Let X be a very general smooth Fano threefold such that rho(X)=1, iota(X)=1, and operatornameg(X)=9 or operatornameg(X)=10. The non-cylindricity conjecture. Then X is not cylindrical. The source presents this as a belief following results showing cylindricity for special threefolds containing lines of specified types; the assertion remains open for very general members.

The non-cylindricity conjecture for very general Fano threefolds of genus 9 or 10

The conjecture, stated as a belief by Cheltsov, Park, Prokhorov, and Zaidenberg in 2020, says that a very general smooth Fano threefold with Picard rank one, index one, and genus $9$ or $10$ is not cylindrical.

Known results

Cylindricity was known for a codimension-one family, including threefolds containing lines with normal bundle $\mathcal{O}_{\mathbb{P}^{1}}(1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-2)$ (Cheltsov, Park, Prokhorov, Zaidenberg, 2020).

May 2026 claimed proof

The preprint Cylinders in Fano threefolds of genus $9$ and $10$ claims that every smooth such threefold with $\operatorname{Pic}(X)=\mathbb{Z}[-K_X]$ is cylindrical. Its construction uses double projection from a line and purportedly finds a singular associated surface; this would refute the non-cylindricity conjecture, including its very general cases. The claim is unverified: no independent proof, correction, retraction, or reported gap was found.

Current status (as of August 2026): a preprint claims cylindricity for all relevant threefolds, contradicting the conjecture, but the claimed proof has no independent verification and the problem remains unsettled.

The non-cylindricity conjecture for very general Fano threefolds of genus 9 or 10

Primary source

Progress summary

Solutions 0