MathDBThe weak F-equivalence conjecture for nonsingular toric weak Fano varieties
The weak F-equivalence conjecture for nonsingular toric weak Fano varieties
Let be a nonsingular toric weak Fano -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 -folds. Weak F-equivalence conjecture. Any nonsingular toric weak Fano -fold is weakly-F-equivalent to the -dimensional projective space .
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.
Progress summary
No proof or counterexample has appeared; a 2025 classification adds useful evidence but leaves the conjecture open.
The conjecture, stated as Conjecture 6.20 in a 1999 preprint, asserts that every nonsingular toric weak Fano variety can be transformed into projective space using the allowed equivariant operations and flops. That source does not prove it.
Known results
- The analogous F-equivalence statement is proved for nonsingular toric Fano threefolds.
- The corresponding four-dimensional Fano result is proved except for the del Pezzo and pseudo-del Pezzo cases.
- A 2025 classification covers smooth weak Fano toric varieties of Picard rank , but does not establish the conjecture.
June 2025 rank- classification
A systematic computational study lists isomorphism classes in dimension and in dimension , and analyzes equivariant blowdowns and flops among them. It supplies partial computational evidence, not a proof or counterexample to the full conjecture.
Current status (as of August 2026): The conjecture remains open; the rank- classification is partial progress, and no proof or counterexample is recorded.
Sources
Sources & referencesView supporting material
Primary source
Hiroshi Sato, “Toward the classification of higher-dimensional toric Fano varieties”, arXiv:math/9911022 (1999).
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