PreprintCC BY 4.0HAL
The elusive relatively K-unstable Delzant octagon: a numerical search
Abstract
Relative K-stability of toric varieties can be tested through explicit Donaldson-Futaki computations, yet finding unstable examples remains challenging. We develop a reproducible symbolic-numeric framework to investigate all fifteen families of Delzant octagons corresponding to smooth toric surfaces of Picard rank six. While numerical optimization identifies several apparently destabilizing configurations, exact rational reconstruction systematically removes the observed instability. We explain why Wang and Zhou's 2014 tentative construction of a relatively K-unstable octagon does not allow to find an explicit example.
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BibTeX
@misc{delcroix2026elusive,
title = {{The elusive relatively K-unstable Delzant octagon: a numerical search}},
author = {Thibaut Delcroix and Bijan Mohammadi},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10019},
eprint = {2609.10019},
archivePrefix = {arXiv},
url = {https://hal.umontpellier.fr/hal-05743531v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The elusive relatively K-unstable Delzant octagon: a numerical search
AU - Thibaut Delcroix
AU - Bijan Mohammadi
PY - 2026
DA - 2026/09/09
PB - HAL
DO - 10.48550/arxiv.2609.10019
UR - https://hal.umontpellier.fr/hal-05743531v1
AB - Relative K-stability of toric varieties can be tested through explicit Donaldson-Futaki computations, yet finding unstable examples remains challenging. We develop a reproducible symbolic-numeric framework to investigate all fifteen families of Delzant octagons corresponding to smooth toric surfaces of Picard rank six. While numerical optimization identifies several apparently destabilizing configurations, exact rational reconstruction systematically removes the observed instability. We explain why Wang and Zhou's 2014 tentative construction of a relatively K-unstable octagon does not allow to find an explicit example.
ER -