Sharp Bounds and Canonical Blow-ups for Toric Projectivization
Abstract
For every dimension n≥3, we construct a family Σn(a,b) of smooth complete fans with Picard number five and fixed combinatorial type. For integers a,b≥1, the minimum number of new rays in a smooth projective refinement is min(a,b), attained by ordinary invariant codimension-two blow-ups. If projective simplicial refinements with maximal-cone multiplicity at most an integer μ≥1 are allowed, the minimum is ⌈min(a,b)/μ⌉. The threefold lower bound depends only on subdivisions of four two-dimensional cones. Toric wedges preserve this minimum while raising the dimension, and a selective support-function criterion gives the matching upper bound. We also construct a projectivization algorithm natural under lattice isomorphisms. For smooth input, it principalizes an intrinsic monomial ideal by ordinary blow-ups with smooth, possibly disconnected centers. Running a fixed functorial principalization algorithm over ℚ gives a sequence of fans valid over every field. An intrinsic resolution procedure treats arbitrary complete rational fans. Finally, Oda's threefold shows why symmetry can prevent a canonical ordering of the components of a center, even when their simultaneous blow-up is canonical.
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BibTeX
@misc{bakhtary2026sharp,
title = {{Sharp Bounds and Canonical Blow-ups for Toric Projectivization}},
author = {Parsa Bakhtary},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.25013},
eprint = {2605.25013},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.25013},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp Bounds and Canonical Blow-ups for Toric Projectivization
AU - Parsa Bakhtary
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2605.25013
UR - https://arxiv.org/abs/2605.25013
AB - For every dimension n≥3, we construct a family Σ_n(a,b) of smooth complete fans with Picard number five and fixed combinatorial type. For integers a,b≥1, the minimum number of new rays in a smooth projective refinement is min(a,b), attained by ordinary invariant codimension-two blow-ups. If projective simplicial refinements with maximal-cone multiplicity at most an integer μ≥1 are allowed, the minimum is ⌈min(a,b)/μ⌉. The threefold lower bound depends only on subdivisions of four two-dimensional cones. Toric wedges preserve this minimum while raising the dimension, and a selective support-function criterion gives the matching upper bound. We also construct a projectivization algorithm natural under lattice isomorphisms. For smooth input, it principalizes an intrinsic monomial ideal by ordinary blow-ups with smooth, possibly disconnected centers. Running a fixed functorial principalization algorithm over ℚ gives a sequence of fans valid over every field. An intrinsic resolution procedure treats arbitrary complete rational fans. Finally, Oda's threefold shows why symmetry can prevent a canonical ordering of the components of a center, even when their simultaneous blow-up is canonical.
ER -