Sharp Bounds and Canonical Blow-ups for Toric Projectivization

Abstract

For every dimension $n\ge 3$, we construct a family ${\Sigma }_n(a,b)$ of smooth complete fans with Picard number five and fixed combinatorial type. For integers $a,b\ge 1$, 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 $\mu \ge 1$ are allowed, the minimum is $\lceil \min (a,b)/\mu \rceil$. 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 $\mathbb{Q}$ 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.

arXiv comment22 pages (19-page article and 3-page supplement), 3 figures; ancillary verification code. v4: substantially expanded and retitled; sharp projectivization bounds in every dimension, multiplicity tradeoff, and canonical projectivization via functorial principalization. Original basis-dependent codimension-two construction retained