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

Cheltsov, Park, Prokhorov, and Zaidenberg formulated the conjecture in 2020: very general smooth Fano threefolds of genus $9$ or $10$ with Picard rank one and index one are 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 double-projection argument allegedly produces a singular associated surface and hence a cylinder, which would refute the conjecture even for very general members. The proof remains unverified; no independent verification, gap report, correction, or retraction was found.

Current status (as of August 2026): the conjecture is contradicted by a preprint claiming cylindricity for all relevant threefolds, but that claimed proof is unverified, so the mathematical question remains unsettled.

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

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