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The weak F-equivalence conjecture for nonsingular toric weak Fano varieties

About 27 years old · traced to

Let XX be a nonsingular toric weak Fano dd-fold. Two such varieties are weakly-F-equivalent if they are connected by a sequence of equivariant blow-ups, blow-downs, and flops through nonsingular toric weak Fano dd-folds. Weak F-equivalence conjecture. Any nonsingular toric weak Fano dd-fold is weakly-F-equivalent to the dd-dimensional projective space Pd\mathbf P^{d}.

This is the weak Fano analogue of the proposed classification conjecture for nonsingular toric Fano varieties, allowing flops in addition to equivariant blow-ups and blow-downs. The source supplies no evidence of a resolution, so the conjecture remains open.

References

Primary source

Hiroshi Sato, “Toward the classification of higher-dimensional toric Fano varieties”, arXiv:math/9911022 (1999).

birational-geometrytoric-geometryweak-fano-varieties
Posted

Progress summary

Refreshed
Claimed solved

A new preprint claims the conjecture is false in three or more dimensions, while the two-dimensional case is reportedly proved.

The conjecture says every nonsingular toric weak Fano variety can be reduced to projective space using the allowed toric modifications. It was formulated by Sato in the 1999-era literature and was previously open for weak Fano varieties.

Known results

  • Sato proved the conjecture in dimension d=2d=2.
  • The related toric Fano, rather than weak Fano, conjecture was proved in dimensions d=3d=3 and d=4d=4.

August 2026 counterexample claim

A preprint claims that for every d≥3d\ge 3 there is a nonsingular projective toric weak Fano dd-fold not weakly FF-equivalent to Pd\mathbb{P}^d. Its proposed obstruction is a ray count: the constructed class has rd=(2d+1d)−1r_d=\binom{2d+1}{d}-1 rays, while Pd\mathbb{P}^d has d+1d+1; the proof is currently unverified. The authors also report assistance from GPT-5.6 Sol.

Current status (as of August 2026): the d=2d=2 case is reported proved, while a preprint claims disproof in every dimension d≥3d\ge 3 but its counterexamples and proof remain unverified.

Sources

Solutions 0

No solutions have been posted yet.