A Basis-Canonical Projectivization Algorithm for Smooth Complete Toric Varieties

We give an explicit projectivization algorithm for smooth complete toric varieties. More precisely, after fixing an ordered lattice basis, every smooth complete fan $\Sigma$ admits a basis-canonical refinement $\widehat{\Sigma}$ that is smooth, complete, and projective, such that the toric morphism $X_{\widehat{\Sigma}}\to X_\Sigma$ factors as a finite sequence of ordinary blow-ups along smooth torus-invariant centers of codimension two. The existence of projectivizations of this kind is classical; the point of the present note is the elementary and deterministic construction. The algorithm attaches to $\Sigma$ a projective wall-arrangement fan and refines $\Sigma$ to a smooth fan $\widehat{\Sigma}=\Gamma(\Sigma)$ subordinate to this arrangement by a sign-adaptation procedure that uses only codimension-two centers. The resulting fan is automatically projective: it refines a projective fan and dominates $\Sigma$ by a projective morphism, and we exhibit an explicit strictly convex support function as the sum of the pulled-back support function of the arrangement and a small relatively ample perturbation. In particular the construction stops at the wall-adaptation stage: every blow-up center has codimension exactly two. The construction is uniform in all dimensions.