Basis-Canonical Projectivization for Smooth Complete Toric Varieties

Abstract

We give an explicit projectivization algorithm for smooth complete toric varieties in arbitrary dimension $n\ge 2$. After fixing an ordered lattice basis, every smooth complete fan~$\Sig$ admits a basis-canonical refinement~$\wSig=\Gam(\Sig)$ that is smooth, complete, projective, and obtained from~$\Sig$ by star subdivisions of two-dimensional cones. Equivalently, $X_{\wSig}\to X_\Sig$ is a finite sequence of ordinary toric blow-ups along smooth invariant centers of codimension two. The algorithm first constructs a projective wall-arrangement fan by extending the spans of the codimension-one cones of~$\Sig$ to central hyperplanes. It then sign-adapts~$\Sig$ to this arrangement by repeatedly subdividing bad two-cones of maximal weight. A lexicographic badness profile gives termination, while projectivity follows from a wall-bend sandwich argument combining a support function pulled back from the arrangement with a relatively ample perturbation. The construction is canonical relative to the chosen ordered basis and requires no additional projectivizing refinement after wall-adaptation. We illustrate the procedure on Oda's non-projective threefold and compare its deterministic length with a separate threefold whose minimal ordinary invariant projectivization length is exactly two.