Poisson blow-ups and the adjoint quotient
Abstract
We leverage Polishchuk's Poisson blow-up criterion in the context of algebro-geometric integrable systems. In more detail, one may associate an integrable system τ:X⟶B to each affine Poisson scheme X over ℂ. We prove that the blow-ups of X along fibers of τ are Poisson schemes occurring in a family X×B̃⟶B, where X×B̃ is itself a Poisson scheme. This result is subsequently specialized to the adjoint quotient τ:g⟶g/ /G=:c of a finite-dimensional complex semisimple Lie algebra g with integrating algebraic group G. We show that the family g×c̃⟶c is flat, conical, and equipped with a canonical Poisson Hamiltonian G-variety structure. We also obtain Poisson-geometric results on the fibers of this family, which are blow-ups of g along regular adjoint orbit closures.
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BibTeX
@misc{crooks2026poisson,
title = {{Poisson blow-ups and the adjoint quotient}},
author = {Peter Crooks and Iva Halacheva},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30185},
eprint = {2608.30185},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30185},
}
RIS
TY - UNPB
TI - Poisson blow-ups and the adjoint quotient
AU - Peter Crooks
AU - Iva Halacheva
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30185
UR - https://arxiv.org/abs/2608.30185
AB - We leverage Polishchuk's Poisson blow-up criterion in the context of algebro-geometric integrable systems. In more detail, one may associate an integrable system τ:X⟶B to each affine Poisson scheme X over ℂ. We prove that the blow-ups of X along fibers of τ are Poisson schemes occurring in a family X×B̃⟶B, where X×B̃ is itself a Poisson scheme. This result is subsequently specialized to the adjoint quotient τ:g⟶g/ /G=:c of a finite-dimensional complex semisimple Lie algebra g with integrating algebraic group G. We show that the family g×c̃⟶c is flat, conical, and equipped with a canonical Poisson Hamiltonian G-variety structure. We also obtain Poisson-geometric results on the fibers of this family, which are blow-ups of g along regular adjoint orbit closures.
ER -