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PreprintCC BY 4.0arXiv

Nontransitivity of braid group actions on exceptional bases in K0(ℙn) for n=4,6

  • Roman Eliseev
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Abstract

Let X=ℙn, where p=n+1 is prime. Consider the orbit of the standard numerical exceptional basis in K0(X) under mutations, sign changes, and isometries of the Euler form. We construct an invariant of this orbit: for every basis from the orbit, the square of rank of all basic vectors art congruent to $1$ modulo p. For ℙ4 and ℙ6, we present explicit numerical exceptional bases violating this congruence; hence the action is not transitive. We also establish a stronger obstruction modulo p2 and use Polishchuk's theorem to show that no vector of the constructed bases is the class of an exceptional object. Finally, we prove that for every prime p, the category Db(Cohℙp−1ℂ) does not contain a pair of mutually orthogonal exceptional objects.

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Cite

BibTeXRIS
BibTeX
@misc{eliseev2026nontransitivity,
  title = {{Nontransitivity of braid group actions on exceptional bases in $K_0(\mathbbP^n)$ for $n=4,6$}},
  author = {Roman Eliseev},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31413},
  eprint = {2609.31413},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31413},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Nontransitivity of braid group actions on exceptional bases in K_0(ℙ^n) for n=4,6
AU  - Roman Eliseev
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31413
UR  - https://arxiv.org/abs/2609.31413
AB  - Let X=ℙ^n, where p=n+1 is prime. Consider the orbit of the standard numerical exceptional basis in K_0(X) under mutations, sign changes, and isometries of the Euler form. We construct an invariant of this orbit: for every basis from the orbit, the square of rank of all basic vectors art congruent to $1$ modulo p. For ℙ^4 and ℙ^6, we present explicit numerical exceptional bases violating this congruence; hence the action is not transitive. We also establish a stronger obstruction modulo p^2 and use Polishchuk's theorem to show that no vector of the constructed bases is the class of an exceptional object. Finally, we prove that for every prime p, the category D^b(Cohℙ^p−1_ℂ) does not contain a pair of mutually orthogonal exceptional objects.
ER  -