Demazure Models of Algebraic Tori in Dimension 4
Abstract
A Demazure model of an algebraic k-torus T split by a finite Galois extension of fields L/k and splitting group G=Gal(L/k) is a smooth projective TL-toric variety XΣ such that the natural action of G on TL extends to XΣ. A smooth projective model of the algebraic k-torus T can then be taken to be the quotient of a Demazure model by its splitting group. The existence of Demazure models of algebraic k-tori was determined by Brylinski and further refined by Colliot-Thélène, Harari and Skorobogatov. Voskresenskii and Kunyavskii's birational classifications of algebraic k-tori in dimensions 2 and 3 involved constructions of Demazure models of the algebraic k-tori corresponding to maximal finite subgroups of GL(r,ℤ) for r=2,3. We discuss some explicit constructions of Demazure models of algebraic k-tori, making connections with the defining integral representations of their splitting groups. The constructions determine smooth projective Demazure models of the algebraic tori corresponding to maximal finite subgroups of GL(4,ℤ). Since projective toric varieties are determined by lattice polytopes, our constructions focus on some highly symmetric families of polytopes, such as root polytopes and central transportation polytopes.
Related
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BibTeX
@misc{lemire2026demazure,
title = {{Demazure Models of Algebraic Tori in Dimension 4}},
author = {Nicole Lemire},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30482},
eprint = {2609.30482},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30482},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Demazure Models of Algebraic Tori in Dimension 4
AU - Nicole Lemire
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30482
UR - https://arxiv.org/abs/2609.30482
AB - A Demazure model of an algebraic k-torus T split by a finite Galois extension of fields L/k and splitting group G=Gal(L/k) is a smooth projective T_L-toric variety X_Σ such that the natural action of G on T_L extends to X_Σ. A smooth projective model of the algebraic k-torus T can then be taken to be the quotient of a Demazure model by its splitting group. The existence of Demazure models of algebraic k-tori was determined by Brylinski and further refined by Colliot-Thélène, Harari and Skorobogatov. Voskresenskii and Kunyavskii's birational classifications of algebraic k-tori in dimensions 2 and 3 involved constructions of Demazure models of the algebraic k-tori corresponding to maximal finite subgroups of GL(r,ℤ) for r=2,3. We discuss some explicit constructions of Demazure models of algebraic k-tori, making connections with the defining integral representations of their splitting groups. The constructions determine smooth projective Demazure models of the algebraic tori corresponding to maximal finite subgroups of GL(4,ℤ). Since projective toric varieties are determined by lattice polytopes, our constructions focus on some highly symmetric families of polytopes, such as root polytopes and central transportation polytopes.
ER -