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On a conjecture of Fujino and Sato

This paper provides short, dimension-independent proofs of Fujino and Sato's results on non-projective toric varieties and extends their factorization conjecture to weak Mori Dream Spaces by utilizing the Cox ring theory and the GKZ secondary fan decomposition to organize small modifications.

Original authors: Michele Rossi

Published 2026-02-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Michele Rossi

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

The Big Picture: Navigating a Labyrinth of Shapes

Imagine you are an architect trying to design a building. In the world of mathematics (specifically algebraic geometry), these "buildings" are called varieties. Some are simple and easy to work with (projective), while others are twisted, knotted, or incomplete (non-projective).

For a long time, mathematicians Fujino and Sato asked a specific question: "If we have a twisted, non-projective building, can we always fix it by making small, precise cuts and swaps until it becomes a nice, projective building?"

They suspected the answer was "yes," but proving it required a lot of heavy, complicated machinery.

Michele Rossi's paper is like finding a master key or a GPS that makes this proof incredibly short and clear. He shows that the answer is indeed "yes," and he does it by using a map called the Secondary Fan.

The Key Concepts (Translated)

1. The "Building" (The Variety)

Think of a Toric Variety as a complex 3D shape made by gluing together simple cones (like ice cream cones).

Projective: A building that is "complete" and sits nicely in a standard space. It's stable and easy to navigate.
Non-Projective: A building that is "broken" or twisted. It might be missing a wall, or the angles are all wrong, making it impossible to view from a single, perfect angle.

2. The "Fix" (Flips and Flops)

How do you fix a twisted building? You don't tear it down. Instead, you perform Flips.

Analogy: Imagine a room with a door that leads to a dead end. A "flip" is like swapping the door to a different wall so the room opens up into a hallway. You change the shape slightly, but the total number of rooms and the overall structure remain mostly the same.
Fujino and Sato conjectured that if you keep doing these "door swaps" (flips), you will eventually reach a perfect, projective building.

3. The "Map" (The Secondary Fan / GKZ Decomposition)

This is the most important part of Rossi's paper.

The Old Way: Trying to prove the conjecture was like trying to walk through a dark forest, step by step, checking every tree to see if you were going the right way. It was slow and depended on the size of the forest (dimension).
Rossi's Way: He pulls out a Satellite Map (the Secondary Fan).
- Imagine the "Moving Cone" as a giant, glowing territory.
- This territory is divided into distinct Chambers (like rooms in a mansion).
- The Rule: Each "Room" (Chamber) represents a different version of your building.
- Projective Buildings live in the "Full-Sized Rooms" (Full-dimensional chambers).
- Twisted Buildings live in the "Hallways" or "Walls" (faces of the rooms).

4. The "Cox Ring" (The Blueprint)

Rossi uses a tool called the Cox Ring. Think of this as the Universal Blueprint for the building.

No matter how you swap the doors (flips), the Blueprint stays the same. It just tells you which "Room" (Chamber) you are currently in.
Because the Blueprint is so powerful, it proves that you can always find a path from a "Hallway" (twisted building) to a "Full-Sized Room" (perfect building) just by crossing a wall.

The Main Discovery: The "Wall-Crossing" Strategy

Rossi's paper simplifies the proof of Fujino and Sato's conjecture into a single, elegant idea:

The Map Exists: Every twisted building corresponds to a specific spot on the map (the Secondary Fan).
The Goal: We want to get to a "Projective" spot (a full room).
The Action: If you are currently on a "Wall" (a non-projective building), the map shows that this wall is just the edge of a big, full room.
The Flip: By performing a "Flip" (crossing the wall), you step into that big room.
The Result: You are now in a projective building!

Rossi proves that this isn't just a lucky guess; it's a mathematical law. As long as you have a "Blueprint" (Cox Ring) and a "Map" (Secondary Fan), you can always navigate from a twisted shape to a perfect one by crossing walls.

The Extension: Beyond the "Toric" World

The paper doesn't stop there. Fujino and Sato only looked at "Toric" buildings (shapes made of cones). Rossi says, "Wait a minute, this logic works for almost any shape, not just cones!"

He extends the map to a broader category called Weak Mori Dream Spaces.

Analogy: Imagine the "Toric" buildings were just houses made of Lego. Rossi proved that the same navigation rules apply even if you are building with clay or wood, as long as the structure has a certain "flexibility" (being Q-factorial).
The Conclusion: If you have a twisted shape of this type, there is always a specific "door swap" (a D-flip) that will turn it into a perfect, projective shape.

Summary in One Sentence

Michele Rossi showed that the complex problem of fixing twisted geometric shapes is actually just a simple navigation task: if you have the right map (the Secondary Fan) and the right blueprint (the Cox Ring), you can always find a path of "door swaps" (flips) to turn a broken shape into a perfect one, without needing to worry about how big or complicated the shape is.

1. Problem Statement

The paper addresses two central problems in the birational geometry of toric varieties and their generalizations:

The Existence of Projective Models: Fujino and Sato previously proved that any complete toric variety $X$ is birational to a projective $\mathbb{Q}$ -factorial toric variety $X'$ via a map that is an isomorphism in codimension 1.
The Factorization Conjecture: Fujino and Sato conjectured that any complete $\mathbb{Q}$ -factorial toric variety can be connected to a projective $\mathbb{Q}$ -factorial toric variety via a finite sequence of flips, flops, and anti-flips. While verified for specific low-dimensional cases, a general proof was lacking.

The author aims to provide a unified, conceptual proof for these results that avoids dimension restrictions and extends the theory beyond the toric setting to Weak Mori Dream Spaces (wMDS).

2. Methodology

Rossi employs the machinery of Cox rings and Mori Dream Spaces (MDS), combined with the geometry of the secondary fan (GKZ decomposition).

Cox Ring and MDS Framework: The paper utilizes the theory developed by Hu and Keel, and extended by Arzhantsev et al., which characterizes varieties where the Cox ring is finitely generated.
- A Mori Dream Space (MDS) is a projective variety with a finitely generated Cox ring.
- A Weak Mori Dream Space (wMDS) is a complete, $\mathbb{Q}$ -factorial variety with a finitely generated Cox ring (not necessarily projective).
Secondary Fan (GKZ Decomposition): The core geometric tool is the decomposition of the moving cone ( $\text{Mov}(X)$ $Mov (X)$ ) into chambers.
- Each full-dimensional chamber corresponds to the nef cone ( $\text{Nef}(X')$ ) of a specific birational model $X'$ .
- The walls between chambers correspond to small $\mathbb{Q}$ -factorial modifications (sQm), which are isomorphisms in codimension 1.
- Crossing a wall corresponds to a flip, flop, or anti-flip.
Wall-Crossing Strategy: The author argues that if a variety is non-projective, its nef cone is a proper face of a full-dimensional chamber in the secondary fan. By choosing a divisor class in the relative interior of the difference between the projective chamber and the non-projective nef cone, one can construct a specific flip that moves the variety into the projective chamber.

3. Key Contributions

A. Conceptual Proof of Fujino–Sato Theorem 1.1

Rossi provides a short, dimension-independent proof that every complete toric variety $X$ admits a projective $\mathbb{Q}$ -factorial model $X'$ isomorphic to $X$ in codimension 1.

Logic: Any complete toric variety admits a small $\mathbb{Q}$ -factorial resolution $X'$ . Since $X'$ is a wMDS, the general theory guarantees the existence of a small modification $f: X' \dashrightarrow X''$ where $X''$ is a projective MDS.
Significance: This bypasses the need for explicit toric combinatorics by relying on the structural properties of Cox rings.

B. Proof of the Fujino–Sato Conjecture

The paper proves the conjecture that any complete $\mathbb{Q}$ -factorial toric variety can be connected to a projective one via flips/flops.

Mechanism: If $X$ is non-projective, $\text{Nef}(X)$ is a face of a full-dimensional chamber $\gamma$ in the secondary fan. There exists a projective model $X'$ such that $\gamma = f^*\text{Nef}(X')$ .
The Flip: By selecting a divisor $D$ with a class in the relative interior of $f^*\text{Nef}(X') \setminus \text{Nef}(X)$ , the author constructs a $D$ -flip $f: X \dashrightarrow X'$ .
Result: This establishes a direct path (a single flip, or a sequence thereof) from any non-projective model to a projective one.

C. Extension to Weak Mori Dream Spaces (wMDS)

The most significant theoretical extension is generalizing these results from toric varieties to complete $\mathbb{Q}$ -factorial weak Mori Dream Spaces.

Theorem: If $X$ is a non-projective complete $\mathbb{Q}$ -factorial wMDS, there exists a divisor $D$ and a $D$ -flip $f: X \dashrightarrow X'$ such that $X'$ is a (projective) Mori Dream Space.
Implication: This demonstrates that the "chamber structure" of the movable cone is the fundamental driver of these birational transformations, independent of the specific toric combinatorics.

4. Results and Examples

Theoretical Results:
- Theorem 2.1: Establishes the equivalence between being a wMDS and admitting a small modification to an MDS.
- Theorem 3.1: Proves that any non-projective wMDS admits a flip to a projective MDS.
Illustrative Example (Example 3.2):
- The author constructs a specific rank-3 toric variety $X(\Sigma)$ associated with a prism in $\mathbb{R}^3$ .
- $X$ is complete but non- $\mathbb{Q}$ -factorial.
- The paper details 8 possible simplicial subdivisions (small resolutions) $\Sigma_i$ .
- Using the Gale dual matrix (weight matrix $Q$ ), the author computes the secondary fan.
- The example visualizes how the nef cones of the resolutions correspond to chambers in the secondary fan. It explicitly shows how a non-projective resolution (where the nef cone is a ray) can be flipped to a projective resolution (where the nef cone is a full-dimensional chamber) by crossing a wall defined by a specific divisor class.

5. Significance

Unification: The paper unifies the study of toric varieties and general MDS under a single framework: the geometry of the secondary fan and the moving cone.
Simplification: It replaces complex, case-by-case toric proofs with a streamlined, conceptual argument based on the existence of chambers in the movable cone.
Generality: By removing restrictions on dimension and Picard number, and extending the results to wMDS, the paper broadens the scope of the Fujino–Sato conjecture significantly.
Navigation Tool: The secondary fan is presented as an efficient "navigation tool" for moving between birational models, clarifying the role of wall-crossing in the Minimal Model Program (MMP).

In summary, Rossi demonstrates that the existence of projective models and the connectivity of birational models via flips are natural consequences of the Cox ring structure and the chamber decomposition of the moving cone, applicable well beyond the realm of toric geometry.