Absolutely universal projective embeddings of long-root geometries of orthogonal type
Abstract
Let G be a split Chevalley group of type Bn (n ≥ 3) or Dn (n ≥ 4), let L be its Lie algebra and Γ its long-root geometry, namely Γ is the line-grassmannian of the polar space associated to G. It is well known that L hosts a projective embedding of Γ, called the adjoint embedding, which realizes Γ as a projective variety in the projective geometry of the underlying vector space of L. It is also known that Γ admits the absolutely universal projective embedding, which is such that all projective embeddings of Γ can be obtained from it as projections. When the field F over which G is defined is finite, the universal embedding of Γ is just the same as the adjoint embedding. The same is not necessarily true when F is infinite. Indeed, as we shall prove in this paper, the universal projective embedding of Γ lives in a direct sum K⊕ L for an F-vector space K, the dual of which is isomorphic to the derivation space Der(F) of F, which in general is non-trivial. Accordingly, the adjoint embedding of Γ is universal if and only if Der(F) is trivial, namely F is either perfect of positive characteristic or algebraic over its minimum subfield.
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BibTeX
@misc{pasini2026absolutely,
title = {{Absolutely universal projective embeddings of long-root geometries of orthogonal type}},
author = {Antonio Pasini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33384},
eprint = {2610.33384},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33384},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Absolutely universal projective embeddings of long-root geometries of orthogonal type
AU - Antonio Pasini
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33384
UR - https://arxiv.org/abs/2610.33384
AB - Let G be a split Chevalley group of type B_n (n ≥ 3) or D_n (n ≥ 4), let L be its Lie algebra and Γ its long-root geometry, namely Γ is the line-grassmannian of the polar space associated to G. It is well known that L hosts a projective embedding of Γ, called the adjoint embedding, which realizes Γ as a projective variety in the projective geometry of the underlying vector space of L. It is also known that Γ admits the absolutely universal projective embedding, which is such that all projective embeddings of Γ can be obtained from it as projections. When the field F over which G is defined is finite, the universal embedding of Γ is just the same as the adjoint embedding. The same is not necessarily true when F is infinite. Indeed, as we shall prove in this paper, the universal projective embedding of Γ lives in a direct sum K⊕ L for an F-vector space K, the dual of which is isomorphic to the derivation space Der(F) of F, which in general is non-trivial. Accordingly, the adjoint embedding of Γ is universal if and only if Der(F) is trivial, namely F is either perfect of positive characteristic or algebraic over its minimum subfield.
ER -