Minkowski decompositions and universal equivariant deformations of toric pairs
Abstract
Let Xσ be the affine normal toric variety associated with a full-dimensional strongly convex rational polyhedral cone σ, and let m be a primitive degree with slice Pm=σ∩[m=1]. We construct an explicit flat algebraic family whose completion is universal for equivariant deformations in all degrees −jm, j≥1, simultaneously. We prove that the irreducible components of the reduced base correspond bijectively to maximal lattice-friendly Minkowski decompositions of Pm, and describe the induced family on each component. When m∈σ∨, we also obtain a formally universal equivariant family for the pair (Xσ,V(χm)). These results extend earlier miniversality and component theorems to arbitrary affine normal toric varieties and arbitrary primitive degrees.
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BibTeX
@misc{filip2026minkowski,
title = {{Minkowski decompositions and universal equivariant deformations of toric pairs}},
author = {Matej Filip},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10230},
eprint = {2610.10230},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10230},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Minkowski decompositions and universal equivariant deformations of toric pairs
AU - Matej Filip
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10230
UR - https://arxiv.org/abs/2610.10230
AB - Let X_σ be the affine normal toric variety associated with a full-dimensional strongly convex rational polyhedral cone σ, and let m be a primitive degree with slice P_m=σ∩[m=1]. We construct an explicit flat algebraic family whose completion is universal for equivariant deformations in all degrees −jm, j≥1, simultaneously. We prove that the irreducible components of the reduced base correspond bijectively to maximal lattice-friendly Minkowski decompositions of P_m, and describe the induced family on each component. When m∈σ^∨, we also obtain a formally universal equivariant family for the pair (X_σ,V(χ^m)). These results extend earlier miniversality and component theorems to arbitrary affine normal toric varieties and arbitrary primitive degrees.
ER -