Counterexamples to Sato's Weak F-Equivalence Conjecture and a Gorenstein Refinement

Abstract: We disprove Sato's weak (F)-equivalence conjecture for nonsingular projective toric weak Fano varieties in every dimension (d \geq 3). Our counterexamples are smooth projective crepant models of centered reflexive simplices. The key input is a rigidity property of ray polytopes: if (X_Σ) is nonsingular and complete and (-K_{X_Σ}) is nef, then every nonzero lattice point of (P_Σ=\operatorname{Conv}(G(Σ))) is a primitive ray generator. For our models, this rules out every weak-Fano-preserving equivariant blow-up and blow-down throughout the flop class. We then introduce Gorenstein weak (F)-equivalence, generated by projective toric birational zigzags through normal projective Gorenstein toric weak Fano varieties, and formulate a corresponding refinement of Sato's conjecture. We prove this refined conjecture in dimensions (d \leq 3), as well as for the family of counterexamples constructed above in every dimension. Finally, we show that the refined conjecture implies the inclusion-connectivity of reflexive (d)-polytopes modulo unimodular equivalence, which is known for (d \leq 4) and remains open for (d \geq 5). The results were developed with the assistance of GPT-5.6 Sol.

• The paper constructs, for every dimension d≥3, a smooth projective toric weak Fano variety with exactly \(\binom{2d+1}{d}-1\) rays, proving it cannot be weakly F-equivalent to \(\mathbb{P}^d\).
• The authors establish ray-polytope rigidity using reflexive-polytope maximality and lattice saturation, showing that smooth toric blow-ups, contractions, and flops cannot remove the ray-count obstruction.
• The paper introduces Gorenstein weak F-equivalence, proves the refined conjecture in dimensions at most three, and explicitly connects the counterexample family to \(\mathbb{P}^d\) in every dimension through singular Gorenstein weak Fano intermediates.

Overview and main results

This paper disproves Sato's weak $F$-equivalence conjecture for nonsingular projective toric weak Fano varieties in every dimension $d\geq 3$, then proposes a Gorenstein refinement that survives the counterexamples. Recall that a nonsingular projective variety is weak Fano when $-K_X$ is nef and big, and that Sato's weak $F$-equivalence is the relation generated by toric blow-ups along invariant orbit closures, their inverses, and toric flops, with every intermediate model required to remain nonsingular, projective, toric, and weak Fano. Sato conjectured that every such variety is weakly $F$-equivalent to $\mathbb{P}^d$; he proved this in dimension two, while his three-dimensional theorem concerns the smaller class of smooth toric Fano threefolds. The failure established here therefore begins precisely at the weak Fano threefolds left open by his positive results.

The main negative result is uniform in dimension: for every $d\geq 3$ there exists a smooth projective toric weak Fano $d$-fold $X_d$, a crepant model of the centered reflexive simplex $\Delta_d$, such that every variety weakly $d\geq 3$0-equivalent to $d\geq 3$1 has exactly

$d\geq 3$2

rays, whereas the fan of $d\geq 3$3 has $d\geq 3$4. Since toric flops preserve ray sets, no chain of admissible smooth moves can change this count. The paper then introduces Gorenstein weak $d\geq 3$5-equivalence, generated by projective equivariant birational morphisms through normal projective Gorenstein toric weak Fano varieties (Cartier canonical class with nef and big anticanonical), proves the corresponding refined conjecture for $d\geq 3$6, and verifies it explicitly for the counterexample family in all dimensions. The authors state that the initial construction was produced with AI assistance ("GPT-5.6 Sol") and subsequently verified by the authors.

The centered simplex and its smooth crepant models

In the lattice $d\geq 3$7, the centered reflexive simplex is

$d\geq 3$8

Three arithmetic facts drive everything downstream: $d\geq 3$9 has exactly $-K_X$0 lattice points; the origin is its unique interior point; and every nonzero boundary lattice point of $-K_X$1 is primitive. Each facet has normalized volume $-K_X$2 and exactly $-K_X$3 relative-interior lattice points.

The smooth model is built via the alcove triangulation. Under unimodular changes of coordinates $-K_X$4 identifies with the alcoved polytope $-K_X$5; the affine braid arrangement cuts it into unimodular simplices using every lattice point, and coherence (in the sense of Lam–Postnikov) makes the triangulation coherent. Restricting to $-K_X$6 and coning gives a complete fan $-K_X$7 whose rays are precisely the nonzero lattice points of $-K_X$8. Projectivity requires care: coherence supplies convex bends across new walls inside each facet cone, while a large multiple of an integral strictly convex support function on the face fan dominates bends across old walls. Nonsingularity follows because facet simplices extend to bases of the saturated sublattice $-K_X$9, forcing maximal cones to be unimodular. The resulting $F$0 satisfies

$F$1

with $F$2 projective and crepant.

A subtle but consequential point is interpretational: the authors read Sato's "equivariant blow-up" as blow-ups along invariant orbit closures only. They note that under a broader reading allowing blow-ups along arbitrary invariant subschemes the problem changes—though their counterexample still disproofs the conjecture even allowing arbitrary smooth invariant centers, since a single such blow-up would suffice under the broad interpretation. This caveat should be weighed against the strength of the disproof.

Ray-polytope rigidity

The technical core is a structure theorem for ray polytopes. For a complete Gorenstein toric variety with $F$3 nef, setting $F$4, the paper establishes:

1. $F$5;
2. for every maximal cone $F$6 with anticanonical Cartier datum $F$7, $F$8, i.e., the cone section is the convex hull of $F$9 and the rays of $F$0;
3. $F$1 is reflexive and $F$2 refines the face fan of $F$3;
4. if $F$4 is nonsingular, the lattice-saturation identity $F$5 holds;
5. facets of $F$6 carry unimodular triangulations by all their lattice points, induced by $F$7.

Item (4) is where smoothness is decisive: it forces integral barycentric coefficients summing to at most one in any maximal cone, so interior lattice points of the cone section are rays. In the singular case the identity fails—the face fan of $F$8 itself has only $F$9 rays despite $\mathbb{P}^d$0's many boundary points—and this failure is exactly what enables the singular moves used later.

Two maximality results convert these structural facts into obstructions. First, if a reflexive polytope $\mathbb{P}^d$1 contains $\mathbb{P}^d$2, then $\mathbb{P}^d$3: the dual polytope $\mathbb{P}^d$4 contains exactly $\mathbb{P}^d$5 nonzero lattice points, so any reflexive $\mathbb{P}^d$6 must equal it. Second, given this, no nontrivial projective equivariant birational morphism onto $\mathbb{P}^d$7 can have a smooth weak Fano source: subdividing would enlarge the ray polytope beyond $\mathbb{P}^d$8, contradicting maximality, hence the ray set cannot grow. Dually, no disjoint contraction from $\mathbb{P}^d$9 preserves nefness of the target's anticanonical: deleting a vertex $d\geq 3$0 of $d\geq 3$1 forces the adjacent edge points $d\geq 3$2 to survive (their divisors meet $d\geq 3$3 via the full-lattice-point facet triangulation), producing a facet at lattice distance $d\geq 3$4—impossible for a reflexive polytope. The disjointness hypothesis is essential and covers simultaneous blow-downs along finitely many pairwise disjoint invariant centers, including inverses of ordinary blow-ups of smooth invariant loci.

Together with the fact that Sato flops preserve ray sets verbatim, these propositions prove the main theorem: the entire weak $d\geq 3$5-equivalence class of $d\geq 3$6 consists of varieties with $d\geq 3$7 rays, excluding $d\geq 3$8. Notably, the obstruction applies not just to single elementary moves but to composites: any projective birational morphism with source or target in the class is trivial.

Gorenstein weak F-equivalence and the three-dimensional theorem

The refined relation replaces Sato's elementary smooth moves with arbitrary projective torus-equivariant birational morphisms between members of $d\geq 3$9, the normal projective toric varieties with Cartier $d$0 and $d$1 nef and big. This deliberately avoids prescribing a list of elementary singular moves—an arbitrary star subdivision on a singular variety may be a weighted rather than ordinary blow-up, and Gorenstein weak Fanos need not be $d$2-factorial. The comparison lemma shows Sato's relation embeds in the Gorenstein one: the key step replaces each Sato flop $d$3 over its common small contraction $d$4 by the zigzag $d$5, proving via the extremal primitive relation that $d$6 and that $d$7 remains Gorenstein weak Fano. The proof uses the generally singular common contraction rather than Sato's auxiliary common blow-up, which is not known to stay weak Fano.

Every member of $d$8 admits a projective crepant anticanonical morphism to $d$9 for the reflexive ray polytope $X_d$0, reducing connectivity questions to reflexive polytopes. For $X_d$1 the refined conjecture follows from two classification inputs: Kreuzer–Skarke's result that the $X_d$2 three-dimensional reflexive polytopes form one connected web under inclusions, and Fredrickson's theorem that nested three-dimensional reflexive polytopes admit compatible MPCP subdivisions. Since MPCP subdivisions of face fans are smooth in dimension three (by Pick's theorem on facet triangles), each inclusion yields a zigzag

$X_d$3

through Gorenstein weak Fano threefolds, giving $X_d$4 for every $X_d$5. A related but distinct projection-based $X_d$6-relation of Kasprzyk–Katzarkov–Przyjalkowski–Sakovics also connects three-dimensional reflexive polytopes.

Explicit links for the counterexample family

The counterexamples themselves become equivalent to projective space under the Gorenstein relation, via an explicit family of reflexive simplices interpolating between $X_d$7 and the standard simplex $X_d$8. For subsets $X_d$9 define

$\Delta_d$0

Each $\Delta_d$1 is reflexive, verified by explicit primitive support forms. The crucial geometric fact is a star-subdivision identity: when $\Delta_d$2, the vertex $\Delta_d$3 lies in the relative interior of the facet of $\Delta_d$4 opposite $\Delta_d$5, and the ray through $\Delta_d$6 lies in the relative interior of the cone $\Delta_d$7 of $\Delta_d$8; consequently

$\Delta_d$9

The common star subdivision defines a projective Gorenstein weak Fano $d\geq 3$00 linking $d\geq 3$01 and $d\geq 3$02. Flipping vertices one at a time gives

$d\geq 3$03

in every dimension, without invoking any classification. Thus the family obstructing Sato's conjecture lies in the connected component of $d\geq 3$04 once singular intermediates are permitted.

Consequences and the higher-dimensional obstruction

Although weaker than Sato's conjecture after restriction to smooth varieties, the Gorenstein conjecture remains strong: its validity in dimension $d\geq 3$05 implies that the graph $d\geq 3$06 of $d\geq 3$07-dimensional reflexive polytopes modulo unimodular equivalence, with edges given by inclusions, is connected. The argument transports cocharacter lattices along the zigzag and uses ray-polytope monotonicity under equivariant morphisms, together with reflexivity of intermediate ray polytopes. This connectivity statement is known for $d\geq 3$08 (Miura for polygons; Kreuzer–Skarke in dimensions three and four) and is open for $d\geq 3$09—so the conjecture cannot currently be attacked, even indirectly, above dimension four. The relation also parallels class-preserving connectivity questions from the Sarkisov program (Brown–Buczyński–Kasprzyk; Miura), though it is not itself a Sarkisov statement since intermediates need not be Mori fibre spaces.

The three-dimensional proof does not extend directly: Fredrickson's compatibility theorem for MPCP subdivisions is special to dimensions two and three. In dimension four, intersecting face fans of nested reflexive polytopes can produce a ray whose generator lies outside both polytopes, hence noncrepant for both anticanonical models—a naive common subdivision exits $d\geq 3$10. Whether this obstruction can be circumvented for general reflexive polytopes, as it was for the centered-simplex family, is the central open question.

Limitations and open questions

Several caveats bound the results. First, the disproof of Sato's conjecture depends on interpreting equivariant blow-ups as blow-ups along invariant orbit closures; the authors argue this matches Sato's intent, but acknowledge the broader reading leads to a different problem. Second, Proposition on no-admissible-blowdowns requires pairwise disjoint exceptional divisors and says nothing about contractions whose exceptional primes intersect. Third, the three-dimensional theorem relies on complete classifications (the $d\geq 3$11 polytopes) and on Fredrickson's compatibility theorem, neither available in higher dimension; the four-dimensional noncrepant-ray phenomenon is an explicit obstruction, not merely a gap in technique. Finally, the equivalence between the Gorenstein conjecture and reflexive-polytope connectivity runs only one way: connectivity of $d\geq 3$12 is necessary, and sufficiency would require constructing compatible Gorenstein weak Fano refinements for every inclusion chain, which fails naively already in dimension four.

Conclusion

The paper settles Sato's weak $d\geq 3$13-equivalence conjecture negatively in all dimensions $d\geq 3$14 via a single explicit family—the smooth crepant models of centered reflexive simplices—using a lattice-saturation property of ray polytopes that holds only under smoothness. It then reframes the problem in the Gorenstein category, where the same family becomes connected to projective space by explicit star-subdivision links and the conjecture holds through dimension three. The refinement implies, and is plausibly equivalent to up to the refinement-construction problem, connectivity of the web of reflexive polytopes, leaving the cases $d\geq 3$15 as the natural next target.