four classes are realized by (commutative) Fano threefolds X_22,V_5,Q^3,ℙ^3, respectively. The fifth is realized by the non-commutative family M_72 and cannot be realized by a full exceptional collection on a smooth projective threefold.
ER  -