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Canonical Projectivization of Smooth Complete Toric Varieties

This paper proves that every smooth complete toric variety can be canonically transformed into a projective one through a finite sequence of toric blow-ups along smooth invariant centers by combining a projective wall-arrangement fan, a sign-adaptation algorithm, and a barycentric derived subdivision.

Original authors: Parsa Bakhtary

Published 2026-05-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Parsa Bakhtary

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect working with a very specific type of building material: Toric Varieties. In the world of mathematics, these are complex geometric shapes built from a grid of points (a lattice).

Some of these shapes are "smooth" (no sharp, jagged edges) and "complete" (they are closed loops, like a sphere rather than an open sheet). However, there is a catch: some of these smooth, complete shapes are non-projective.

Think of "projective" as the ability to fit the shape perfectly inside a standard, finite room (like a box) without it stretching off to infinity or getting distorted. In math, being projective is a very desirable property because it makes the shape much easier to study and measure.

The Problem:
For a long time, mathematicians knew that if you had a smooth, non-projective shape, you could eventually turn it into a projective one. But the old methods were messy. They were like trying to fix a crooked house by first smashing it into a pile of rubble (introducing singularities/rough edges) and then carefully rebuilding it. The intermediate steps were ugly and broken.

The big question was: Can we fix a smooth, non-projective shape into a projective one without ever breaking its smoothness? Can we do it by only making small, careful additions, like adding a new room or a wall, while keeping the whole structure perfectly smooth the entire time?

The Solution (The Paper's Claim):
Parsa Bakhtary says yes. This paper provides a step-by-step, "canonical" (standardized) recipe to turn any smooth, complete toric variety into a projective one, ensuring that every single step in the process is a smooth, clean operation.

Here is the recipe, explained with everyday analogies:

The Three-Step Recipe

The author uses a three-part strategy to fix the shape. Think of it as a renovation project.

1. The "Wall Arrangement" (The Blueprint)

First, the author looks at the shape and draws a set of invisible "walls" (hyperplanes) based on the shape's existing features.

Analogy: Imagine your house has some crooked windows. You draw a grid of perfectly straight lines on the floor that align with the best possible orientation for those windows. This grid is called the Wall-Arrangement Fan.
Goal: This grid represents the "perfect" projective version of the shape. It's the target we want to reach.

2. The "Sign-Adaptation" (The Tweaking)

Now, the shape doesn't quite fit the grid yet. Some parts of the shape cross over the "walls" in the wrong direction (mathematically, the signs of the coordinates don't match).

Analogy: Imagine a tree branch growing across a property line. To fix this without cutting the tree down (which would break the "smoothness"), you gently guide the branch.
The Move: The author uses a specific algorithm to perform blow-ups. In geometry, a "blow-up" is like taking a sharp corner or a crossing point and replacing it with a small, smooth curve or a new wall.
The Magic: The paper proves that you can fix these "bad crossings" by only adding new walls along codimension-two centers.
- Simple translation: If you are in 3D space, you are fixing the shape by adding new "curves" (lines) rather than tearing down whole "surfaces." It's a very precise, surgical fix.
- The author calls this the Sign-Adaptation Algorithm. It systematically fixes every "bad" crossing until the shape aligns perfectly with the "Wall-Arrangement" grid. Crucially, the shape remains smooth after every single fix.

3. The "Barycentric Wrapper" (The Final Polish)

Even after the shape aligns with the grid, it might still be a bit "lumpy" or not quite projective in the strictest sense.

Analogy: Imagine you have a rough-hewn stone that fits in your blueprint, but it's not a perfect cube. You take the stone and slice it into tiny, perfect pyramids from the center out. This is called a Barycentric Subdivision.
The Move: The author takes the now-aligned shape and performs a final round of "blow-ups" at the centers of its faces and edges.
The Result: This final step forces the shape to become projective. It's like putting a final, perfect coat of paint and a frame on the picture.

Why This Matters

The paper claims that this process is canonical. This means it's not a random guess. If you give the computer the same starting shape and the same "ordered list of directions" (lattice basis), it will always produce the exact same result.

Key Takeaways for the General Audience:

No Breaking Allowed: Unlike old methods that broke the shape to fix it, this method keeps the shape smooth and perfect at every single step.
Surgical Precision: The fixes are done by adding small, specific elements (like adding a new curve to a surface) rather than massive reconstruction.
Universal: It works for shapes in any number of dimensions (2D, 3D, or higher).
The "Bad" Objects: The paper identifies the specific "bad" parts of the shape (called "bad curves" or "bad two-cones") that prevent it from being projective and shows exactly how to fix them one by one.

In short, the paper provides a guaranteed, step-by-step manual to turn any smooth, closed geometric shape into a projective one, ensuring the shape never loses its smoothness during the transformation.

Technical Summary: Canonical Projectivization of Smooth Complete Toric Varieties

Problem Statement
The paper addresses a fundamental question in toric geometry: given a smooth complete toric variety $X_\Sigma$ (defined by a smooth complete fan $\Sigma$ in a lattice $N \cong \mathbb{Z}^n$ ), can one construct a smooth projective toric variety that dominates it via a proper birational toric morphism? While the existence of such a dominating variety is classical (via the Toric Chow Lemma followed by resolution of singularities), the classical approach typically passes through singular intermediate varieties. The specific challenge addressed here is whether this projectivization can be achieved entirely through a sequence of smooth intermediate varieties, where every morphism is an ordinary blow-up along a smooth torus-invariant center. This relates to the "smooth stellar factorization" problem and distinguishes the work from the Weak Factorization Theorem, which allows both blow-ups and blow-downs.

Methodology
The author, Parsa Bakhtary, proposes a uniform, constructive algorithm that refines the input fan $\Sigma$ into a smooth projective fan $\hat{\Sigma}$ in three distinct stages. The construction is canonical once an ordered lattice basis of $N$ is fixed.

Wall-Arrangement Fan Construction:
The first step constructs a reference fan $\mathcal{A}_\Sigma$ , termed the "wall-arrangement fan." For every codimension-one cone $\tau \in \Sigma$ , the author extends the supporting hyperplane to a central hyperplane in $N_\mathbb{R}$ . The arrangement of these hyperplanes defines a complete fan $\mathcal{A}_\Sigma$ .
- Property: $\mathcal{A}_\Sigma$ is a complete, projective fan that refines $\Sigma$ . However, $\mathcal{A}_\Sigma$ is generally not smooth or simplicial.
Sign-Adaptation Algorithm (The Wall-Adaptation Stage):
The goal is to refine $\Sigma$ into a fan $\Gamma(\Sigma)$ that is "subordinate" to $\mathcal{A}_\Sigma$ (i.e., $\Gamma(\Sigma) \preceq \mathcal{A}_\Sigma$ ) while maintaining smoothness.
- Mechanism: The algorithm iterates through the wall normals $m_i$ defining $\mathcal{A}_\Sigma$ . For a given normal $m$ , a two-dimensional cone $\langle u, v \rangle$ is defined as " $m$ -bad" if the generators lie on opposite sides of the hyperplane $\ker(m)$ (i.e., $m(u) \cdot m(v) < 0$ ).
- Operation: The algorithm performs a finite sequence of star subdivisions on $m$ -bad two-cones. Specifically, for a bad cone $\langle u, v \rangle$ , it inserts the ray generated by $u+v$ .
- Termination: A weight function $w_m(\tau) = |m(u)| + |m(v)|$ is used to prove termination. Each subdivision strictly reduces the maximum weight of bad cones or reduces their count.
- Result: This stage produces a smooth fan $\Gamma(\Sigma)$ where every cone lies entirely within a single chamber of the wall arrangement (i.e., all ray generators of any cone lie in the same closed half-space defined by each wall normal). This ensures $\Gamma(\Sigma) \preceq \mathcal{A}_\Sigma$ . Crucially, every step is a blow-up along a smooth codimension-two invariant center.
Barycentric Wrapper (The Projectivity Stage):
Although $\Gamma(\Sigma)$ is smooth and refines the projective fan $\mathcal{A}_\Sigma$ , it is not necessarily projective itself.
- Operation: The author applies a barycentric derived subdivision, denoted $\text{Bar}(\Gamma(\Sigma))$ . This involves inserting rays corresponding to the barycenters (sums of generators) of all cones of dimension $k \geq 2$ , processed in decreasing order of dimension.
- Result: By Lemma 6.4, if a smooth fan $\Phi$ refines a projective fan $\Delta$ , then the barycentric subdivision $\text{Bar}(\Phi)$ is projective. This step forces projectivity while preserving smoothness.

Key Contributions and Results

Theorem 1.1 (Canonical Toric Projectivization): The paper proves that for any smooth complete fan $\Sigma$ in $N_\mathbb{R}$ ( $n \geq 2$ ), there exists a canonical finite sequence of toric blow-ups:
$X_{\hat{\Sigma}} = X_r \to X_{r-1} \to \dots \to X_0 = X_\Sigma$
such that:
1. $X_{\hat{\Sigma}}$ is smooth and projective.
2. Each morphism is the blow-up of a smooth torus-invariant center of codimension $\geq 2$ .
3. The construction is canonical relative to a fixed ordered lattice basis.
Uniformity: The proof is uniform across all dimensions $n$ , avoiding dimension-specific case analyses (unlike some prior approaches to non-projective threefolds).
Dimension Three Specialization: In dimension three, the "bad" objects resolved by the wall-adaptation stage are invariant curves (corresponding to 2-cones), and the barycentric stage resolves invariant curves and torus-fixed points (3-cones). This aligns with the "bad curve" philosophy observed in non-projective Moishezon threefolds.
Length Estimates: The paper provides estimates for the number of blow-ups required. The total length decomposes into an adaptation cost (related to the number of rays added to satisfy wall conditions) and a barycentric cost (related to the number of cones in the adapted fan). For threefolds, a specific formula $L_{full} = 6f_1(\Gamma) - f_1(\Sigma) - 10$ is derived.

Significance and Claims
The paper claims to provide a constructive solution to the projectivization problem that maintains smoothness throughout the entire process.

Contrast with Classical Methods: Unlike the classical Toric Chow Lemma followed by desingularization (which may introduce singular intermediate fans), this method ensures every intermediate variety is smooth.
Contrast with Weak Factorization: Unlike the Weak Factorization Theorem (Włodarczyk), which decomposes birational maps into blow-ups and blow-downs, this construction achieves the result using blow-ups only.
Contrast with Q-Factorial Modifications: The approach differs from frameworks using flips, flops, or Q-factorial modifications (e.g., Fujino–Sato), which often result in varieties isomorphic in codimension one rather than a dominating variety.
Canonicity: The construction is deterministic once a lattice basis is chosen, resolving the "bad" subvarieties (those crossing projectivity walls) in a systematic order.

The author notes that the smoothness hypothesis is essential; if the input fan is not smooth, the sum of generators $u+v$ may not be primitive, preventing the direct application of the sign-adaptation algorithm. The work is presented as a direct, unweighted construction for smooth fans, distinct from weighted versions of factorization conjectures.