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PreprintCC BY 4.0arXiv

The E6 normal forms: semisimple deformations, geometry, DT invariants, and the E6,∞ order

  • Shengyong Pan
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Abstract

We study the completed Jacobi algebras of x3+xy3+p(y) for ordyp=n≥4. The explicit normal-form morsification, the filtered length and centre proofs, and the cyclic terminal identities give finite free semisimple deformations. For every parameter of finite order we construct a polynomial curve neighbourhood and a local projective small contraction; the resulting geometric GV and Donaldson--Thomas calculations identify the block multiplicities. The one-dimensional limit E6,∞ is studied separately as an order: we determine its normalization, conductor, radical-idealizer chain, and maximal overorders. The canonical chain yields a nine-dimensional finite quotient whose Ringel dual is isomorphic to its opposite. An expanded construction gives a choice-independent $22$-dimensional quotient and a $40$-dimensional Ringel dual; a corner quotient relates the two finite constructions. The length and centre formulas are proved by a filtered comparison with a homogeneous matrix order and by terminal identities at the exceptional endpoint.

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Cite

BibTeXRIS
BibTeX
@misc{pan2026normal,
  title = {{The $E_6$ normal forms: semisimple deformations, geometry, DT invariants, and the $E_6,\infty$ order}},
  author = {Shengyong Pan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13745},
  eprint = {2610.13745},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13745},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The E_6 normal forms: semisimple deformations, geometry, DT invariants, and the E_6,∞ order
AU  - Shengyong Pan
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13745
UR  - https://arxiv.org/abs/2610.13745
AB  - We study the completed Jacobi algebras of x^3+xy^3+p(y) for ord_yp=n≥4. The explicit normal-form morsification, the filtered length and centre proofs, and the cyclic terminal identities give finite free semisimple deformations. For every parameter of finite order we construct a polynomial curve neighbourhood and a local projective small contraction; the resulting geometric GV and Donaldson--Thomas calculations identify the block multiplicities. The one-dimensional limit E_6,∞ is studied separately as an order: we determine its normalization, conductor, radical-idealizer chain, and maximal overorders. The canonical chain yields a nine-dimensional finite quotient whose Ringel dual is isomorphic to its opposite. An expanded construction gives a choice-independent $22$-dimensional quotient and a $40$-dimensional Ringel dual; a corner quotient relates the two finite constructions. The length and centre formulas are proved by a filtered comparison with a homogeneous matrix order and by terminal identities at the exceptional endpoint.
ER  -