1 Introduction↩︎

A fundamental question in toric geometry asks: given a smooth complete toric variety \(X_\Sigma\), can one find a smooth projective toric variety dominating it via a proper birational toric morphism? In dimension two this is already settled, since every smooth complete toric surface is projective; the first genuinely nontrivial case is dimension three.

The bare existence of such a dominating projective toric variety is not new. One route is the toric Chow lemma, which gives a projective refinement of a complete fan, followed by toric resolution of singularities; this proves existence but may pass through singular intermediate fans. More directly for smooth complete toric varieties, Bonavero recalls, citing Morelli, that every such toric variety becomes projective after a finite sequence of blow-ups along smooth toric subvarieties [1], [2]. The same projectivization principle appears in the toric Moishezon step in the exposition of Abramovich–Matsuki–Rashid after Morelli: after star subdivisions one obtains a fan which is part of a projective fan [3]. Thus the contribution of this note is not the existential statement by itself, but a direct, basis-canonical, and elementary algorithm for producing one such projectivization.

The construction is designed to keep the geometry transparent. First we associate to \(\Sigma\) a projective wall-arrangement fan \(\mathcal{A}_\Sigma\) by extending every codimension-one cone of \(\Sigma\) to a central hyperplane. Second, we refine \(\Sigma\) to a smooth fan \(\Gamma(\Sigma)\) subordinate to \(\mathcal{A}_\Sigma\) by resolving exactly the two-dimensional cones that cross one of these walls. This wall-adaptation stage uses only codimension-two centers. The output \(\Gamma(\Sigma)\) is then automatically projective: it refines the projective fan \(\mathcal{A}_\Sigma\) and at the same time dominates \(\Sigma\) by a projective toric morphism, and these two facts combine to produce an explicit strictly convex support function on \(\Gamma(\Sigma)\). No additional refinement step is needed to force projectivity, and the morphism to \(X_\Sigma\) is a sequence of ordinary toric blow-ups along smooth invariant centers of codimension exactly two.

Basis-canonical convention. Throughout the construction an ordered lattice basis \(B\) of \(N\) is fixed. It orients each wall normal by the first nonzero coordinate, orders the wall normals, and breaks ties among equally weighted bad two-cones. A construction below is called \(B\)-canonical, or basis-canonical, if it is determined by \(\Sigma\) and this ordered basis. The wall-arrangement fan itself is intrinsic to \(\Sigma\); the basis only chooses a deterministic order for the subsequent operations.

The construction is \[\label{eq:construction} \Sigma\;\xrightarrow{\mathrm{sign-adapt}}\; \widehat{\Sigma}= \Gamma(\Sigma),\tag{1}\] where \(\Gamma(\Sigma)\) is obtained from \(\Sigma\) by a finite deterministic sequence of star subdivisions along two-dimensional cones (the wall-adaptation stage). The auxiliary wall-arrangement fan \(\mathcal{A}_\Sigma\) does not appear in the output; it is used only to organize the wall-adaptation and to certify projectivity of \(\Gamma(\Sigma)\) in Section 6.

Relationship to prior work. The theorem should be read as a constructive refinement of the classical toric Moishezon–Morelli projectivization statement, not as a first existence theorem. Compared with the Chow-then-desingularize approach, it never leaves the smooth category and it gives a prescribed sequence of centers. Compared with the Morelli–Abramovich–Matsuki–Rashid factorization machinery and the related weak-factorization literature [3]–[5], it avoids cobordisms and \(\pi\)-desingularization: the only operation is the Euclidean sign-adaptation move on a bad two-cone, and projectivity of the output is read off directly from the wall arrangement. The factorization framework is designed for the more general problem of factoring birational maps, whereas the present note solves the easier problem of constructing one projective smooth dominator. The tradeoff is that the output is generally far from minimal.

This is also different from the projectivization-by-small-modification framework of Fujino–Sato and Rossi. Fujino–Sato prove that every complete toric variety has a projective \(\mathbb{Q}\)-factorial toric model isomorphic in codimension one; in the smooth threefold case with Picard number at most five, their projectivizing sequences consist of flops or anti-flips, within a broader flips/flops/anti-flips framework [6]. Rossi develops the related Cox-ring and GKZ picture [7]. Those results change the variety in codimension one by small birational maps; here the final variety dominates the original by blow-ups. Adiprasito–Pak prove a much stronger recent common-stellar-subdivision theorem, including a weighted form of Oda’s strong factorization conjecture and a PL common-subdivision theorem [8]. Their theorem is broader and deeper; the present note is more specialized and records a concrete unweighted projectivization algorithm for smooth complete fans.

Organization. Section 2 fixes conventions. Section 3 constructs \(\mathcal{A}_\Sigma\). Section 4 proves the sign-adaptation lemma. Section 5 handles sequential adaptation. Section 6 proves that \(\Gamma(\Sigma)\) is projective. Section 7 assembles the proof. Section 8 gives a computed threefold run. Sections 9–11 discuss specializations, length estimates, and further remarks.

2 Conventions↩︎

Let \(N \cong \mathbb{Z}^n\) be a lattice with dual lattice \(M = \mathop{\mathrm{Hom}}(N, \mathbb{Z})\). Set \(N_\mathbb{R}= N \otimes_\mathbb{Z}\mathbb{R}\) and \(M_\mathbb{R}= M \otimes_\mathbb{Z}\mathbb{R}\). A cone is always a strongly convex rational polyhedral cone in \(N_\mathbb{R}\). A fan is a collection of cones closed under taking faces and with pairwise intersections being faces.

For primitive lattice vectors \(u_1, \ldots, u_k \in N\), write \(\langle u_1, \ldots, u_k \rangle\) for the cone \(\mathbb{R}_{\ge 0} u_1 + \cdots + \mathbb{R}_{\ge 0} u_k\). A cone is simplicial if its primitive generators are linearly independent, and smooth (or unimodular) if they extend to a lattice basis of \(N\). A fan is smooth if every cone is smooth.

A fan \(\Gamma\) refines a fan \(\Sigma\), written \(\Gamma\preceq \Sigma\), if every cone of \(\Gamma\) is contained in a cone of \(\Sigma\). A complete fan is projective if it admits a strictly upper convex piecewise-linear support function.

For a smooth cone \(\sigma = \langle u_1, \ldots, u_k \rangle\) in a fan \(\Sigma\), the star subdivision of \(\Sigma\) at the ray \(\mathbb{R}_{\ge 0}(u_1 + \cdots + u_k)\) replaces every cone of \(\Sigma\) containing \(\sigma\) by the cones obtained by removing one generator of \(\sigma\) at a time and inserting \(u_1 + \cdots + u_k\). Geometrically, this is the toric blow-up of the orbit closure \(V(\sigma)\); see [9].

3 The wall-arrangement fan↩︎

This is the toric analogue of Danilov’s approach to the toric Chow lemma [10]: extending each codimension-one cone of a complete fan to a full hyperplane produces a projective subdivision.

4 The sign-adaptation algorithm↩︎

5 Sequential adaptation and monotonicity↩︎

6 Projectivity of \(\Gamma(\Sigma)\)↩︎

The fan \(\Gamma(\Sigma)\) is smooth, complete, and refines the projective arrangement fan \(\mathcal{A}_\Sigma\). We show that it is itself projective, so that the construction terminates at \(\Gamma(\Sigma)\) with no further work. The mechanism is the standard one by which an ample class on a base and a relatively ample class for a morphism combine to an ample class on the total space; here both ingredients are visible at the level of support functions, and the linearity of the wall-bend functional makes the combination completely explicit.

The argument is a small general “sandwich” lemma. Throughout this section, until the final application, let \(\Sigma\) be a complete fan, let \(\mathcal{A}\) be a complete projective fan with \(\mathcal{A}\preceq\Sigma\), and let \(\Gamma\) be a smooth complete fan with \(\Gamma\preceq\mathcal{A}\) such that the refinement morphism \(X_\Gamma\to X_\Sigma\) is projective. In the application, \(\mathcal{A}=\mathcal{A}_\Sigma\) and \(\Gamma=\Gamma(\Sigma)\).

Wall bends↩︎

Let \(\Phi\) be a smooth complete fan in \(N_\mathbb{R}\) and let \(h\colon N_\mathbb{R}\to\mathbb{R}\) be a \(\Phi\)-linear support function, that is, a function whose restriction to each maximal cone is linear. Let \(\tau\in\Phi(n-1)\) be a wall, shared by the two maximal cones \(\sigma_+=\langle\tau,a\rangle\) and \(\sigma_-=\langle\tau,b\rangle\), where \(a\) and \(b\) are the generators of \(\sigma_+\) and \(\sigma_-\) not lying on \(\tau\), and \(\tau=\langle r_1,\dots,r_{n-1}\rangle\). Because \(\Phi\) is smooth, the \(n+1\) primitive generators \(a,b,r_1,\dots,r_{n-1}\) satisfy a unique integral wall relation \[\label{eq:wallrel} a+b=\sum_{i=1}^{n-1} c_i\,r_i,\qquad c_i\in\mathbb{Z},\tag{2}\] in which the coefficients of \(a\) and \(b\) are both \(1\); this is the smooth case of the wall relation [9]. Define the bend of \(h\) across \(\tau\) by \[\label{eq:bend} \delta_\tau(h):=\sum_{i=1}^{n-1} c_i\,h(r_i)-h(a)-h(b).\tag{3}\] By construction \(\delta_\tau\) is a linear functional of the values of \(h\) on the rays of \(\Phi\). Writing \(m_+,m_-\in M_\mathbb{R}\) for the linear forms with \(h|_{\sigma_\pm}=m_\pm\), and using 2 together with \(m_+|_\tau=m_-|_\tau\), one obtains the equivalent expression \[\label{eq:bend2} \delta_\tau(h)=(m_+-m_-)(b),\tag{4}\] which is well defined because \(m_+-m_-\) vanishes on \(\mathop{\mathrm{span}}(\tau)\). We fix the orientation of 3 for which a strictly upper convex support function has all bends positive; with this convention, \(h\) is a strictly upper convex support function for \(\Phi\), equivalently the support function of an ample divisor on the projective variety \(X_\Phi\), if and only if \(\delta_\tau(h)>0\) for every wall \(\tau\in\Phi(n-1)\) [9]. The same functional governs the relative notion: a \(\Gamma\)-linear support function \(g\) is strictly convex relative to \(\Sigma\) if \(\delta_\tau(g)>0\) for every wall \(\tau\in\Gamma(n-1)\) whose relative interior lies in the interior of a maximal cone of \(\Sigma\), and a refinement morphism \(X_\Gamma\to X_\Sigma\) is projective if and only if such a \(g\) exists [9].

Two families of walls↩︎

Since \(\Gamma\preceq\mathcal{A}\), the relative interior of each wall \(\tau\in\Gamma(n-1)\) is contained in the relative interior of a unique cone \(\delta\) of \(\mathcal{A}\), and \(\dim\delta\ge\dim\tau=n-1\), so \(\dim\delta\in\{n-1,n\}\). Call \(\tau\) inherited if \(\dim\delta=n-1\), in which case \(\tau\) lies in a wall of \(\mathcal{A}\), and interior if \(\dim\delta=n\), in which case the relative interior of \(\tau\) lies in the interior of a maximal cone of \(\mathcal{A}\). Every wall of \(\Gamma\) is of exactly one of these two types.

7 Proof of the main theorem↩︎

8 A computed threefold run↩︎

We illustrate the wall-adaptation stage on a genuinely non-projective smooth complete toric threefold. Let \(X_\Sigma\) be Oda’s example, the smooth complete threefold of Picard number \(4\) occurring as case [7-5] of Fujino–Sato [6] and in Oda’s book [12], with ray generators \[\begin{align} v_1 &= (1,0,0), & v_2 &= (0,1,0), & v_3 &= (0,0,1), & v_4 &= (-1,-1,-1),\\ v_5 &= (-1,-1,0), & v_6 &= (0,-1,-1), & v_7 &= (-1,0,-1), \end{align}\] and ten maximal cones \[\begin{gather} \langle v_1,v_2,v_3\rangle,\; \langle v_1,v_2,v_7\rangle,\; \langle v_1,v_3,v_6\rangle,\; \langle v_1,v_6,v_7\rangle,\; \langle v_2,v_3,v_5\rangle,\\ \langle v_2,v_5,v_7\rangle,\; \langle v_3,v_5,v_6\rangle,\; \langle v_4,v_5,v_6\rangle,\; \langle v_4,v_5,v_7\rangle,\; \langle v_4,v_6,v_7\rangle. \end{gather}\] This fan is smooth and complete, with \(f\)-vector \((f_1,f_2,f_3)=(7,15,10)\).

Non-projectivity. The fan \(\Sigma\) admits the following elementary obstruction to projectivity. Consider the three walls \(\langle v_1,v_7\rangle\), \(\langle v_2,v_5\rangle\), \(\langle v_3,v_6\rangle\), whose wall relations in the sense of 2 are \[v_2+v_6=v_1+v_7,\qquad v_3+v_7=v_2+v_5,\qquad v_1+v_5=v_3+v_6.\] For any support function \(h\) on \(\Sigma\), the corresponding bends \[\begin{align} \delta_1&=h(v_1)+h(v_7)-h(v_2)-h(v_6),\\ \delta_2&=h(v_2)+h(v_5)-h(v_3)-h(v_7),\\ \delta_3&=h(v_3)+h(v_6)-h(v_1)-h(v_5) \end{align}\] satisfy \(\delta_1+\delta_2+\delta_3=0\) identically, as one checks by cancelling terms. Strict upper convexity would require \(\delta_1,\delta_2,\delta_3>0\), which is incompatible with their vanishing sum. Hence \(X_\Sigma\) carries no ample divisor and is non-projective.

For the standard ordered basis of \(\mathbb{Z}^3\), the distinct wall normals of \(\Sigma\) are, in canonical order, \[\begin{gather} (0,0,1),\;(0,1,-1),\;(0,1,0),\;(1,-1,-1),\;(1,-1,0),\\ (1,-1,1),\;(1,0,-1),\;(1,0,0),\;(1,1,-1). \end{gather}\] Applying Construction 3 gives the following complete count of wall-adaptation blow-ups.

For example, the first wall normal \(m=(0,0,1)\) has the unique bad cone \(\langle v_3,v_6\rangle\) in the initial fan. The algorithm inserts \[s=v_3+v_6=(0,-1,0),\] and the two affected maximal cones \(\langle v_1,v_3,v_6\rangle\) and \(\langle v_3,v_5,v_6\rangle\) are subdivided by replacing \(\langle v_3,v_6\rangle\) with \(\langle v_3,s\rangle\) and \(\langle s,v_6\rangle\). Each new maximal cone is smooth because the operation is a star subdivision at the sum of two generators of a smooth two-cone.

The full computation gives \[f_1(\Gamma)=32,\qquad f_2(\Gamma)=90,\qquad f_3(\Gamma)=60,\] so the wall-adaptation stage inserts \(32-7=25\) rays, matching the table. The inserted rays, in algorithm order, are \[\begin{array}{r@{\,=\,}c@{\qquad}r@{\,=\,}c@{\qquad}r@{\,=\,}c} s_1&(0,-1,0) & s_2&(-2,-1,-1) & s_3&(-1,0,0)\\ s_4&(0,1,1) & s_5&(-2,0,-1) & s_6&(1,1,1)\\ s_7&(0,-1,1) & s_8&(1,0,1) & s_9&(1,1,0)\\ s_{10}&(2,1,1) & s_{11}&(-1,-1,-2) & s_{12}&(0,0,-1)\\ s_{13}&(0,-1,-2) & s_{14}&(-1,0,1) & s_{15}&(1,-1,-2)\\ s_{16}&(1,0,-1) & s_{17}&(1,2,1) & s_{18}&(-1,-2,-1)\\ s_{19}&(1,-1,1) & s_{20}&(0,1,-1) & s_{21}&(-2,1,-1)\\ s_{22}&(-1,1,0) & s_{23}&(1,-1,0) & s_{24}&(1,1,2)\\ s_{25}&(2,-1,1) & \multicolumn{4}{c}{} \end{array}\]

By Lemma 7 the resulting smooth complete fan \(\widehat{\Sigma}=\Gamma\) is projective. The following explicit support function also certifies ampleness directly. Write \(h(r)=H(r)/2\) on the rays \(r\in\{v_1,\ldots,v_7,s_1,\ldots,s_{25}\}\), where \[\begin{array}{c|rrrrrrr} r&v_1&v_2&v_3&v_4&v_5&v_6&v_7\\ \hline H(r)&-10&-13&-15&-11&-16&-9&-10 \end{array}\] and \[\begin{array}{c|rrrrrrrrrrrrr} r&s_1&s_2&s_3&s_4&s_5&s_6&s_7&s_8&s_9&s_{10}&s_{11}&s_{12}&s_{13}\\ \hline H(r)&-13&-20&-13&-15&-21&-14&-22&-12&-18&-22&-14&-12&-19 \end{array}\] \[\begin{array}{c|rrrrrrrrrrrr} r&s_{14}&s_{15}&s_{16}&s_{17}&s_{18}&s_{19}&s_{20}&s_{21}&s_{22}&s_{23}&s_{24}&s_{25}\\ \hline H(r)&-25&-25&-20&-25&-22&-21&-24&-25&-19&-15&-22&-25. \end{array}\] With the wall-bend convention of 3 , all \(90\) wall bends of this \(h\) are positive; their distinct values are \[\frac{1}{2},\;1,\;\frac{3}{2},\;2,\;\frac{5}{2},\;3,\;\frac{7}{2},\;4,\;\frac{9}{2}.\] Thus the minimum bend is \(\tfrac12\). The ancillary script verify_oda_threefold.py reproduces the run, prints the final \(60\) maximal cones, checks smoothness and completeness, verifies adaptation to all nine wall normals, and verifies these \(90\) positive bends using exact rational arithmetic. The entire projectivization is therefore the sequence of \(25\) codimension-two blow-ups recorded above, with no subsequent steps.

A remark on the Fujino–Sato list. The case [8-10] of [6], there denoted \(Z_{10}\), is by contrast already projective, so it is not a source of non-projective test inputs; the present example uses case [7-5], which is non-projective as shown above. This count should not be read as a minimality statement: the example illustrates the determinism and uniformity of the algorithm, and ad hoc projectivizations of a particular threefold can be much shorter.

9 Specialization to dimension three↩︎

10 Length estimates↩︎

Write \(f_k(\Phi)=|\Phi(k)|\). The algorithm gives an explicit, though generally nonminimal, count of the blow-ups needed to reach a projective fan. Each star subdivision adds exactly one ray, and for a smooth complete toric variety \(\rho(X_\Phi)=f_1(\Phi)-n\), so the number of blow-ups equals the Picard-rank increase from \(X_\Sigma\) to \(X_{\widehat{\Sigma}}\). Since the only stage is wall-adaptation, this number is \[L=f_1(\Gamma(\Sigma))-f_1(\Sigma),\] in every dimension. For the threefold of Section 8 this is \(32-7=25\).

The adaptation cost for a fixed wall normal is governed locally by a Euclidean descent on \(m\)-weights. In dimension two, the cost for a single bad cone with \(m\)-values \((a,-b)\) is the sum of the partial quotients in the continued fraction of \(a/b\). In higher dimensions the same local descent operates, but link effects can create and remove bad cones in neighboring maximal cones, so a closed-form sum over the original bad cones is not available from this argument. Finiteness and computability are exactly the content of Lemma 2. One may of course stop the algorithm early if an intermediate fan is already projective; the estimate above concerns the explicit full sequence.

11 Remarks↩︎

Acknowledgments↩︎

The author used Claude (Anthropic) and ChatGPT (OpenAI) during manuscript development for exploratory discussion, drafting assistance, proof-checking prompts, and computational sanity checks. The final literature positioning, verification of the arguments, and responsibility for all mathematical claims are the author’s.

References↩︎